Topology at IU

Historical and Current Seminars

Monday September 18, 2000 in Wylie 101: Frank Quinn, 4-dimensional surgery and subexponential growth
Wednesday September 27, 2000 in Rawles Hall 104: Zhenghan Wang, Distribution of Jones polynomials at root of unities.
Thursday September 28, 2000 in Swain East 140: Dennis Sullivan, Operations in the free loop space of manifolds (joint seminar w/ Differential Geometry)
Friday October 13, 2000 in Rawles Hall 104: Uwe Kaiser
Wednesday October 25, 2000 in Rawles Hall 104: Kim Pearson, The Gromov-Lawson-Rosenberg conjecture for cocompact Fuchsian groups
Wednesday November 15, 2000 in Rawles Hall 104: Jon Short, Weakened group topologies and their locally isometric completions.
Wednesday February 21, 2001 in Rawles Hall 104: Prof. Eiji Ogasa, The intersection of spheres in a sphere and a new geometric interpretation of the Arf invariant; This talk is based on the speakers paper math.GT/0003089 in http://xxx.lanl.gov/ Let S^3_i be a 3-sphere embedded in the 5-sphere S^5 (i=1,2). Let S^3_1 and S^3_2 intersect transversely. Then the intersection C of S^3_1 and S^3_2 is a disjoint collection of circles. Thus we obtain a pair of 1-links, C in S^3_i (i=1,2), and a pair of 3-knots, S^3_i in S^5 (i=1,2). Conversely let (L_1,L_2) be a pair of 1-links and (X_1,X_2) be a pair of 3-knots. It is natural to ask whether the pair of 1-links (L_1,L_2) is obtained as the intersection of the 3-knots X_1 and X_2 as above. We give a complete answer to this question. We ask a question to generalize this result. (Problem from section 1 of the above paper.) Two copies of this paper have been placed on the bookshelf in Rawles Hall 107, close to the picture of Einstein. I can be found in Rawles Hall 411 and frequently in the lounge, Rawles Hall 107. Please drop by if you would like to discuss this topic further.
Wednesday February 28, 2001 in Rawles Hall 104: Taehee Kim, Non-slice Knots with Vanishing Casson-Gordon Invariants I
Wednesday March 7, 2001 in Rawles Hall 104: Taehee Kim, Non-Slice Knots with Vanishing Casson-Gordon Invariants II
Wednesday March 28, 2001 in Rawles Hall 104: Des Sheiham, Invariants of Boundary Link Cobordism
Wednesday May 2, 2001 in Rawles Hall 104: Diarmuid Crowley, The classification of highly connected (8k-1)-manifolds - current results
Wednesday July 25, 2001 in Rawles Hall 316: sepanski, Seiberg-Witten invariants of circle bundles
Thursday October 25, 2001 in RH104: Alejandro Adem, Representations and K-theory of the Braid Groups
Thursday November 15, 2001 in Rawles Hall: Bob Oliver, Actions of finite groups on acyclic 2-complexes
Thursday November 29, 2001 in Rawles Hall: Jim McClure, Operads and cosimplicial spaces
Wednesday January 9, 2002 in Rawles Hall: Chuck Livingston, Concordance Genus of Knots; The concordance genus of a knot K is the minimum genus among all knots concordant to K. I will discuss relations between the concordance genus and classically studied knot invariants. The computation of the concordance genus for low crossing-number knots will also be described.
Wednesday January 16, 2002 in Rawles Hall 104: Jan Jaworowski, Periodic maps on vector space bundles
Wednesday January 23, 2002 in RH 104: Charles Delman, Laminar 3-manifolds: an introduction
Wednesday January 30, 2002 in RH 104: Charles Delman, Laminar 3-manifolds
Wednesday February 6, 2002 in RH 104: Scott Baldridge, Classification of symplectic 4-manifolds which admit circle actions with fixed points
Wednesday February 13, 2002 in RH 104: Paul Kirk, The Atiyah-Patodi-Singer rho invariant for manifolds with boundary
Wednesday February 20, 2002 in RH 104: Zhenghan Wang, Curves on surfaces and topological field theories
Wednesday February 27, 2002 in Rawles Hall: Ayelet Lindenstrauss, The Hochschild homology of fatpoints; I will talk about a calculation of the Hochschild homology of the ring k[x_1,...,x_n]/{frak M}^a, where M is the ideal (x_1,...,x_n) and k is a characteristic zero field. Using simplicial methods, this Hochshild homology can be described in terms of the homology of tori relative to their “a-fat” diagonals (all points in the torus, exhibited as a product of circles, where a or more coordinates are the same).
Wednesday March 6, 2002 in RH 104: Charles Livingston, Knotting Contractible 4-manifolds in R^4
Wednesday March 27, 2002 in RH 104: Dusan Repovs, New Results on 2-polyhedra in 3- and 4-manifolds; We shall present a very broad survey of results, starting from the mid 1960's, addressed at a very general audience, concerning the geometric topology of polyhedra and the main applications. Here are the key topics to be covered in the talk: 2-dimensional PL topology: We shall consider the problem of resolving arbitrary 2-polyhedra by fake surfaces and its application to getting a reduction of the classical Whitehead asphericity conjecture to the case of special polyhedra (Casler, Ikeda, Piergallini-Matveev, Wright, Hog-Angeloni-Metzler, Repovs-Skopenkov). 3-dimensional PL topology. We shall study spines of topological 3-manifolds and an interesting application of the classical knot theory in producing nonhomeomorphic 3-manifolds with equivalent spines (Mitchell-Przytycki-Repovs, Cavicchioli-Lickorish-Repovs, Glock-Hog-Angeloni) as well as problems of thickenings (Brodsky-Repovs-Skopenkov). 4-dimensional PL topology. We shall investigate embeddings of 2-polyhedra into topological 4-manifolds and an interesting application of the classical Bing shrinking techniques to the problem of different regular neighborhoods of codimension 2 embeddings (Geoghegan-Mihalik, Lasheras, Salihov, Oniscenko-Repovs-Skopenkov, Bestvina-Daverman-Venema-Walsh). We shall also list several open problems and open conjectures from this and related areas of geometric topology.
Wednesday April 3, 2002 in RH 104: Jim Davis, Gromov-Lawson-Rosenberg Conjecture for Cocompact Fuchsian Groups; We give nasc for a closed manifold of dimension greater than for 4 whose fundamental group is cocompact Fuchsian to admit a metric of positive scalar curvature.
Wednesday April 10, 2002 in RH 104: Diarmuid Crowley, Quadratic Z modules, Kreck monoids and the classification of simply
Wednesday April 17, 2002 in RH 104: Ben Himpel, Casson's invariant and gauge theory
Wednesday April 24, 2002 in RH 104: Paul Kirk, SU(3) Casson Invariant
Wednesday May 15, 2002 in Rawles Hall 104: Scott Carter, Dynamical quandle cocycles and twist-spun knots.; There has been a burst of activity recently in quandle theory. Andruskiewitsch and Grana (among others) are developing quandle cohomology is relation to classifying certain pointed Hopf Algebras. They have defined a general quandle cohomology theory that encompasses those given by Fenn and Rourke and generalizations by me and my collaborators (CJKLS) and Ohtsuki. One notion of Andruskiewtsch and Grana that is particularly helpful is that of a dynamical 2-cocycle. Angela Harris (who is currently a master's degree student at University of South Alabama), Masahico Saito and I are looking into the following. Starting from a Fox coloring of a classical knot we indicate how to extend the coloring to a coloring using a dynamical 2-cocycle. We then use this extension to give colorings of various twist-spuns of the original knot. Our purpose is to generalize the results of Shima and Satoh on the minimal number of triple points for the 2 and 3 twist-spin trefoils. The talk can be self-contained at an introductory level if graduate students are present.
Wednesday September 11, 2002 in Rawles Hall 104: Ernesto Lupercio, Geometry and Topology of Orbifolds I; Orbifolds are geometric objects that generalize the concepts of manifolds and of finite groups. Just as a Manifold is a topological space with additional structure, an orbifold includes a topological space and the local information of a group action on a manifold. Orbifolds appear naturally in algebraic geometry and in physics, especially in the phenomenon of Mirror Symmetry. In the first talk I will introduce the category of orbifolds, and natural cohomology Theories on them. I will also introduce Gerbes and Twisted K-theory.
Thursday September 12, 2002 in Rawles Hall 104: Ernesto Lupercio, Geometry and Topology of Orbifolds II; In the second talk I will explain how the Loop Orbifold provides a natural framework for the study of topological and algebraic geometrical questions for orbifolds.
Wednesday September 18, 2002 in Rawles Hall 104: Jae Choon Cha, Concordance and rational knots
Wednesday September 25, 2002 in Rawles 104: Scott Baldridge, Seiberg-Witten invariants of 4-manifolds with S^1 actions; Abstract: In this talk we show how to calculate the SW invariant of any b_+>1 closed, smooth, oriented 4-manifold with an effective circle action. Applications of this theorem to questions in symplectic geometry and topology will be discussed.
Wednesday October 2, 2002 in Rawles Hall: Scott Baldridge, Seiberg-Witten invariants of 4-manifolds with S^1 actions, continued; Contuation of Sept. 25 seminar
Thursday October 10, 2002 in Rawles Hall 104: Diarmuid Crowley, Modified L-groups for modified surgery; One of the striking features of the classical surgery obstruction groups is that they depend only upon the fundamental group and its orientation character. The main aim of this talk is to present new L-groups for Kreck's modified surgery theory (at least in the simply connected case) which depend upon higher homotopy groups of the manifolds to be classified. As an application, we show how the new L-groups are used in the classification of simply connected 7-manifolds (with torsion free second homology). A corollary of this classification is that the set of homotopy types of simply connected 7-manifolds with rank(H_2(M)) = r is infinite for r>1.
Wednesday October 23, 2002 in Rawles Hall 104: Chuck Livingston, Knotted Contractible 4-Manifolds; Mazur constructed examples of smooth compact contractible 4-manifolds. His examples (and perhaps all such manifolds) smoothly embed in S^4 with simply connected complement. I will describe Lickorish's construction of large families of such 4-manifolds that possess embedding in S^4 with non-simply connected complement.
Thursday October 31, 2002 in Rawles Hall 104: Alejandro Adem, Fixity and Group Actions
Wednesday November 13, 2002 in Rawles Hall: Vahagn Minasian, THH and TAQ for E_infty - algebras; @algebra, Hochschild Homology (HH) and Andr'e-Quillen Homology (AQ) are homology theories for associative and associative commutative algebras respectively. We discuss the topological analogues of these theories (THH and TAQ), give some comparison results and show how THH and TAQ lead to two notions of 'etale in topological categories. We show an 'etale descent lemma for THH that generalizes the corresponding result from algebra, and use it to prove a topological analogue of the HKR theorem.
Wednesday November 20, 2002 in Rawles Hall: Taehee Kim, New Obstructions to Doubly Slicing Knots
Wednesday December 4, 2002 in Rawles Hall: Pat Gilmer, Integral Bases for TQFT modules
Wednesday December 11, 2002 in Rawles Hall: Ayelet Lindenstrauss, Calculus of Functors, 119
Wednesday January 22, 2003 in Rawles Hall: Chuck Livingston, Knotted Contractible Manifolds II; In a previous seminar I described Lickorish's result that any perfect group of deficiency 0 arises as the fundamental group of the complement of a contractible manifold in S^4. In this talk I will describe the construction of contractible manifolds with complements having large negative deficiency. Applications to homology 4-spheres will also be presented.
Wednesday January 29, 2003 in Rawles Hall: John Mcatee, Intrinsic knotting and linking of graphs; Summary: A graph G is intrinsically knotted if every embedding of G in R^3 contains a nontrivial knot. G is intrinsically linked if every embedding into R^3 contains a non-split link. We will describe some examples of intrinsically knotted and linked graphs and survey the major results about this subject.
Wednesday February 19, 2003 in Rawles Hall 104: Zhenghan Wang, On TQFT representations of mapping class groups; We will discuss the generic Jones represenations of the braid groups and their relations with TQFT representations of the braid groups. The question whether the generic Jones representationis faithful is still open.
Wednesday April 2, 2003 in Rawles Hall 104: Tibor Macko, Automorphisms of infinite-dimensional real projective space; Useful information about automorphisms of RP^infty can be obtained from the so called block structure space of RP^infty. The block structure space of a compact topological manifold X is a space which has as its k-th homotopy group the group of manifold structures on X x Delta^k relative to the boundary, i.e. the group of homotopy equivalences from a manifold M to X x Delta^k which are homeomorphisms on the boundary . By assigning the block structure space of RP(V) to the finite-dimensional real vector space V we obtain a continuous functor from the category of vector spaces to (based) spaces. In this talk I will present the study of this functor from the point of view of the orthogonal calculus of M. Weiss, which is designed for the functors of this type. More concretely I will show that it is a linear functor. Tools used in the proof are mainly surgery theory and Poincare embeddings. Application of this result is an attractive description of the block structure space of RP^infty, which is a colimit of the block structure spaces of finite-dimensional real vector spaces.
Tuesday April 8, 2003 in Rawles Hall: Andrew Ranicki, Algebraic transversality; Topological transversality of manifolds is a geometric cutting technique which deforms maps on manifolds to ensure that certain preimages are submanifolds. For example, this is the geometric ingredient of the Mayer-Vietoris exact sequence for the homology of a union, and the Seifert-Van Kampen theorem expressing the fundamental group of a union of spaces as an amalgamated free product or HNN extension. Algebraic transversality considers modules and quadratic forms over amalgamated free products and HNN extensions, cutting up such objects by algebraic mimicry of the topological model. This type of algebra has applications in the topology of high-dimensional manifolds with infinite fundamental group.
Wednesday April 9, 2003 in Rawles Hall 104: Stefan Friedl, The eta invariant and slicing obstructions; We classify the metabelian unitary representations of $\pi_1(M_K)$, where $M_K$ is the result of zero-surgery along a knot $K\subset S^3$. We show that certain eta-invariants associated to metabelian representations $\pi_1(M_K)\to U(k)$ vanish for slice knots and that even more eta-invariants vanish for ribbon knots. This sliceness obstruction turns out to be at least as strong as the Casson-Gordon obstruction. It turns out that eta-invariants can in many cases be easily computed for satellite knots. We make use of this to study the relation between the eta-invariant sliceness obstruction, eta-invariant ribboness obstruction and the $L^2$-eta invariant obstruction recently introduced by Cochran, Orr and Teichner. In particular we give an example of a knot which has zero Casson-Gordon invariants, zero eta-invariant sliceness obstruction and zero metabelian $L^2$-eta invariants, but is not ribbon.
Wednesday April 23, 2003 in Rawles Hall 104: Ayelet Lindenstrauss, On the algebraic K-theory of a square-zero extension of ${\bf F}_p$; I will be discussing joint work with Randy McCarthy of UIUC in which we calculate the algebraic K-theory of a square-zero extension of F_p by a finite F_p-vector space after completion at p. The case of a one-dimensional vector space, namely that of the dual numbers over F_p, was done by Hesselholt-Madsen. We use a variant of topological cyclic homology whose construction is motivated by Goodwillie calculus.
Wednesday November 5, 2003 in Rawles Hall 104: Razvan Gelca, Recursive Relations of Colored Jones Polynomials
Wednesday November 12, 2003 in Rawles Hall 104: Jim Davis, Fixity and Free Group Actions on Products of Spheres
Wednesday February 18, 2004 in Rawles Hall 104: Shmuel Weinberger, Manifolds with Excessive Symmetry
Thursday March 4, 2004 in Rawles Hall 104: Taehee Kim, Filtrations of knots and knot concordance; A report on recent joint work withTim Cochran.
Wednesday March 10, 2004 in Rawles Hall 104: Diarmuid Crowley, Recognizing open subsets of the n-sphere; I will report on joint work in progress with Steve Ferry which aims to characterise those open manifolds which are the compliments of finite polyhedrons K in S^n (assuming for now that K is simply connected and that 2k + 2 < n). The talk will rapidly move to measuring the twistedness of twisted doubles and the relationship of this to high codimension link theory (a la Sato and Levine).
Tuesday April 6, 2004 in RH 316: Jerry Kaminker, Amenability and the Novikov conjecture
Wednesday April 14, 2004 in Rawles Hall 104: Noah Salvaterra, Khovanov Homology
Wednesday April 21, 2004 in Rawles Hall 104: Allan Edmonds, Surfaces in the 3-torus; I will characterize those maps of a closed surface to the 3-torus that are homotopic to embeddings.
Monday May 3, 2004 in Probably RH 104: Oleg Viro, Algebraic Knot Theory
Friday May 14, 2004 in Rawles Hall 104: W. Jaco, The Homeomorphism Problem for 3-Manifolds; The solution to the homeomorphism problem for 3-manifolds will be presented–there is now an algorithm to determine whether two given 3-manifolds are homeomorphic.
Friday May 21, 2004 in Rawles Hall 104: Diarmuid Crowley, The inertia groups of HP^2; I shall present a computation of the group of isotopies of quaternionic projective space (up to a Z/2 ambiguity). Along the way, I will explain how this fits into a broader framework for manifold classification and also give an explicit representative for an exotic self-diffeomorphism of S^7.
Thursday May 27, 2004 in Rawles Hall 104: Tobia Hagge, The necessity of Reidemeister move II
Friday May 28, 2004 in Rawles Hall 104: Kiyoshi Igusa, Semi-invariants, pictures and the homology of nilpotent groups
Thursday June 3, 2004 in Rawles Hall 104: Cornelia Van Cott, From Seifert circles to braids
Tuesday August 10, 2004 in Rawles Hall 316: Stefan Friedl, New examples of topologically slice knots
Wednesday October 6, 2004 in Rawles Hall 104: P. Kirk, The SU(3) Casson invariant for Brieskorn Spheres
Wednesday October 13, 2004 in Rawles Hall 104: P. Kirk, The SU(3) Casson invariant for Brieskorn Spheres
Wednesday October 20, 2004 in Rawles Hall 104: Jeremy Brookman, Splitting homotopy equivalences along codimension 1 submanifolds; Abstract: In the 1970s, Cappell defined the obstruction groups UNil_n, containing the obstruction to splitting a homotopy equivalence of (n-1)-dimensional manifolds along a codimension 1 submanifold. Separate consideration was given to the cases of n being odd and n being even. I shall describe the algebraic definition of the UNil groups and the construction of the splitting obstruction, including a new definition when n is odd using the algebraic theory of surgery of Ranicki.
Wednesday October 27, 2004 in Rawles Hall 104: Jeremy Brookman, Splitting homotopy equivalences along codimension 1 submanifolds (continued); In the 1970s, Cappell defined the obstruction groups UNil_n, containing the obstruction to splitting a homotopy equivalence of (n-1)-dimensional manifolds along a codimension 1 submanifold. Separate consideration was given to the cases of n being odd and n being even. I shall describe the algebraic definition of the UNil groups and the construction of the splitting obstruction, including a new definition when n is odd using the algebraic theory of surgery of Ranicki.
Wednesday November 3, 2004 in Rawles Hall 104: Jeremy Brookman, Splitting homotopy equivalences along codimension 1 submanifolds (continued)
Wednesday November 10, 2004 in Rawles Hall 104: Qayum Khan, Stable homeomorphisms of 4-manifolds with infinite fundamental group; I.Hambleton and M.Kreck (1993) showed for closed topological 4-manifolds $X$ and $Y$ with finite fundamental group that: if there exists a homeomorphism between $X # r(S^2\times S^2)$ and $Y # r(S^2times S^2)$ for some $r$, then there exists a homeomorphism between $X$ and $Y$. Such $X$ and $Y$ in the hypothesis are called “stably homeomorphic”. There is an additional hypothesis that $X$ itself is internally a connected sum $X_0 # S^2\times S^2$. I extend their theorem to the case of infinite (good) fundamental group, under certain ring-theoretic conditions. The motivating example comes from understanding the tangential homotopy type of the connected sum of two copies of real projective 4-space along a homotopy 3-sphere, which has fundamental group the infinite dihedral group.
Wednesday December 1, 2004 in Rawles Hall 104: Yuhan Lim, SU(2) Seiberg-Witten Invariants.
Wednesday December 8, 2004 in Rawles Hall 104: Chuck Livingston, Twisted Alexander Polynomials of Periodic Knots; I will describe joint work with Hillman and Naik giving extensions of Murasugi's results on the Alexander polynomials of periodic knots the setting of twisted polynomials.
Tuesday January 25, 2005 in Rawles Hall 104: Jae Choon Cha, Homology Equivalences and Algebraic Closures; We discuss the relationship between algebraic closures and homology equivalences, and some applications to homology cobordisms of odd-dimensional manifolds and link concordance.
Monday January 31, 2005 in Rawles Hall 104: Slavik Jablan
Monday January 31, 2005 in Rawles Hall 104: Slavik Jablan
Wednesday February 2, 2005 in Rawles Hall 104: Julia Viro, Lines and circles meeting links; We will consider low bounds for the number of lines meeting given 4 disjoint smooth closed curves in the real projective 3-space in a given cyclic order. Similarly, we estimate the number of circles meeting in a given cyclic order given 6 disjoint smooth closed curves in Euclidean 3-space. The estimations are formulated in terms of linking numbers of the curves and based on a study of a surface swept by projective lines meeting 3 given disjoint smooth closed curves and a surface swept by circles meeting 5 given disjoint smooth closed curves. These results admit generalizations to higher dimensions and more complicated patterns of intersections. Lines and circles can be replaced by configurations of curves of other kinds; linking numbers, by Vassiliev invariants.
Thursday February 3, 2005 in Rawles Hall 104: Oleg Viro, Khovanov homology of classical and virtual links; In 1999 Khovanov introduced homology groups for classical links such that coefficients of the Jones polynomial appear as alternating sums of the ranks of these groups. In the talk, down to earth adaptation of this construction will be discussed. It is generalized to a class of virtual links (or abstract Gauss diagrams). The original chain complex defining the Khovanov homology is huge, but can be replaced by its deformation retract which is essentially smaller than the original one.
Wednesday February 23, 2005 in Rawles Hall 104: Jim Davis, A two component link with Alexander polynomial one is concordant to the Hopf link
Wednesday March 2, 2005 in Rawles Hall 104: Chuck Livingston, The Hausmann-Weinberger invariant of Z^n; The Hausmann-Weinberger invariant of a finitely presented group is the minimum Euler characteristic of a closed 4-manifold have that group as its fundamental group. In this talk I will describe my joint work with Paul Kirk computing this invariant for finitely generated free abelian groups.
Wednesday March 9, 2005 in Rawles Hall 104: Paul Kirk, Symplectic four-manifolds with prescribed fundamental groups
Wednesday March 30, 2005 in Rawles Hall 104: Swatee Naik, Realizing Ozsvath-Szabo Invariants
Tuesday April 5, 2005 in Rawles Hall 104: Cornelia Van Cott, Braid Groups and Manifolds; A brief survey talk about braid groups
Wednesday April 27, 2005 in Rawles Hall 104: Jennifer Franko, A Representation Arising from the Yang-Baxter Equation
Thursday July 21, 2005 in Rawles Hall 104: Constance Leidy, Higher order linking forms: theorem and conjecture
Wednesday August 3, 2005 in Rawles Hall 104: Eli Grigsby, Torsions and Knot Floer Homology; The knot Floer homology groups of a knot K in a closed, connected, oriented, 3-manifold Y were developed several years ago by Ozsvath and Szabo and independently by Rasmussen. They proved that if K is a knot in S^3, the Euler characteristic of the knot Floer homology group gives the Alexander polynomial of K. This relationship is most easily understood via the relationship of both invariants to the torsion of the infinite cyclic cover of the knot complement. We will warm up by explaining this relationship, and go on to prove a generalization of this result which holds for a nullhomologous knot in a rational homology sphere. Along the way, we will focus on the example of the preimage of a knot in its double branched cover.
Thursday August 4, 2005 in Rawles Hall 104: Eli Grigsby, Torsions and Knot Floer Homology (continued); The knot Floer homology groups of a knot K in a closed, connected, oriented, 3-manifold Y were developed several years ago by Ozsvath and Szabo and independently by Rasmussen. They proved that if K is a knot in S^3, the Euler characteristic of the knot Floer homology group gives the Alexander polynomial of K. This relationship is most easily understood via the relationship of both invariants to the torsion of the infinite cyclic cover of the knot complement. We will warm up by explaining this relationship, and go on to prove a generalization of this result which holds for a nullhomologous knot in a rational homology sphere. Along the way, we will focus on the example of the preimage of a knot in its double branched cover.
Monday September 5, 2005 in Rawles Hall 104: Oleg Viro, Khovanov Homology
Tuesday September 13, 2005 in RH316: Chuck Livingston, Mapping Class Group Seminar - Organizational Meeting
Thursday September 15, 2005 in Rawles Hall 104: Mike Mandell, Ring spectra and module spectra
Wednesday September 21, 2005 in Rawles Hall 104: Mike Mandell, Ring spectra and module spectra (continued)
Wednesday September 28, 2005 in Rawles Hall 104: Mike Mandell, Ring spectra and module spectra (continued)
Wednesday October 5, 2005 in Rawles Hall 104: Eric Wilson, Exotic Lens Spaces
Wednesday October 12, 2005 in Rawles Hall 104: Eric Wilson, Exotic Lens Spaces
Wednesday October 19, 2005 in Rawles Hall 104: Michael McCooey, Concordance of group actions on the 4-sphere
Tuesday October 25, 2005 in Rawles Hall 104: Qayum Khan, L-theory of type III virtually cyclic groups; Given an arbitrary group, the Farrell-Jones isomorphism conjecture states roughly that its L-theory is given by its virtually cyclic subgroups and how they fit together. There are three types, and those groups which surject onto the infinite dihedral group with finite kernel are what I call type III. The computation of their L-groups reduces to a computation of associated unitary nilpotent groups UNil. I will discuss some progress in this direction, following recent work of Connolly, Davis, Kozniewski, and Ranicki.
Wednesday October 26, 2005 in Rawles Hall 104: Diarmuid Crowley, Stable diffeomorphism and Kreck's monoid $l_{2q+1}$.
Thursday October 27, 2005 in Rawles Hall 104: Howard Masur, Complexes of curves in low dimensional topology and geometry; One can associate to a surface of genus g with n boundary components a number of different complexes. Among these are the curve complex, the arc complex, the pants complex, and for the boundary of a handlebody, the meridian disc complex. The combinatorics and geometry of these complexes play a role in the geometry and topology of 3 manifolds. I will discuss some of these interactions.
Wednesday November 2, 2005 in Rawles Hall 104: Bjorn Jahren, Topological classification of free involutions on S^1xS^n; The recent calculations of UNil-groups now makes a complete such classification possible. This is based on a classification of the possible homotopy types of quotients, and we show that the only homotopy types that occur are those of four linear examples. However, some of these homotopy types can be realized by infinitely many different conjugacy classes of involutions.
Wednesday November 16, 2005 in Rawles Hall 104: Allan Edmonds, The center conjecture for equifacetal simplices; Equifacetal (geometric) simplices exist in abundance and are characterized by having a vertex transitive isometry group. This implies that such a simplex has a unique center. The center conjecture proposes that the converse is true. On the one hand, the center conjecture is true in general if one adopts a sufficiently weak definition of "center". On the other hand, a more precise conjecture is true at least through dimension six for the most stringent definition of "center".
Thursday December 1, 2005 in Rawles Hall 104: Sam Lewellen
Monday January 30, 2006 in Ballantine Hall 242: Radmila Sazdanovic, Knot Theory Program: LinKnot; The Mathematica-based Windows knot theory program LinKnot is the extension of the program Knot2000 (K2K) written by M.Ochiai and N.Imafuji. LinKnot is the knot theory program that works with knots and links (KLs) given in the Conway notation. Conway symbols are the input used for creating Dowker codes or P-data (the main input for K2K functions). Instead of a graphical input or Dowker codes, for the first time in a computer program you can use human-comprehensive Conway notation of KLs represented as a Mathematica string and work with links, and not only with knots. For all KLs there is no restriction on the number of crossings. The program provides also the complete data base of alternating KLs with at most 12 crossings, and non-alternating KLs with at most 11 crossings, and the data base of basic polyhedra with at most 20 crossings. By using it, it is possible to draw KLs, calculate all polynomial invariants of KLs, work with braids, reduce KLs, etc. For the first time, it is possible to compute unknotting and unlinking numbers, calculated according to Bernhard-Jablan Conjecture. For all alternating KLs you can compute minimum Dowker codes, find all non-isomorphic projections, work with the graphs of KLs, compute linking numbers, breaking and splitting numbers, signatures, and many other KL invariants. The main property of the program is a possibility to use it as a tool in experimenting with KLs, for computing properties connected with infinite classes of KLs (KL families) and make new conjectures in knot theory. For example, except the famous Nakanishi-Bleiler example of the knot 514 with the unknotting number projection gap, by using LinKnot we discovered an infinite collection of such KLs.
Tuesday January 31, 2006 in Rawles Hall 104: Slavik Jablan, Basic Polyhedra
Tuesday January 31, 2006 in Ballantine Hall 221: Radmila Sazdanovic, On Properties of Chromatic Graph Cohomology
Thursday February 2, 2006 in Rawles Hall 104: Laurentiu Maxim, Higher-order Alexander invariants of plane curve complements; I will survey various Alexander-type invariants of plane curve complements, with an emphasis on obstructions on the type of groups that can arise as fundamental groups of complements to affine curves.
Wednesday February 22, 2006 in Rawles Hall 104: L. Nicolaescu, Counting Morse functions; I will describe several equivalence relations on the set of excellent Morse functions on a compact smooth manifold M (excellent = at most one critical point per level set). I will describe Arnold's count of equivalence classes in the case M= S^1, and then I will explain how to count these classes when M= S^2. This mathematical journey will take us in unexpected places.
Wednesday March 1, 2006 in Rawles Hall 104: Jan Jaworowski, Equivariant Maps of Manifolds; The talk is related to the paper by Yasuhiro Hara (Y. Hara, The degree of equivariant maps, Topology and its Appl. vol 148, 2004, p. 113-1210). I take a slightly different approach from that used by Hara and prove a little more. The talk will be very elementary, certainly suitable for graduate students.
Monday March 6, 2006 in Ballantine 011: Vaughan Jones, Tangles: linear and quadratic
Wednesday March 22, 2006 in Rawles Hall 104: J. E. McClure, E-infinity Structures on Surgery Spectra
Tuesday March 28, 2006 in Rawles Hall 104: Matt Hedden, Recent work regarding smooth knot concordance invariants; In the past few years, several new invariants of smooth knot concordance have been discovered. These invariants have arisen in very different contexts: some with analytical origins and others defined combinatorially with close ties to representation theory . I'll survey these invariants and discuss generalizations which should exist. I will probably focus most attention on two concordance invariants, denoted s(K) and tau(K), associated to a knot, K. tau(K) was discovered by Ozsvath and Szabo, and independently by Rasmussen, while s(K) was discovered by Rasmussen. They are perhaps the most notable of the recently discovered invariants and were conjectured to be equal. In fact, they are not and I'll discuss the first counterexamples to this conjecture, jointly discovered with Philip Ording.
Wednesday March 29, 2006 in Rawles Hall 104: Ed Swartz, Face enumeration on manifolds; In the over 100 years since the discovery of the Euler-Poincar? formula there have been tremendous advances in the understanding of the geometry and topology of manifolds. However, the enumerative properties of triangulations remain mysterious. For instance, the minimum number of faces needed to produce a closed $d$-manifold with nonzero first Betti number was unknown until recently. We will survey what is known and sample a variety of new results and techniques.
Monday April 3, 2006 in Rawles Hall 104: Danielle O'Donnol, Intrinsically n-linked complete bipartite graphs
Tuesday April 4, 2006 in Rawles Hall 104: Zhenghan Wang, Classification of TQFTs
Wednesday April 5, 2006 in Kirkwood 200: Stavros Garoufalidis, Asymptotic expansions in quantum topology; A celebrated invariant of knots in 3-space is the Jones polynomial. Using the Jones polynomial one can construct "analytic" invariants of 3-manifolds, where "analytic" has a precise number-theoretic meaning. In the talk we will discuss the relation between quantum invariants and their better known classical analogues: such as the algebrotoplogical invariants of the knot complements and the riemannian (mostly hyperbolic) invariants of complements.
Thursday April 6, 2006 in Rawles Hall 104: Denis Auroux, Near symplectic structures and Lefschetz pencils on smooth 4-manifolds; Every compact smooth 4-manifold with nonzero b+ carries a near-symplectic structure, i.e. a closed 2-form which degenerates in a specific way along a union of circles. The study of these structures is largely motivated by Taubes' program to interprete Seiberg-Witten invariants in terms of enumerative geometry in this context. In this talk, we will describe an extension to the near-symplectic situation of Donaldson's construction of Lefschetz pencils (this is joint work with S. Donaldson and L. Katzarkov). More precisely, we will define the appropriate notion of singular Lefschetz pencil, state the existence result, and discuss some simple topological considerations and examples.
Wednesday April 26, 2006 in Rawles Hall 104: P. Kirk, Symplectic 4-manifolds with arbitrary fundamental group near the BMY line
Wednesday May 3, 2006 in Rawles Hall 104: Shmuel Weinberger, Spaces of Translations; The simplest nontrivial dynamical systems are arguably irrational rotation on the circle. In this talk, I will try to generalize their classification to other (almost as simple) translation systems.
Friday May 26, 2006 in Rawles Hall 104: Vladimir Turaev, A generalization of several classical invariants of links; We extend several classical invariants of links in the 3-sphere to links in so-called quasi-cylinders. These invariants include the linking number, the Seifert form, the Alexander module, the Alexander-Conway polynomial and the Tristram-Levine signatures. This is joint work with D. Cimazoni.
Tuesday May 30, 2006 in Rawles Hall 104: Stefan Friedl, Symplectic products and subgroup separability; A folklore conjecture says that if a product $S^1\times N^3$ is symplectic, then $N^3$ fibers over $S^1$. We will give a proof for this conjecture in the case that $N$ has vanishing Thurston norm, and we will reduce this conjecture to a conjecture on Kleinian groups. (This is joint work with Stefano Vidussi).
Wednesday September 6, 2006 in Rawles Hall 104: Nansen Petroysan, Hirsch rank and actions of solvable groups; My goal will be to outline the proof of the following theorem: There must be a bound on the Hirsch rank of a solvable group if acts cellularly on a finitely dominated, finite dimensional complex such that there is a bound on the Hirsch ranks of the isotropy subgroups. I will then describe some applications and examples to free and proper group actions on spaces such as (S^n) x (R^k). If time permits, I will discuss the case when k=1 connecting it to a conjecture made by Mislin and Talelli in the early 90s. It states that any torsion-free group with periodic cohomology must have finite cohomological dimension, i.e. it must act freely and cellularly on a finite dimensional contractible complex.
Saturday September 16, 2006 in IUPUI - LD 229: Mike Mandell, Joint Purdue-IUPUI-IUB topology seminar: Toward Quillen-Steenrod Opns for BP
Saturday September 16, 2006 in IUPUI - LD 229: Chuck Livingston, Joint Purdue-IUPUI-IUB topology seminar: The Geography of 4-manifolds with specified groups
Thursday October 5, 2006 in Rawles Hall 104: Maria Basterra, Cohomology of highly structured ring spectra
Wednesday October 11, 2006 in Rawles Hall 104: David Cimasoni, Dimers on surface graphs and spin structures; In the 1960's, W. Kasteleyn showed that the partition function for the dimer model on a planar graph is equal to the Pfaffian of an N by N matrix, where N is the number of vertices of the graph. He also stated that for any graph G, the partition function can be written as a linear combination of 4^g Pfaffians, where g denotes the genus of a surface S where G can be embedded. A formal combinatorial proof of this fact was only given in 1999 by Gallucio-Loeb and Tesler. In this talk, we shall explain a geometric proof of this fact. In our formula for the partition function, each of the 4^g terms corresponds to a spin structure on S, and each coefficient in the linear combination is equal to the Arf invariant of the corresponding spin structure. This is joint work with Nicolai Reshetikhin.
Wednesday October 18, 2006 in Rawles Hall 104: Steve Mitchell, Schubert varieties in the affine Grassmannian; The loop space of a compact Lie group is homotopy equivalent to a certain infinite-dimensional projective algebraic variety, the "affine Grassmannian". This variety shares many of the beautiful features of ordinary Grassmannians; for example, it has a decomposition into Schubert cells whose closures are ordinary projective varieties. There are two striking differences, however: The affine Grassmannian is a topological group, and the principal bundle defining it as a homogeneous space is topologically trivial. In this largely expository talk I will present a topologist's perspective on the affine Grassmannian, and discuss some ongoing work involving its Schubert varieties: For example, which of these are nonsingular?
Saturday October 21, 2006 in SE 140: Vigleik Angeltveit, Two notions of a simplicial space up to homotopy
Saturday October 21, 2006 in SE 140: Tom Goodwillie, Differential operators in homotopy calculus
Saturday October 21, 2006 in SE 140: Ciprian Manolescu, A combinatorial description of knot Floer homology
Saturday October 21, 2006 in SE 140: Vladimir Turaev, Homotopy and cobordism of words
Sunday October 22, 2006 in SE 140: Stefan Schwede, On the homotopy groups of symmetric spectra
Sunday October 22, 2006 in SE 140: Craig Westerland, Free loop spaces and Koszul duality
Sunday October 22, 2006 in SE 140: Marc Culler, Mod p homology and hyperbolic volume
Wednesday October 25, 2006 in Rawles Hall 104: Jan Jaworowski, The degree of maps of free G-manifolds and the average
Tuesday October 31, 2006 in Rawles Hall 104: Cornelia Van Cott, Braid length and its relationship to the number of braid strands; Every knot can be represented by infinitely many different braids. In 2004, Gittings discovered some surprising examples of knots for which these braid representatives become simpler as the number of strands is increased. We will study this occurrence and discuss several general relationships between braid length and the number of braid strands.
Thursday November 2, 2006 in Rawles Hall 104: Matthew Ando, K(1)-localizations of topological modular forms and p-adic modular forms; We will explain how to construct a map of E-infinity ring spectra MString –> tmf. An important part of the story is the relationship between p-adic modular forms and K(1)-localizations of tmf. We shall see that the relationship between the Atiyah-Bott-Shapiro orientation of KO and the String orientation of tmf is another part of the story about Bernoulli numbers and Eisenstein series.
Wednesday November 8, 2006 in Rawles Hall 104: Christian Kassel, Braid groups and automorphisms of the free groups
Thursday November 16, 2006 in Rawles Hall 104: Jennifer Franko, Quantum Computing, Braids and Representations; Braid group representations can be used to describe the actions of the quantum bits in a topologicial quantum computer. The quantum bits themselves are encoded in states of quasi particles, called anyons. Trajectories of these quasi particles in 3-dimensional space-time form braids the closure of which are knots or links. Any invertible matrix which satisfies the Yang Baxter Equation can be used to obtain representations of the braid group. During this talk we will consider unitary solutions to the Yang Baxter equation and the construction of representations arising from these solutions as well link invariants these representations yield.
Tuesday November 28, 2006 in Rawles Hall 104: Jae Choon Cha, Witt class defect invariants of 3-manifolds and links
Wednesday November 29, 2006 in Rawles Hall 104: Marta Asaeda, TQFT and operator algebras
Wednesday December 6, 2006 in Rawles Hall 104: Koby Rubinstein, Sets of homotopy types in Sobolev spaces
Wednesday January 10, 2007 in Rawles Hall 104: Paul Kirk, An interesting small symplectic 4-manifold
Thursday January 18, 2007 in Rawles Hall 104: Anar Akhmedov, Construction of new symplectic cohomology S^2 times S^2; In this talk we present new examples of symplectic manifolds with same integral cohomology as S^2 times S^2. We also discuss the generalization of these examples as well as its application in the construction of simply connected 4-manifolds with small Euler characteristic.
Saturday February 10, 2007 in IUPUI LD229: Richard Hind, Joint IU-IUPUI-Purdue seminar: Symplectic embeddings in Euclidean space; We will derive new estimates for the Gromov width of certain domains in R^4, that is, on the maximal radius of balls which can be symplectically embedded into the domain. Time permitting we will also discuss some analogous questions in R^6.
Saturday February 10, 2007 in IUPUI LD 229: Kevin Costello, Joint Purdue-IUPUI-IUB topology seminar: Renormalisation and the Batalin-Vilkovisky formalism; The Batalin-Vilkovisky formalism is a machine used by physicists to quantise gauge theories. I'll give an introduction to the BV formalism, and discuss a new approach to renormalisation in this context. If there's time, I'll discuss some applications to Chern-Simons theory.
Thursday March 8, 2007 in Rawles Hall 104: Christopher Dwyer, A completion theorem for twisted equivariant K-theory
Monday April 16, 2007 in Swain West 217: Matt Hedden, Knot Floer homology and algebraic curves
Monday April 16, 2007 in Swain East 245: Teena Gerhardt, Equivariant Homotopy and Algebraic K-theory
Tuesday April 17, 2007 in Rawles Hall 104: P. Heck, Z. Su, C. Van Cott, Three Talks
Wednesday April 18, 2007 in Rawles Hall 104: Paul Kirk, Small Symplectic 4-Manifolds
Wednesday April 25, 2007 in Rawles Hall 104: J. Yazinski, Immersed curves in the plane and Reidemeister move 2
Thursday August 2, 2007 in Rawles Hall 316: Kate Kearney, Eight Faces of the Poincare Homology Sphere; The Poincare Homology Sphere is a 3-manifold that arises in many different ways. This talk will discuss eight such ways, as presented in a paper with the same title as this talk, written by Kirby and Scharlemann.
Wednesday August 29, 2007 in Rawles Hall 104: Matt Hedden, On a conjecture regarding knots in the three-sphere on which surgery yields lens spaces
Thursday August 30, 2007 in Rawles Hall 104: Nadya Shirokova, The categorification of Vassiliev theory; Many recent theories, e.g. Khovanov, Ozsvath-Szabo homologies, can be considered as categorifications of classical invariants. We introduce a program of classification of such theories. We consider them as constructible sheaves on the spaces of manifolds, extend them to the singular locus, introduce the definition of the theory of finite type and construct our first examples. The resulting theory can be considered as the categorification of Vassiliev theory.
Wednesday September 12, 2007 in Rawles Hall 104: Pierre Py, Joint Topology-Dynamics Seminar: Quasi-morphisms and Hamiltonian diffeomorphisms; A quasi-morphism on a group $\Gamma$ is a map $\phi : \Gamma \to \mathbb{R}$ such that there exist a constant $C$ with $\vert \phi(xy)-\phi(x)-\phi(y) \vert \le C$, for all $x,y \in \Gamma$. Recently, Gambaudo and Ghys, as well as Entov and Polterovich discovered that many groups of Hamiltonian diffeomorphisms admit non-trivial quasi-morphisms, although they do not admit any non-trivial homomorphism to $\mathbb{R}$. I would like to survey and compare all these constructions.
Wednesday September 26, 2007 in Rawles Hall 104: Kevin Pilgrim, An algebraic formulation of Thurston's characterization of rational functions
Wednesday October 3, 2007 in Rawles Hall 104: Kari Ragnarsson, Completion of G-spectra and stable maps between classifying spaces; For a finite group G, completion at the augmentation ideal I(G) of the Burnside ring A(G) features in cohomological completion theorems such as the Atiyah–Segal completion theorem and Carlsson's theorem (formerly the Segal Conjecture). Completion of G-spectra at ideals of the Burnside ring was introduced by Greenlees–May to obtain topological versions of these theorems, letting the completion take place at the level of spectra rather than cohomology groups. When G is a p-group, May–McClure showed that I(G)-adic completion coincides with p-completion (modulo some basepoint issues) but in general I(G)-adic completion seems very complicated.
In this talk I will describe how, for the purposes of I(G)-completion, one may replace a G-spectrum E by a related G-spectrum E' whose isotropy groups are all groups of prime power order. The I(G)-adic completion of the latter is often more approachable, admitting a description as a colimit of completions at primes. As an application I will give a simplified description of the function spectrum F(BG,BK) for a finite group and a compact Lie group K.
Wednesday October 10, 2007 in Rawles Hall 104: Mihai Staic, Hopf Algebras, Braids, and Knot Invariants
Thursday October 11, 2007 in Rawles Hall 104: Sergei Chmutov, Thistlethwaite's theorem for virtual links; Regions of a link diagram can be colored in black and white in a checkerboard manner. Putting a vertex in each black region and connecting two vertices by an edge if the corresponding regions share a crossing yields a planar graph. In 1987 Thistlethwaite proved that the Jones polynomial of the link can be obtained by a specialization of the Tutte polynomial of this planar graph. I will explain a generalization of this theorem to virtual links. In this case the graph will be a ribbon graph, which means that it will be embedded into a (higher genus, possibly non-oriented) surface. For such graphs we use a generalization of the Tutte polynomial discovered recently by B. Bollobas and O. Riordan. This is a joint work with Jeremy Voltz.
Wednesday October 31, 2007 in Rawles Hall 104: Nathan Geer, Renormalized quantum invariants; In this talk I will discuss a renormalization of the Reshetikhin-Turaev quantum invariants, by "fake quantum dimensions." In the case of simple Lie algebras these "fake quantum dimensions" are proportional to the genuine quantum dimensions. More interestingly I will discuss two examples where the genuine quantum dimensions vanish but the "fake quantum dimensions" are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras defined by Patureau-Mirand and myself. These invariants are multivariable and generalize the multivariable Alexander polynomial. The second example, is a hierarchy of invariants arising from nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaev's quantum dilogarithm invariants of links. This is joint work with Bertrand Patureau-Mirand and Vladimir Turaev.
Thursday November 1, 2007 in Rawles Hall 104: Chris Douglas, Higher Clifford Algebras; Real K-Theory is 8-periodic. This periodicity can be seen algebraically from the periodicity of Clifford algebras — I will discuss how the notion of Clifford algebra arises naturally from thinking about how to define K-theory. It turns out that Clifford algebras form a 2-category, and the generator of that 2-category has order 8. We are interested in the analogous story for elliptic cohomology — I will describe the geometric approach to elliptic cohomology introduced by Segal, Stolz, and Teichner, and explain the role of what we call "higher Clifford algebras". These objects ought to form a 3-category — we introduce such a 3-category and show that its generator has order at least 24. This is joint work with Arthur Bartels and Andre Henriques.
Thursday November 1, 2007 in Rawles Hall 104: Matt Mauntel, Algebraic K-Theory
Wednesday November 28, 2007 in Rawles Hall 104: Frank Quinn, An approach to the Farrell-Jones conjecture; The conjecture describes the algebraic K-theory of group rings via an assembly map involving only the K-theory of virtually cyclic groups. Special cases have been obtained slowly and with difficulty by Farrell-Jones, Bartels-Lueck-Reich, Farrell-Linnell, Davis-Lueck and others. We describe an approach to the general result that has led to a number of curious ideas and should at least give some injectivity results.
Wednesday December 5, 2007 in Rawles Hall 104: Chuck Livingston, Finite Representations of Knot Groups
Thursday December 13, 2007 in Rawles Hall 104: David Roshenthal, On the algebraic K-theory of groups with finite asymptotic dimension; In this joint work with Arthur Bartels, it is proved that the assembly maps in algebraic K-and L-theory with respect to the family of finite subgroups is injective for groups with finite asymptotic dimension that admit a finite model for the universal space for proper actions. The result also applies to certain groups that admit only a finite dimensional model for the universal space. In particular, it applies to discrete subgroups of virtually connected Lie groups.
Wednesday January 9, 2008 in Rawles Hall 104: Patricia Hersh, Regular cell complexes modeling Bruhat intervals; We give a new criterion for determining whether a finite CW complex is regular. This involves both combinatorial conditions on the closure poset and also topological conditions on the codimension one cell incidences. As an application, we prove a conjecture of Fomin and Shapiro answering a question of Bjorner, namely the question of finding naturally arising regular CW complexes with the Bruhat intervals as their closure posets. To this end, we confirm a conjecture of Fomin and Shapiro that certain stratified totally positive spaces known to have the proper closure posets are regular CW decompositions; this involves showing that an intriguing map of Lusztig yields the characteristic maps.
Tuesday January 15, 2008 in Rawles Hall 104: Allan Edmonds, The Plus Construction for Manifolds; This is a short coda to our seminar of Fall, 2007, in which I will interpret the plus construction in the category of manifolds. One nice consequence is that topological or PL homology n-spheres, n>=5, bound contractible manifolds.
Wednesday February 13, 2008 in Rawles Hall 104: Neza Mramor-Kosta, The degree of maps between manifolds with free group actions; The degree of a map between two manifolds with free group actions is in some cases determined by the classifying maps of the two manifolds. In particular this is true for equivariant maps, and for maps which are in certain ways close to being equivariant. We will describe some classes of such maps, and the resulting formulas for the degree.
Wednesday February 20, 2008 in Rawles Hall 104: Chuck Livingston, Twisted Alexander Polynomials and Knot Slicing
Tuesday April 8, 2008 in Rawles Hall 104: Andrew Ranicki, The geometric Hopf invariant; This is a joint project with Michael Crabb (Aberdeen). The geometric Hopf invariant of a stable map F:Sigma^kX to Sigma^kY is a Z_2-equivariant map h_{R^k}(F) which “counts the double points” of F. The homotopy class of h_{R^k}(F) is the primary obstruction to F being homotopic to the k-fold suspension Sigma^kF_0 of an unstable map F_0:X to Y. The geometric Hopf invariant has applications to double points of immersions of manifolds, and surgery obstruction theory (including the non-simply-connected case).
Wednesday April 9, 2008 in Rawles Hall 104: Danny Ruberman, Knot concordance and Heegaard Floer Homology of Branched covers
Thursday April 10, 2008 in Rawles Hall 104: Danny Ruberman, Rohlin's invariant and periodic-end Dirac operators
Wednesday April 16, 2008 in Rawles 104: Jamie Pommersheim, The geometry of summation; The classical Euler-Maclaurin formula expresses the sum of a function $f$ of one variable as the integral $f$ plus terms involving derivatives of $f$ at the endpoints. This formula may be viewed as expressing the sum $f$ over the integral points in a one dimensional polytope (i.e., a line segment) in terms of certain integrals over the faces of the polytope. In this talk, we explore higher dimensional analogs of Euler-Maclaurin formula, which allow one to sum functions over polytopes of any dimension. An important special case is summing the constant function $f=1$, i.e., counting the number of lattice points inside the polytope. Much progress on this problem has been made in the last 20 years using the theory of toric varieties. After discussing some of these results, we will introduce the Euler-Maclaurin formulas of Berline and Vergne, as well as joint work of the Garoufalidis and the speaker.
Thursday April 24, 2008 in Rawles Hall 104: Jim McClure, A convenient category of $C^\infty$ manifolds.; Joint with Gerd Laures. Gluing of smooth manifolds has always been a delicate topic because it is only well defined if one chooses “gluing data” (such as vector fields). But in fact one can expand the category of smooth manifolds with boundary in such a way that gluing is canonically defined and is a categorical pushout. Moreover, if we redefine the word “set” then gluing becomes strictly associative (not just up to isomorphism). Our category also has Cartesian products and has the same bordism groups as the usual category of smooth manifolds with boundary.
Wednesday May 28, 2008 in Rawles Hall 104: Charles Livingston, The concordance genus of knots
Wednesday September 17, 2008 in Rawles Hall 104: Matvei Libine, Equivariant cohomology and Berline-Vergne localization formula
Wednesday September 17, 2008 in Rawles Hall 104: Matvei Libine, Equivariant cohomology and Berline-Vergne localization formula; The notion of equivariant cohomology plays an important role in symplectic geometry, algebraic geometry, representation theory and other areas of mathematics. Let G be a compact Lie group acting on a topological space M. For the topologists, the equivariant cohomology of M is defined to be the ordinary cohomology of the space (M x E)/G, where E is any contractible topological space on which G acts freely. (This definition does not depend on the choice of E.) If M is a smooth manifold there is an alternative way of defining the equivariant cohomology groups of M involving de Rham theory. My goal is to explain the Berline-Vergne localization formula (which is a generalization of the Duistermaat-Heckman Theorem) expressing integrals of equivariant forms as sums over fixed points. In a future talk I will describe my own generalization of this formula.
Wednesday September 24, 2008 in Rawles Hall 104: Bert Guillou, Massey products and the motivic fundamental group of the punctured projective line; We will discuss, following a suggestion of S. Bloch, the construction of certain mixed Tate motives associated to the projective line minus the points 0, 1, and infinity. In the end, this will reduce to the question of producing a cellular approximation of a certain dg-module over a dga, and we will see that the construction of this cell module is controlled by Massey products in the motivic cohomology of the ground field.
Wednesday October 8, 2008 in Rawles Hall 104: Allan Edmonds, A survey of group actions on 4-manifolds; A survey of almost 50 years of locally linear, compact and finite group actions on topological 4-manifolds, with an emphasis on history, guiding questions, and open problems, and highlighting some challenges for the existence of smooth actions. A version of a talk presented at the conference on Transformation Groups in Topology and Geometry, University of Massachusetts at Amherst, July 2008.
Wednesday October 15, 2008 in Rawles Hall 104: Allan Edmonds, Survey of group actions on 4-manifolds, Part 2; A continuation of last week's talk, focusing mainly on existence and classification of finite group actions on 4-manifolds.
Wednesday October 22, 2008 in Rawles Hall 104: Steve Mitchell, The Topology of Birkhoff Varieties; In his study of singularities of systems of ODE's (c.1910), Birkhoff introduced the decomposition $$SL_n {\Bbb C} [z,z^{-1}] =\coprod _{\lambda} P^- \lambda P ,$$ where $P=SL_n {\Bbb C} [z]$, $P^-=SL_n {\Bbb C} [z^{-1}]$, and $\lambda$ ranges over the diagonal matrices $diag (z^{a_1},...z^{a_n})$ with $a_1 \geq a_1 \geq ...\geq a_n$ and $\sum a_i =0$. The Birkhoff decomposition can be reinterpreted as a classification of holomorphic vector bundles over ${\Bbb P} ^1$ (due to Grothendieck, 1958). Yet another interpretation involves passing to the {\it affine Grassmannian} ${\cal L} =SL_n {\Bbb C} [z,z^{-1}] /P$, which has the structure of an infinite-dimensional projective variety. Then the $P^-$-orbits define the {\it Birkhoff stratification} of ${\cal L}$. The closure $Z_\lambda$ of an orbit is a {\it Birkhoff variety}. These are analogous to the “opposite” or “dual” Schubert varieties in an ordinary Grassmannian, except that instead of having finite dimension, $Z_\lambda$ has finite codimension. In this talk I will describe recent joint work with Luke Gutzwiller on the topology of Birkhoff varieties. Our main result is that the inclusion of a Birkhoff variety in the affine Grassmannian is a homotopy equivalence. This result holds in all Lie types, and also for a finer version of the Birkhoff decomposition that I will define.
Wednesday November 12, 2008 in Rawles Hall 104: Yazinski, Fundamental Groups of Symplectic 4-Manifolds
Wednesday December 3, 2008 in Rawles Hall 104: Kate Ponto, Traces and fixed points; We can associate an integer called the Lefschetz number to each endomorphism of a closed manifold. The Lefschetz fixed point theorem states that the Lefschetz number of an endomorphism with no fixed points is zero. This theorem is a consequence of the identification of the Lefschetz number, a global invariant, with a local invariant. There are many proofs of this identification, one uses the trace in a symmetric monoidal category. For closed smooth manifolds of dimension at least three, the Lefschetz number of an endomorphism can be refined to an invariant that is zero if and only if the endomorphism has no fixed points. A major step in the proof of this result is the identification of a local invariant with a global invariant. I will explain how to prove this identification using the trace in a bicategory. I will also explain how to use this perspective to define equivariant and fiberwise fixed point invariants.
Wednesday December 10, 2008 in Rawles Hall 104: Daniel Ramras, Gauge theory and homotopical representation theory; To every representation rho: Gamma –> G of a discrete group Gamma into a Lie group G, there is an associated map of classifying spaces Brho: BGamma –> BG, which classifies the G-bundle associated to rho. When Gamma admits a compact manifold as its classifying space, this correspondence can be studied using gauge theory. The resulting computations show interesting relationships between representation spaces of discrete groups, cohomology, and K-theory, and suggest analogies with the Atiyah-Segal Theorem and the Quillen-Lichtenbaum conjectures.
Tuesday December 16, 2008 in Rawles Hall 104: Victor Ostrik, Graded fusion categories; This is a joint work with P.Etingof and D.Nikshych. An additive tensor category C is called graded by a group G if there is a direct sum decomposition C=oplus_{g in G}C_g with C_g otimes C_h subset C_{gh}. This definition implies that a component _e corresponding to the unit element e of G is a tensor subcategory of C and we say that C is a G-graded extension of C_e. In this talk I will describe a theory which aims to describe all graded extensions of a given tensor category in a class of fusion categories (which are in a sense simplest additive tensor categories).
Wednesday January 14, 2009 in Rawles Hall 104: John Klein, Higher torsion invariants.; will outline the current state of the theory of higher Reidemeister torsion, and discuss examples and open problems.
Wednesday January 21, 2009 in Rawles Hall 104: Chuck Livingston, The Stable 4-Genus of Knots; The 4-genus g(K) of a knot K is the minimal genus of a surface bounded by K in the 4-ball. The limit g(nK)/n as n goes to infinity exists and is called the stable 4-genus. I will discuss examples and properties of this invariant.
Wednesday February 4, 2009 in Rawles Hall 104: John Bryden, The linking form conjecture for 3-manifolds; The linking form conjecture states that any non-degenerate, symmetric bilinear form on a finite abelian group (into Q/Z) is isomorphic to the linking form of a Seifert manifold that is a rational homology sphere. As an immediate consequence it would show that any closed, connected, oriented 3-manifold has linking form isomorphic to the linking form of a Seifert manifold. This may be somewhat surprising. This conjecture arose when Florian Deloup and I were investigating a certain class of quantum invariants. What is more surprising is that the proof arises from homotopy theory.
Wednesday February 18, 2009 in Rawles Hall 104: John Bryden, The linking form conjecture for 3-manifolds; The linking form conjecture states that any non-degenerate, symmetric bilinear form on a finite abelian group (into Q/Z) is isomorphic to the linking form of a Seifert manifold that is a rational homology sphere. As an immediate consequence it would show that any closed, connected, oriented 3-manifold has linking form isomorphic to the linking form of a Seifert manifold. This may be somewhat surprising. This conjecture arose when Florian Deloup and I were investigating a certain class of quantum invariants. What is more surprising is that the proof arises from homotopy theory.
Wednesday February 25, 2009 in Rawles Hall 104: Mike Mandell, Localization Sequences in THH and TC; I will outline the relationship of THH and TC to algebraic K-theory and describe the localization results referred to in Hesselholt's abstract for his lecture next week.
Wednesday March 4, 2009 in Rawles Hall 104: Lars Hesselholt, Vanishing of negative K-groups; A conjecture of Weibel predicts that the Bass negative K-groups of a noetherian scheme X of dimension d vanish in degrees i < - d. A few years ago, the conjecture was verified for schemes separated and essentially of finite type over a field of characteristic 0 by Cortinas-Haesemeyer-Schlichting-Weibel. I will discuss recent work with Thomas Geisser which establishes the conjecture for schemes separated and essentially of finite type over a perfect field of characteristic p > 0 under the assumption of resolution of singularities. The proof in characteristic 0 uses a comparison of K-theory to Connes' cyclic homology. Similarly, the proof in characteristic p > 0 uses a comparison of K-theory to the topological cyclic homology of Bokstedt-Hsiang-Madsen. In both cases, the starting point of the proof is a theorem of Haesemeyer which allows one to show that the difference between K-theory and the respective cyclic theories satisfies descent with respect to Voevodsky's cdh-topology. The proofs are completed by careful analysis of the cyclic theories. A central technical component in the proof is the recent extension from rings to exact categories of the definition of the respective cyclic theories by Keller and Blumberg-Mandell.
Thursday March 5, 2009 in Rawles Hall 104: Tim D. Cochran, New Characteristic Series for Groups and the Fractal Nature of Knot Concordance; For each sequence P={p_1(t),...} of polynomials we define a characteristic series of groups, called the derived series localized at P. Given a knot K, such a sequence of polynomials can arise naturally as the orders of certain submodules of the sequence of higher-order Alexander modules of K. These also yield filtrations of the knot concordance group that refine the n-solvable filtration. We show that the quotients of successive terms of these refined filtrations have infinite rank. These give higher-order analogues of the p(t)-primary decomposition of the algebraic concordance group. We also use these to give evidence that the set of smooth concordance classes of knots is a fractal set.
Wednesday March 11, 2009 in Rawles Hall 104: Vladimir Touraev, Bundles over surfaces from quantum viewpoint; A classical problem in the theory of fiber bundles is to detect whether or not the given fiber bundle has a section. When the base of the bundle is a closed orientable surface and the fundamental group G of the fiber is finite, a complete answer is formulated in terms of characteristic classes of the bundle derived from linear representations of G. The proofs use the techniques of 2-dimensional topological quantum field theory.
Wednesday April 1, 2009 in Rawles Hall 104: Jim Davis, Mapping tori of self-homotopy equivalences of lens spaces; Alternate titles include: 1) There are no exotic beasts in Hillman's zoo 2) Homotopy fibrations and a lemma of Gauss
Wednesday April 15, 2009 in Rawles Hall 104: Teena Gerhardt, Algebraic K-theory of the Dual Numbers; Nearly 30 years ago, Soule showed that the abelian group K_n(Z[x]/x^2, (x)) is finitely generated with rank 0 if n is even and 1 if n is odd. We compute this group for n odd and find its order for n even. Further, we generalize these results to the study of the algebraic K-theory of truncated polynomial algebras, K_n(Z[x]/x^e, (x)). This is joint work with Vigleik Angeltveit and Lars Hesselholt.
Wednesday April 22, 2009 in Rawles Hall 104: John Bryden, Group cohomological methods in low dimensional topology
Wednesday April 29, 2009 in Rawles Hall 104: Allan Edmonds, Groups that act pseudofreely on spheres
Thursday April 30, 2009 in Rawles Hall 104: Matt Hedden, Using Heegaard Floer homology to show that Khovanov homology of satellites detects the unknot
Wednesday May 6, 2009 in Rawles Hall 104: Chuck Livingston, Slice knots not concordant to polynomial 1 knots
Wednesday May 20, 2009 in Rawles Hall 104: Qingxiang Zhang, Dehn surgery on two bridge knots and two generator Klienian groups; Let < f, g > be a two generator Klienian group. We define B(f)=tr^2(f)-4, B(g)=tr^2(g)-4,G(f, g)=tr[f, g]-2 where tr(f) and tr(g) denote the traces of representative matrices of f and g respectively and [f, g] denotes the multiplicative commutator fgf^{-1}g^{-1}. These three complex numbers are called the parameters of the two generator group < f, g>, we write (< f, g>)=(G(f, g),B(f), B(g)). We discuss the G(f, g) plane for g is elliptic of order 2 and f is elliptic of order p for p=3, 4, 5, 6, 7.
Wednesday August 19, 2009 in Rawles Hall 104: Rinat Kashaev, Algebraic knot invariants from ideal triangulations.
Tuesday September 29, 2009 in Rawles Hall 104: Pat Gilmer, Topological applications of integral tqft
Wednesday September 30, 2009 in Rawles Hall 104: Vigleik Angeltveit, Uniqueness of Morava K-theory; Any cohomology theory is represented by what is called a spectrum. For each prime p and positive integer n there is a cohomology theory, or spectrum, called the n'th Morava K-theory K(n). When n=1 this is essentially mod p complex K-theory. I will discuss multiplicative structure on K(n). In particular I will discuss how many associative multiplications K(n) has. The main result is that there is essentially only one. To analyze this question I will discuss the notion of an A-infinity structure, a multiplication which is associative up to homotopy and higher homotopies in a precise sense.
Tuesday October 6, 2009 in Rawles Hall 104: Michael Brandenbursky, Knot theory and quasi-morphisms; Quasi-morphisms on a group are real-valued functions which satisfy the homomorphism equation ?up to a bounded error?. They are known to be a helpful tool in the study of the algebraic structure of non-Abelian groups. I will discuss a construction relating a) certain knot and link invariants ? in particular,the ones that come from the knot Floer homology and a Khovanov-type homology, b) braid groups, c) the dynamics of area-preserving diffeomorphisms of a two-dimensional disc, d) quasi-morphisms on the group of all such compactly supported diffeomorphisms of the disc.
Wednesday October 21, 2009 in Rawles Hall 104: Matthew Gelvin, Fusion systems acting on finite sets; Fusion systems are an abstraction of the p-local structure of finite groups: They model the inclusion of a Sylow p-subgroup in an ambient finite group, together with ambient conjugation (or "fusion") data. The work of Broto-Levi-Oliver has shown how these algebraic, almost combinatorial objects are closely related to the homotopy type of the p-completion of classifying spaces of finite groups. In this talk, I will examine how the fundamental notion of group actions on finite sets can be understood in the context of fusion systems. I will try to motivate the notion of a "fusion action system," as well as the basic properties they should satisfy. There is also a strong parallel with the work of BLO: Fusion action systems have a notion of "classifying space," which, in the presence of an actual ambient group G acting on a finite set X, reconstructs the Borel construction EGtimes_G X up to p-completion. I shall finish with an overview of this classifying space story, and suggest some applications to fusion system theory.
Wednesday November 4, 2009 in Rawles Hall 104: David Gepner, Algebraic K-theory and stabilization; We show that algebraic K-theory is the universal additive functor from stable infinity categories to spectra. Generalizing the construction of this universal additive functor reveals that K-theory arises as a special case of the stabilization procedure left adjoint to the forgetful functor from stable infinity categories to exact infinity categories. As an application, we construct (cyclotomic) trace maps and address issues of multiplicative structure. This is joint work with Andrew Blumberg and Goncalo Tabuada.
Wednesday November 11, 2009 in Rawles Hall 104: Rafael Torres, Geography and Botany of Irreducible Symplectic 4-Manifolds with abelian fundamental group.; The recent successes on the existence and (lack of) uniqueness of irreducible (symplectic) smooth 4-manifolds with trivial fundamental group can be extended naturally to the non-simply connected realm. The talk deals with the use of current technology on the study of 4-manifolds with abelian fundamental group.
Wednesday November 18, 2009 in Rawles Hall 104: Brenda Johnson, Calculus of Functors; Goodwillie's calculus of homotopy functors provides a means of analyzing functors of spaces via "polynomial" approximations to the functors. After providing an introduction to the calculus of functors, I will discuss work with Kristine Bauer and Randy McCarthy aimed at understanding and developing this theory in other contexts.
Wednesday December 2, 2009 in Rawles Hall 104: Mike Hill, Equivariant Homotopy and the Kervaire Invariant
Wednesday December 9, 2009 in Rawles Hall 104: Andrew Blumberg, THH of Waldhausen categories and localization
Thursday January 21, 2010 in Rawles Hall 104: Sergei Matveev, Roots and prime decompositions of geometric objects.; A generalization of the Kneser-Milnor theorem on prime decomposition of 3-manifolds.
Thursday January 28, 2010 in Rawles Hall 104: Mihai Staic, Secondary Cohomology and k-invariants; The 2-type of a topological space is determined by the first k-invariant $\kappa^3\in H^3(\pi_1(X),\pi_2(X))$. For a triple $(G,A, \kappa)$ (where G is a group, A is a G-module and $\kappa:G\to A$ is a 3-cocycle) and a G-module B we introduce a new cohomology theory $_2H^n(G,A, \kappa;B)$ which we call the secondary cohomology. We give a construction that associates to a pointed topological space $(X, x_0)$ an invariant $_2\kappa^4\in \, _2H^4(\pi_1(X), \pi_2(X), \kappa^3; \pi_3(X))$. This construction can be seen a 3-type generalization of the classical k-invariant. We also present the relation with the Eilenberg-MacLane space $K(A,2)$.
Wednesday February 24, 2010 in Rawles Hall 104: Burglind Joricke, Analytic knots, satellites and 4-ball genus; Call a knot K in the sphere in complex affine 2-space analytic if it bounds a complex curve in the ball. The genus of the curve is the 4-ball genus of the knot. Let K be an analytic knot. For an arbitrary analytic knot contained in a small tubular neighbourhood of K we give a sharp estimate of the 4-ball genus. The proof uses mainly complex geometry tools. We do not know a proof using knot invariants and we do not know whether the estimate is true for arbitrary quasipositive satellites of quasipositive knots.
Wednesday March 3, 2010 in Rawles Hall 104: Paul Kirk, Instantons, Chern-Simons invariants and knot concordance; I will talk on some results related to the title
Wednesday March 10, 2010 in Rawles Hall 104: Marcy Robertson, Derived Morita Theory for Enriched Symmetric Multicategories; Operads, multicategories, and their representations (also called operadic/multicategorical algebras) play a key role in organizing hierarchies of higher homotopies in any category with a good notion of homotopy theory. In this talk we show how one can generalize work of Toen and Rezk to provide a description of the derived category of any multicategorical algebra. Time permitting, we will discuss applications of this theory to problems in combinatorial representation theory. We do not assume prior knowledge of the theory of operads and multicategories.
Wednesday March 24, 2010 in Rawles Hall 104: Vladimir Touraev, Introduction to knotoids; Knotoids are geometric objects generalizing classical knots. I will introduce knotoids and explain why they are interesting.
Thursday April 15, 2010 in Rawles Hall 104: Chuck Livingston, The nonorientable 4-genus of knots; I will discuss bounds on the minimal first betti number of a nonorientable embedded surface in the 4-ball with boundary a given knot.
Wednesday April 28, 2010 in Rawles Hall 104: Niles Johnson, Power Operations and p-typicality
Thursday May 6, 2010 in Rawles Hall 104: Diarmuid Crowley, Embeddings of 4-manifolds into 7-space; Abstract: (joint with Arkadiy Skopenkov.) I will describe recent results classifying isotopy classes of smooth and piecewise linear embeddings of closed connected smooth 4-manifolds with torsion free integral homology groups into 7-dimensional Euclidean space. The case of S^1 x S^3 is of particular interest. The method is via the modified surgery algorithm for classifying embeddings and involves new calculations with the surgery obstruction monoid l_8(e).
Thursday May 6, 2010 in Rawles Hall 104: Diarmuid Crowley, An introduction to the Manifold Atlas; The Manifold Atlas is a scientific Wiki about manifolds with plans to become a sort of on-line journal. In this short talk I will introduce the Atlas by outlining it's goals, its structure and some aspects of using the Atlas.
Thursday July 15, 2010 in Swain West 217: Kate Kearney, Concordance genus of knots
Tuesday September 7, 2010 in Lindley Hall 019: Eiji Ogasa, Local moves of n-dimensional knots; We consider how $n$-knots are changed and how invariants of $n$-knots are changed if $n$-knots are changed by local moves of $n$-knots. In the $n=1$ case, some results are well known: pass-moves of 1-knots and the Arf invariants, crossing change of 1-knots and the Conway-Alexander polynomial (resp. the Jones polynomial, the Vassiliev invariants). We consider what kind of local moves of $n$-knots ($n\geqq2$) should be considered in order to generalize the $n=1$ case. This talk is mainly based on the speaker's published papers: [1]Local move identities for the Alexander polynomials of high dimensional knots and inertia groups. Journal of knot theory and its ramificatioms vol18, no.4, 2009 531-545, math.GT/0512168, UTMS 97-63. [2]Ribbon-moves of 2-links preserve the $\mu$-invariant of 2-links. Journal of knot theory and its ramificatioms, vol13, no5, 2004, 669-687, math.GT/0004008, UTMS 97-35. [3]Supersymmetry, homology with twisted coefficients and n-dimensional knots. International Journal of Modern Physics A Vol. 21, Nos. 19 & 20 (10 August 2006), pp.4185-4196, hep-th/0311136.
Wednesday September 8, 2010 in Rawles Hall 104: Allan Edmonds, Group actions on aspherical manifolds; This will be a survey of some topics related to finite group actions on aspherical manifolds, somewhat from the point of view of low dimensional topology.
Wednesday September 15, 2010 in Rawles Hall 104: Margaret Doig, Heegaard-Floer homology and knot surgery
Thursday September 23, 2010 in Rawles Hall 104: Heather Dye, Categorifications of the Arrow Polynomial; I will present several categorifications of the arrow polynomial, an invariant of virtual links. An overview of virtual links, the arrow polynomial, and Khovanov homology will be included.
Wednesday October 6, 2010 in Rawles Hall 104: Chuck Livingston, Bing Doubling and the boundary genus of links
Wednesday October 13, 2010 in Rawles Hall 104: Margaret Doig, Heegaard-Floer homology and knot surgery (continued)
Monday October 18, 2010 in Rawles Hall 316: Jonathan Yazinski, Involutions on exotic 4-manifolds
Wednesday October 20, 2010 in Rawles Hall 104: Nathaniel Rounds, Local Poincare Duality
Thursday October 21, 2010 in Rawles Hall 104: Qayum Khan, On isovariant rigidity of CAT(0) manifolds; We discuss Quinn's equivariant generalization of the Borel Conjecture. This concerns cocompact proper actions of a discrete group $\Gamma$ on a Hadamard manifold $X$. We give a complete solution when the action of $\Gamma$ is pseudofree and when $X$ more generally is a CAT(0) manifold. Here, emph{pseudofree} means that the singular set is discrete. A rich class of examples is obtained from crystallographic groups $Gamma$ made out of isometric spherical space form groups $G$. If $\Gamma$ has no elements of order two, then we obtain equivariant topological rigidity of the pair $(X,\Gamma)$. Hence, if $\Gamma$ is torsion-free, then we generalize a recent theorem of A.~Bartels and W.~L"uck, which validates the classical Borel conjecture for CAT(0) fundamental groups. Otherwise, if $\Gamma$ has elements of order two, we show how to parameterize all possible counterexamples, in terms of Cappell's UNil summands of the $L$-theory of infinite dihedral groups. These projects are joint work with Frank Connolly and Jim Davis.
Wednesday November 3, 2010 in Rawles Hall 104: Ben McCarty, An infinite family of Legendrian torus knots distinguished by cube number; For a knot K, the cube number is a knot invariant defined to be the smallest n for which there is a cube diagram of size n for K. There is also Legendrian version of this invariant called the Legendrian cube number. We will show that the Legendrian cube number distinguishes the Legendrian left hand torus knots with maximal Thurston-Bennequin number and maximal rotation number from the Legendrian left hand torus knots with maximal Thurston-Bennequin number and minimal rotation number
Tuesday November 9, 2010 in Rawles Hall 104: Joan Licata, Legendrian contact homology for Seifert fibered spaces; In this talk, I'll focus on Seifert fibered spaces whose fiber structure is realized by the Reeb orbits of an appropriate contact form. A Legendrian knot in such a manifold is described by a specially labeled Lagrangian diagram, and from this data one can compute both the "classical" invariants for Legendrian knots in rational homology three-spheres and also a new invariant which takes the form of a differential graded algebra. This work is joint with J. Sabloff.
Wednesday December 1, 2010 in Rawles Hall 104: Sam Lewallen, Twisted signatures and some topology behind the Jones polynomial; We'll discuss a new relationship between Khovanov homology and SU(2) representations of the knot group. In simple cases, this leads to a definition of the Jones polynomial directly in terms of the SU(2) representation variety. In particular, we show that for 2-bridge knots, the Jones polynomial is determined by certain invariants of flat connections on the knot complement, which are themselves equal to some classical, metacyclic invariants of the knot group.
Wednesday February 9, 2011 in Rawles Hall 104: Jae Choon Cha, Applications of covering link calculus; TBA
Wednesday February 23, 2011 in Rawles Hall 104: Ryan Budney, Symmetry of knots; Abstract: In the 80's Allen Hatcher proved a rather remarkable conjecture of Steve Smale, that says that in principle there is a "uniform" way to straighten all unknots. _Specifically, the path-component of the trivial knot in the space of smooth embeddings of the circle in the 3-sphere has as a deformation-retract the subspace of great circles. _This talk will be about a certain logical extension of these ideas into families of canonical positions for all knot types, developing connections between operads and geometrization of 3-manifolds.
Wednesday March 2, 2011 in Rawles Hall 104: Ayelet Lindenstrauss, The Algebraic K-Theory of the Ring of Formal Power Series; (joint with Randy McCarthy) We study the algebraic K-theory of parametrized endomorphisms of a unital ring R with coefficients in a simplicial R-bimodule M, and compare it with the algebraic K-theory of the ring of formal power series in M over R. Waldhausen defined an equivalence from the suspension of the reduced Nil K-theory of R with coefficients in M to the reduced algebraic K-theory of the tensor algebra T_R(M). Extending Waldhausen's map from nilpotent endomorphisms to all endomorphisms, our map has to land in the ring of formal power series rather than in the tensor algebra, and is no longer in general an equivalence (it is an equivalence when the bimodule M is connected). Nevertheless, the map shows a close connection between its source and its target: it induces an equivalence on the Goodwillie Taylor towers of the two (as functors of M, with R fixed), and allows us to give a formula for the suspension of the invariant W(R;M) (which can be thought of as Witt vectors with coefficients in M, and is what the Goodwillie Taylor tower of the source functor converges to) as the inverse limit, as n goes to infinity, of the reduced algebraic K-theory of T_R(M)/ (M^n).
Wednesday March 9, 2011 in Rawles Hall 104: Adam Levine, A Combinatorial Spanning Tree Model for Knot Floer Homology; We provide an explicit description of a complex, based on spanning trees of the black graph of a diagram of a knot K in S^3, that computes the knot Floer homology of K. The strategy is to iterate Manolescu's unoriented skein exact sequence for knot Floer homology, using twisted coefficients in a power series ring, to form a cube of resolutions in which the only nonzero groups correspond to the connected resolutions. This construction has intriguing similarities with Ozsvath and Szabo's spectral sequence from the reduced Khovanov homology of K to the Heegaard Floer homology of the double branched cover of K, and it suggests an approach toward understanding the behavior of knot Floer homology under mutation. This is joint work with John Baldwin.
Thursday March 24, 2011 in Rawles Hall 104: Philip Hackney, Operations in the spectral sequence of a cosimplicial space
Wednesday March 30, 2011 in Rawles Hall 104: Palanivel Manoharan, Lefschetz fixed point theory on fibrations
Wednesday April 13, 2011 in Rawles Hall 104: Josh Greene, Lattices, graphs, and Conway mutation; I will discuss the proof and consequences of the following result: if D and D' are reduced, alternating diagrams for a pair of links with branched double covers Y and Y', then D and D' are mutants iff Y and Y' are homeomorphic iff Y and Y' have isomorphic Heegaard Floer homology. _The main input is a combinatorial result which characterizes the 2-isomorphism type of a graph in terms of the d-invariant of its lattice of integral flows.
Thursday April 14, 2011 in Rawles Hall 104: Mark Powell, Links not concordant to the Hopf Link
Wednesday April 20, 2011 in Rawles Hall 104: Pat Gilmer, Irreducibility properties of modular representations of mapping _class groups arising in TQFT; We find _decomposition series of length at most two for the modular representations of the mapping class groups of a one-holed surfaces induced by integral TQFT. We given the dimensions of these irreps.
Thursday April 21, 2011 in Rawles Hall 104: Tetsuya Abe, The Rasmussen invariant of a homogeneous knot; homogeneous knot is a generalization of alternating knots and positive knots. In this talk, we determine the Rasmussen invariant of a homogeneous knot. As a corollary, we obtain some characterizations of a positive knot. In particular, we recover Baader_s theorem which states that a knot is positive if and only if it is homogeneous and strongly quasipositive.
Wednesday April 27, 2011 in Rawles Hall 104: Yi-Jen Lee
Thursday April 28, 2011 in Rawles Hall 104: Tetsuya Abe, The Rasmussen invaiant of homogeneous links
Thursday May 5, 2011 in Rawles Hall 104: Matt Hedden, Contact Structures and Heegaard-Floer Homology
Tuesday August 16, 2011 in Swain West 217: Jen Hom, Concordance and the knot Floer complex; We will use the knot Floer complex to define a new smooth knot concordance invariant. Applications include a formula for the Ozsvath-Szabo tau invariant of iterated cable knots, better bounds (in many cases) on the 4-ball genus knots, and a new infinite family of smoothly independent topologically slice knots. We will also discuss various algebraic properties of this new invariant.
Wednesday August 31, 2011 in Rawles Hall 104: Yuanyuan Bao, On knot Floer homology of satellite knots and their genera; Knot Floer homology in the three-sphere is a categorification of the classical Alexander polynomial. Hedden studied the knot Floer homology of the Whitehead double of a knot. We generalize his method to a class of satellite knots, and get similar formulas for the top grading. As an application, we use our results to calculate the Seifert genera of these satellite knots.
Wednesday September 14, 2011 in Rawles Hall 104: Justin Young, Enhanced Bar Cobar Duality
Wednesday September 21, 2011 in Rawles Hall 104: Greg Chadwick, Structured orientations of Thom spectra
Wednesday September 28, 2011 in Rawles Hall 104: Heather Dye, Smoothed Invariants of Virtual Knots; I introduce two types of smoothed invariants. The first smoothed invariant uses linking numbers to construct an invariant vector and its associated polynomial. The second is an invariant of flat virtual knot diagrams. This invariant is a formal sum of diagrams and can be used in conjunction with other invariants to produce a family of invariants. We discuss its construction and a non-trivial example.
Wednesday October 5, 2011 in Rawles Hall 104: Alexis Virelizier, Knots and Hopf diagrams; We will investigate the algebraic structure hidden in knot diagrams and its relationships with quantum invariants.
Tuesday October 11, 2011 in Rawles Hall 104: Zhongtao Wu, Cosmetic Surgery Conjecture on S^3; It has been known over 40 years that every closed orientable 3-manifold is obtained in surgery on a link in S3. However, a complete classification has remained elusive due to the lack of uniqueness of this surgery description. In this talk, we discuss the following uniqueness theorem for Dehn surgery on a nontrivial knot in S3. Let K be a knot in S3, and let r and r' be two distinct rational numbers with different absolute values, then there is no orientation preserving homeomophism between the manifolds obtained by performing Dehn surgery of type r and r', respectively. In particular, this result implies the Knot Complement Theorem of Gordon and Luecke.
Wednesday October 12, 2011 in Rawles Hall 104: P. Kirk, Coisotropic Luttinger Surgery; I will describe a type of symplectic surgery generalizing Luttinger surgery to higher dimensions. As an application, I'll construct an infinite family of non-Kahler symplectic 6-manifolds with vanishing canonical class (joint with Scott Baldridge)
Wednesday October 19, 2011 in Rawles Hall 104: Vladimir Turaev, Fox pairings and generalized Dehn twists
Tuesday November 1, 2011 in Rawles Hall 104: Colin Guillarmou, Chern Simons line bundle over Teichmuller space
Wednesday November 2, 2011 in Rawles Hall 104: Ben Ward, Cyclic A-infinity Algebras and Deligne's Conjecture; First we describe a class of homotopy Frobenius algebras via cyclic operads which we call cyclic $A_\infty$ algebras. We then define a suitable new combinatorial operad which acts on the Hochschild cochains of such an algebra in a manner which encodes the homotopy BV structure. Moreover we show that this operad is equivalent to the cellular chains of a certain topological (quasi)-operad of CW complexes whose constituent spaces form a homotopy associative version of the Cacti operad of Voronov. These cellular chains thus constitute a chain model for the framed little disks operad, proving a cyclic $A_\infty$ version of Deligne's conjecture.
Wednesday November 16, 2011 in Rawles Hall 104: Se-Goo Kim, Splitting Properties of Von Neumann Invariants of Knots
Wednesday November 30, 2011 in Rawles Hall 104: Se-Goo Kim, Splitting Properties of Von Neumann Invariants of Knots, continued
Wednesday December 7, 2011 in Rawles Hall 104: Chuck Livingston, Two torsion in smooth knot concordance; I will discuss an ongoing project with Matt Hedden and Se-Goo Kim building a collection of topologically slice knots that represent two-torsion in the smooth knot concordance group.
Wednesday January 11, 2012 in Rawles Hall 104: David Gepner, Brauer groups of commutative ring spectra; The space of units $GL_1(R)$ of a commutative ring spectrum $R$ is the infinite loop space of a spectrum $gl_1(R)$. Typically, this spectrum is taken to be connective, meaning it has no nonzero negative homotopy groups. However, there are other interesting deloopings of $GL_1(R)$ which carry important algebraic information about R. One in particular has $\pi_{-1} gl_1(R) = \pi_0$Pic(R), the Picard group of R, and $\pi_{-2} gl_1(R) = \pi_0 Br(R)$, the Brauer group of $R$. If $R$ is connective, there is a spectral sequence for computing the homotopy groups of $Pic(R)$ and $Br(R)$ in terms of the homotopy groups of R and the etale cohomology of the multiplicative group on Spec $\pi_0 R$. If $S$ is a $G$-Galois extension of $R$, one obtains a Galois descent spectral sequence which calculates the Picard and Brauer groups of $R$ relative to $S$.
Wednesday March 7, 2012 in Rawles Hall 104: Diarmuid Crowley, Some new results on the Gromoll filtration; Let $\Gamma_n$ denote the group of oriented homotopy n-spheres, n>4. There is a decreasing filtration of $\Gamma_n$ due to Gromoll: $$\Gamma = \Gamma_{n, 2} \supset \Gamma_{n, 3} \supset \cdots \supset \Gamma_{n, n-3} = 0$$ The determination of the Gromoll filtration is a difficult open problem: it seems to be unknown for any n > 4 when Gamma^n eq 0. Moreover, it has many interesting applications in differential topology. For example: 1) Helping determine which manifolds appear as the total spaces of bundles of smooth manifolds. 2) Proving that certain spaces of diffeomorphisms are not homotopy equivalent to finite CW complexes. 3) Hitchin used the fundamental result of Cerf that $\Gamma^n_1 = \Gamma^n_2$ to find non-zero homotopy groups of the space of metrics of positive scalar curvature on spin manifolds. In this talk I will survey methods for understanding the Gromoll filtration: Plumbing bundles and compositions in the homotopy groups of spheres allow one to deeper into the filtration. Whereas stable pseudo-isotopy theory gives an obstruction theory, complete in a range, for moving deeper in the filtration. I will also report on new results for certain Gromoll groups and discuss some applications as well as some open problems. The talk is based on joint work with Thomas Schick and also with Wolfgang Steimle.
Wednesday March 21, 2012 in Rawles Hall 104: Gabriel C. Drummond-Cole, Homotopically trivializing the circle in the framed little disks; An action of the space of framed little disks on a target space induces a circle action on the target space. Kontsevich suggested that an action of the framed little disks along with a trivialization of the circle action could be encapsulated as follows: this data should be the same as an action of the Deligne-Mumford-Knudsen compactified genus zero moduli space. I'll present a rigorous formulation of this statement in the category of topological operads. There will be many pictures.
Tuesday March 27, 2012 in Rawles Hall 104: Efstratia Kalfagianni, Jones polynomials and incompressible surfaces; I will report on joint work with Dave Futer (Temple) and Jessica Purcell (BYU): Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We show that certain coefficients of the polynomial measure how far this surface is from being a fiber in the knot complement; in particular, the surface is a fiber if and only if a particular coefficient vanishes. Our results also yield concrete relations between hyperbolic geometry and colored Jones polynomials: for certain families of links, coefficients of the polynomials determine the hyperbolic volume to within a factor of 4. Our approach is to generalize the checkerboard decompositions of alternating knots. Under mild diagrammatic hypotheses we show that the checkerboard knot surfaces are incompressible, and obtain an ideal polyhedral decomposition of their complement. We employ normal surface theory to establish a dictionary between the pieces of the JSJ decomposition of the surface complement and the combinatorial structure of certain spines of the checkerboard surface (state graphs). Since state graphs have previously appeared in the study of Jones polynomials, our setting and methods create a bridge between the polynomials and the geometry of the surface complement.
Wednesday March 28, 2012 in Rawles Hall 104: Kate Poirier, Compactifying string topology; String topology studies the algebraic topology of the free loop space of a manifold. In this talk, we describe a compact space of graphs and show how this space gives algebraic operations on the singular chains of the free loop space. In particular, our chain level operations induce Cohen and Godin's "positive boundary TQFT" on the homology the free loop space. This project is joint work with Nathaniel Rounds.
Tuesday April 3, 2012 in Rawles Hall 104: Kristen Hendricks, Heegaard-Floer Homology of 2-Fold Branched Covers
Wednesday April 18, 2012 in Rawles Hall 104: Andrew Salch, Twisted deformation theory of modules and Adams spectral sequences; We discuss some applications of the representation theory of Hopf algebras to Adams spectral sequences and the problems of classifying spaces and spectra and computing stable homotopy groups. We then describe a twisted version of the deformation theory of modules, its relationship to Hochschild cohomology, and its use in computing representation semirings of Hopf algebras. We work out some sample computations in the case of Rep(A(1)), the representation semiring of the subalgebra of the 2-primary Steenrod algebra generated by the first two Steenrod squares. Finally, we describe some future directions we hope to take with this work, mainly what kind of relationship we would like these twisted deformation theories to have with a "nonabelian higher-order Hochschild cohomology."
Wednesday August 29, 2012 in Rawles Hall 104: Jim Davis, Every finite group acts freely and homologically trivially on a product of spheres
Wednesday September 12, 2012 in Rawles Hall 104: Jae Choon Cha, Amenable L2-signatures and homology cobordism of 3-manifolds; I will introduce amenable L2-signatures and some applications to homology cobordism of 3-manifolds. In particular I will talk about the notion of hidden torsion in 3-manifold groups in the context of homology cobordism. If time permits, I will also introduce the notion of Whitney tower cobordism of bordered 3-manifolds and related applications. Many results in this talk are joint with Kent Orr.
Wednesday September 26, 2012 in Rawles Hall 104: Vladimir Turaev, Quasi-Poisson structures on representation spaces of surfaces; Given an oriented surface with non-empty boundary S, we introduce a canonical quasi-Poisson bracket on the space of N-dimensional linear representations of pi_1(S) (for all N). Our bracket extends the known Poisson bracket on the GL_N-invariant functions on this space. Our main tools are intersections of loops on the surface and a natural structure of a quasi-Poisson double algebra on the group algebra of pi_1(S).
Wednesday October 10, 2012 in Rawles Hall 104: Maciej Borodzik, Topological approach to semicontinuity of spectra; The spectrum is one of the deepest invariant of an isolated singularity of a complex hypersurface (i.e. a zero set of a polynomial in n variables, where n>1). Its definition involves lots of Hodge theory. One of its most important properties is its semicontinuity under complex deformation. The classical proof of this property requires full machinery of Hodge structures. In the talk, after giving an outline of the theory, I will show, how one can associate a link to a singularity and how can one relate the spectrum to the Levine–Tristram signatures of this link. In dimension n=2 the classical Murasugi inequality for signatures turns out to be enough to prove semicontinuity of spectra. I will also sketch generalizations to higher dimensions. This is joint project with A. Nemethi (Budapest) and A. Ranicki (Edinburgh).
Wednesday October 17, 2012 in Rawles Hall 104: Ralph Kaufman, Feynman Categories; Abstract: In many situations, the structures exhibit common phenomena, like gluings/substitutions and self-gluings/substitutions. Geometric examples are given by gluing surfaces at punctures or boundaries, and algebraic examples include substitutions in functions and contractions of tensors. These phenomena can be described in terms of operads and their cousins. We will give examples and formulate a new categorical framework allowing to treat them all at once. Upshots are universal constructions of secondary algebraic structures such as Lie brackets and Batalin-Vilkovisky algebras.
Tuesday October 23, 2012 in Rawles Hall 104: Stefan Friedl, The virtual fibering theorem for 3-manifolds; In 2007 Agol showed that any irreducible 3-manifold such that its fundamental groups is `virtually RFRS' is virtually fibered. We will give a somewhat different proof using complexities of sutured manifolds. This is joint work with Takahiro Kitayama.
Wednesday October 31, 2012 in Rawles Hall 104: P. Kirk, Instanton homology and traceless character varieties of knots
Wednesday November 7, 2012 in Rawles Hall 104: Y. Fukumoto, Cobordisms of Lens spaces
Thursday November 8, 2012 in Rawles Hall 104: Allan Edmonds, The Euler Characteristic Problem for Aspherical Manifolds-1; It is a classical problem of geometric topology to prove that the sign of the Euler characteristic of a closed aspherical manifold depends only on its dimension. We will discuss this problem and survey a recent result in which it is verified for manifolds arising as hyperbolizations of other manifolds. The proof uses topological combinatorics.
Tuesday November 13, 2012 in Rawles Hall 104: Allan Edmonds, The Euler Characteristic Problem for Aspherical Manifolds-2; This talk will survey the class of aspherical manifolds called Haken n-manifolds, recently developed by B. Foozwell and H. Rubinstein. We will then outline how one can prove that a closed Haken 4-manifold has non-negative Euler characteristic. The proof involves input from topological combinatorics and an apparently new result about the face numbers of certain 4-polytopes.
Wednesday November 28, 2012 in Rawles Hall 104: John Harper, TQ-homology completion of nilpotent structured ring spectra; An important theme in current work in homotopy theory is the investigation and exploitation of enriched algebraic structures on spectra that naturally arise, for instance, in algebraic topology, algebraic K-theory, and derived algebraic geometry. Such structured ring spectra or “geometric rings” are most simply viewed as algebraic-topological generalizations of the notion of ring from algebra and algebraic geometry. This talk will describe recent progress, in joint work with M. Ching, on developing standard tools of the homotopy theory of spaces in this new algebraic-topological context of structured ring spectra, with a special emphasis on recovering algebraic and topological structures from associated homology objects. For operadic algebras in spectra (e.g., associative, commutative, or more generally, E_n ring spectra) the associated homology object is topological Quillen (TQ) homology (a type of derived abelianization of the object). Its theory was begun by Basterra for commutative ring spectra, and has been developed in certain directions by Basterra-Mandell, Goerss-Hopkins, and Mandell, among others. I will explain how a connected non-unital ring spectrum (either associative, commutative, or more generally, an E_n ring spectrum) can be recovered from the inverse limit of a tower of fibrations that by construction encodes all co-opeations on TQ-homology. I will also explain how one can relax the connectivity condition to a nilpotency condition if TQ-homology is suitably adjusted.
Monday January 7, 2013 in Rawles Hall 104: Danielle O'Donnol, Legendrian $\theta$-graphs; We define the classical invariants Thurston-Bennequin number, $tb$, and rotation number, $rot$, for Legendrian graphs. We determine which $tb$ and $rot$ can be realized as the Thurston-Bennequin number and the rotation number of a Legendrian $\theta$-graph with all unknotted cycles. We investigate whether these invariants determine the graph up to Legendrian isotopy. We also defined the transverse push-off of a Legendrian graph and explore whether it contains more information then the classical invariants. (Work done in collaboration with Elena Pavelescu.)
Thursday January 17, 2013 in Rawles Hall 104: Daniel Ramras, Stable representation theory and the geometry of flat connections; In the 1960_s, Atiyah and Segal studied the map R(G) –> K(BG) sending a representation of a compact Lie group G to the induced bundle over BG. We consider a natural generalization of this map to spherical families of (finite-dimensional, unitary) representations of infinite discrete groups. This topological Atiyah-Segal map is closely linked to the natural map Hom(G,U(n)) –> Map(BG,BU(n)), and is thereby related to questions about (families of) flat bundles over BG and spaces of flat connections. On the other hand, this map can be described in terms of a homotopy limit problem for Carlsson_s deformation K-theory functor. This brings methods from stable homotopy theory to bear, leading to various results for (products of) aspherical surfaces, tori, and flat manifolds. Parts of this work are joint with Tom Baird.
Wednesday February 6, 2013 in Rawles Hall 104: Jen Hom, Bordered Floer homology and the Seifert form; We study the bordered Floer homology of the 3-manifold with boundary obtained by cutting S^3 along a Seifert surface for a knot K. In particular, we show that this bordered invariant is a categorification of the Seifert form. This is joint work in progress with Sam Lewallen, Tye Lidman and Liam Watson.
Wednesday February 13, 2013 in Rawles Hall 104: Gwenael Massuyeau, Surgery equivalence relations of 3-manifolds defined by subgroups of the Torelli group; Two compact oriented 3-manifolds M and M' are said to be "Y_k-equivalent" if M' is obtained from M by "twisting" an arbitrary surface E of M with a homeomorphism belonging to the k-th term of the lower central series of the Torelli group of E. The "J_k-equivalence" relation is defined in a similar way using a homeomorphism of E acting trivially on the k-th nilpotent quotient of the fundamental group of E. In this talk, we shall review what is known about these equivalence relations and we will focus on a specific class of 3-manifolds, namely the monoid of homology cylinders. We shall characterize these relations for k=1,2,3 using classical and quantum invariants. (This talk is based on joint works with K. Habiro & J.-B. Meilhan.)
Wednesday February 20, 2013 in Rawles Hall 104: Palanivel Manoharan, Lefschetz theory on Fibre bundles via Gysin homomorphism.
Tuesday February 26, 2013 in Rawles Hall 104: Randy McCarthy, Calculus of Functors: A Case for the Unbased Setting; In the mid-80's, Tom Goodwillie created a Taylor tower for functors of spaces/spectra or other suitable model categories with an initial and final object (not necessarily equal). Most work on the subject during the last 25 years has been concerned with functors from pointed categories (initial and final objects are equal). In this talk, we will discuss some of the basics of this Calculus and examples, specifically an interpretation due to Goodwillie and Waldhausen of DeRham cohomology via Calculus of functors which requires one to work in the unbased setting.
Wednesday February 27, 2013 in Rawles Hall 104: Kevin M. Pilgrim, Mapping class semigroups?; Recent work on classification problems arising in one-dimensional complex analytic dynamics suggest an underlying theory of mapping class semigroups. Let $S^2$ denote the two-sphere, and fix a finite set $P subset S^2$. The set $BrCov(S^2, P)$ of orientation-preserving branched covering maps of pairs $f: (S^2, P) to (S^2, P)$ of degree at least two and whose branch values lie in $P$ is closed under composition and under pre- and post-composition by orientation-preserving homeomorphisms $h: (S^2, P) to (S^2, P)$ fixing $P$ set wise. Composition descends to a well-defined map on homotopy classes relative to $P$, yielding a countable semigroup $BrMod(S^2, P)$. In addition to the semigroup structure, $BrMod(S^2, P)$ is naturally equipped with two commuting actions of the mapping class group $Mod(S^2, P)$, induced by pre- and post-composition. This richer {em biset} structure, and a related circle of constructions, turn out to be extraordinarily useful in this context. They lead to: algebraic invariants of elements of $BrCov(S^2, P)$ and an analog of the Baer-Dehn-Nielsen theorem; analogs of classical Hurwitz classes; conjugacy invariants; canonical decompositions and forms; an analog of Thurston's trichotomous classification of mapping classes; and induced dynamics on Teichm"uller spaces. This talk is based on algebraic and dynamical perspectives growing out of work of L. Bartholdi and V. Nekrashevych, and is based on ongoing conversations with S. Koch, D. Margalit, and N. Selinger. Not being an expert in topology, I invite your comments and welcome your thoughts.
Thursday February 28, 2013 in Rawles Hall 104: Matt Hedden, Taut foliations, left-orderability, and L-spaces
Wednesday March 6, 2013 in Rawles Hall 104: Mark Powell, Link concordance and the Alexander polynomial; TBA
Wednesday March 20, 2013 in Rawles Hall 104: Burglind Juhl-Joricke, From Hilbert's Thirteenth Problem to braid invariants; I will remind Hilbert's 13th Problem and some mathematical questions initiated by it. In particular, I will address problems considered by Gorin and Lin in the 70-ies which lead to the notion of conformal module of conjugacy classes of braids. A discussion of this invariant and applications will be given. The applications concern the question which smooth objects involving braids are isotopic to the respective holomorphic objects. In particular, I will sketch a conceptional proof of a theorem of Gorin, Lin and Zjuzin. The theorem concerns reducibility of algebroid functions with empty discriminant set on annuli. The new proof uses a result of Penner on entropy and the relation between conformal module and entropy.
Monday April 1, 2013 in Swain West 220: Rob Kirby, Morse 2-functions and trisections of 4-manifolds; In joint work with David Gay, we show existence and uniqueness of Morse 2-functions (also known as broken Lefschetz fibrations), and then existence and uniqueness (up to stabilization) of trisections of 4-manifolds (analogous to Heegaard splittings of 3-manifolds).
Wednesday April 10, 2013 in Rawles Hall 104: Chuck Livingston, Splitting the 3-dimensional rational cobordism group
Wednesday May 1, 2013 in Rawles Hall 104: Y. Fukumoto, Furuta's 10/8 theorem
Monday May 20, 2013 in Rawles Hall 104: David Krcatovich, Knot 4-genus bounds from Heegaard Floer homology
Friday July 5, 2013 in RH 104: Paul Kirk, SU(2) representations of knot groups
Wednesday August 21, 2013 in Rawles Hall 104: Stefan Friedl, Splittings of knot groups; We will discuss in which ways knot groups can be written as HNN extensions. This is joint work with Dan Silver and Susan Williams
Wednesday September 4, 2013 in Rawles Hall 104: Taylor McNeill, A Filtration of the Torelli Group
Wednesday September 11, 2013 in Rawles Hall 104: Patrick Haggerty, Connected sum at infinity and 4-manifolds; In this talk, we investigate connected sum at infinity, an operation introduced by Gompf on non-compact manifolds. In defining connected sum at infinity, rays are chosen in each manifold involved, just as a choice of disc is made in each manifold involved in a connected sum. In connected sum, the choice of disc does not affect the resulting manifold, but in connected sum infinity, a different choice of rays could change the resulting manifold. Such a dependence has been demonstrated for 3-manifolds. We present an infinite family of smooth 4-manifolds for which the choice of rays affects the connected sum at infinity operation, and then generalize this family to produce higher dimensional examples. These examples answer affirmatively a conjecture of Siebenmann.
Wednesday September 18, 2013 in Rawles Hall 104: Harry Baik, Laminations on the circle and hyperbolic geometry; We will see that laminations can be used to study groups acting on the circle. In particular, it is possible to characterize Fuchsian groups in terms of topological invariant laminations. It is a geometrization result in the sense that you start with a topological dynamical data and hyperbolic geometry comes out of the picture. Main motivation of this program is Thurston's universal circle theory for tautly foliated 3-manifolds.
Wednesday September 25, 2013 in Rawles Hall 104: Matt Hogancamp, A quasi-local approach to link homology; There exist many categorifications of quantum link invariants, but as yet none of their “colored" versions are functorial under 4-dimensional link cobordisms. In this talk I propose a new categorification of the colored Jones polynomial which is likely to be functorial. The construction uses a new, quasi-idempotent chain complex which categories a multiple of the Jones-Wenzl projector.
Wednesday October 2, 2013 in Rawles Hall 104: Masaaki Suzuki, Meridional and non-meridional epimorphisms between knot groups; An epimorphism between knot groups is called meridional if a meridian is sent to a meridian. We determined whether there exists a meridional epimorphism between knot groups in case of the knots with up to 11 crossings. In this talk, we show some examples of non-meridional epimorphisms too.
Tuesday October 8, 2013 in Rawles Hall 104: Taehee Kim, Character varieties and twisted Alexander polynomials; For a knot, it is unknown if the twisted Alexander polynomials associated to SL(2,C)-representations of the knot group can detect fiberedness of the knot. In this talk, we will address this question from the viewpoint of the character variety of the knot group. This is joint work with Takahiro Kitayama and Takayuki Morifuji.
Wednesday October 9, 2013 in Rawles Hall 104: Vladimir Turaev, Brackets in loop algebras of manifolds; For a smooth oriented manifold M with non-empty boundary, we study the space Omega of loops in M based at a distinguished point of the boundary. Using the ideas of string topology of Chas-Sullivan, we define on the loop algebra H_*(Omega) a double bracket in the sense of Van den Bergh satisfying a version of the Jacobi identity. For dim(M)> 2, this double bracket induces Gerstenhaber brackets in the representation algebras associated with H_*(Omega). This extends to higher dimensions the classical Poisson brackets on the moduli space of representations of the fundamental groups of surfaces.
Wednesday October 16, 2013 in Rawles Hall 104: Chuck Livingston, Projective curves and their singularities via Heegaard-Floer Invariants
Wednesday October 23, 2013 in Rawles Hall 104: M. Mandell, The homotopy theory of cyclotomic spectra; In this talk, I'll explain what a cyclotomic spectrum is and what topological cyclic homology (TC) is. I'll discuss recent work with Blumberg on developing a homotopy theory for cyclotomic spectra and (if time allows) our verification of a conjecture of Kaledin on the representability of TC.
Tuesday November 5, 2013 in Rawles Hall 104: Mark Powell, Link Slicing and Casson Towers; We give new minimal conditions for embedding topological locally flat disks, and apply these results to show some new examples of slice links.
Wednesday November 6, 2013 in Rawles Hall 104: Irina Bobkova, A resolution of the spectrum E^{hS_2^1} at the prime 2.; In this talk we will introduce chromatic approach in stable homotopy theory: the homotopy of the p-local sphere spectrum S is determined by a family of localizations L_{K(n)}S with respect to Morava K-theories K(n). Then we will discuss some computations when p, n=2. Considerable information here can be derived from the action of the Morava stabilizer group on the Lubin-Tate theory. Goerss, Henn, Mahowald and Rezk have constructed a resolution of the K(2)-local sphere at the prime 3 which allows to simplify computations of pi_*L_{K(2)}S. We will discuss a generalization of their work to the prime 2 and construct a resolution of the spectrum E^{hS_2^1}, which is closely related to the K(2)-local sphere at the prime 2.
Thursday November 7, 2013 in Rawles Hall 104: Phillip Egger, On the existence/non-existence of a v_2^{32} self-map on A_1 at the prime 2; Using the Adams spectral sequence, Adams proved that the mod 2 Moore spectrum RP^2 admits a degree 8 v_1 self-map, which is therefore v_1^4. Behrens, Hill, Hopkins and Mahowald proved that the cofiber M(1,4) of this map admits a v_2^{32} self-map. We expect that the smash product of RP^2 with CP^2 admits a v_1 self-map and that the cofiber A_1 of this map also admits a v_2^{32} self-map. Purpose of the talk will be to explain everything in an elementary fashion from computational point of view and given hands on computation of examples.
Wednesday November 20, 2013 in Rawles Hall 104: Jeff Meier, Distinguishing topologically and smoothly doubly slice knots; The purpose of this talk is to show the existence of an infinite family of slice knots that are topologically doubly slice, but not smoothly doubly slice. First, I will present some sufficient conditions for a knot to be topologically doubly slice, and show how the knots in question satisfy these criteria. Then, I will outline how the correction terms coming from Heegaard Floer homology can be used to obstruct these knots from being smoothly doubly slice. Along the way, we will encounter some very interesting open questions related to the study of doubly slice knots.
Wednesday January 8, 2014 in Rawles Hall 104: Andrew Ranicki, Surgery on braids; The mathematical theory of braids was founded by Emil Artin: in fact, his 1947 Annals paper was written while he was in Bloomington. The representation theory of the braid groups is an important tool of low-dimensional topology. On the other hand, surgery theory has been an important tool of high-dimensional topology. The closure of a braid is a link. The object of this talk is to describe a surgery-theoretic approach to braids, building on the work of Gambaudo and Ghys (2005) and Bourrigan (2013) on the nonadditivity of the signatures of the closures of braids.
Wednesday February 19, 2014 in Rawles Hall 104: Heather Dye, The loop invariant; This talk is based on the paper _The Three Loop Isotopy and Framed Isotopy Invariants of Virtual Knots and Links" and is joint work with Micah Chrisman (Monmouth University). We introduce a finite type invariant of virtual knots called the three loop invariant. This invariant is a Gauss diagram formula where regions of the diagram are enhanced with the relative index of the arrows. We investigate its properties with respect to symmetries and connected sums. The invariant is an analog of the Grishanov-Vassiliev finite type invariant of degree two.
Wednesday February 26, 2014 in Rawles Hall 104: Burglind Juhl Joericke, Braid invariants and elliptic fiber bundles; Take a smooth fiber bundle over an open Riemann surface whose fibers are tori (elliptic fiber bundle). We consider the question whether the bundle is isotopic to a holomorphic bundle. When the open Riemann surface is an annulus the isotopy class of the bundle is determined by an isotopy class of self-homeomorphisms of a fiber (more precisely by its conjugacy class). The answer to the question is given in terms of a single invariant, the conformal module of the conjugacy class. It turns out that this invariant is inverse proportional to the entropy of the conjugacy class. We also give a result for elliptic fiber bundles over a torus with a hole.
Wednesday March 5, 2014 in Rawles Hall 104: Jae Choon Cha, A Topological Approach to Cheeger-Gromov Universal Bounds
Wednesday March 12, 2014 in Rawles Hall 104: Rafael Zentner, Computations in spectral sequences from Khovanov homology via partial functoriality; We use a weak functoriality result to draw conclusions about the Kronheimer-Mrowka spectral sequence from reduced Khovanov homology to reduced instanton knot Floer homology and the Ozsvath-Szabo spectral sequence from reduced Khovanov homology with Z/2 coefficients to the Heegaard-Floer-hat homology of the double branched cover. In particular, we obtain results for the first mentioned spectral sequence for the (4,5) torus knot, and for both spectral sequences for three-stranded pretzel knots.(joint with Andrew Lobb)
Wednesday March 26, 2014 in Rawles Hall 104: Noah Snyder, Topological field theories, homotopy fixed points, and algebraic structures; The cobordism hypothesis gives a classification of local topological field theories with values in some (algebraic) n-category. Part of this theorem is that there's an O(n) action on the space of framed TFTs and that homotopy fixed points for actions of certain subgroups of O(n) classify TFTs with other topological structures (spin, oriented, etc.). This gives a dictionary between topological structures and algebraic structures, which I will explain via some examples in this talk. In 3-dimensions this is part of joint work with Chris Douglas and Chris Schommer-Pries, though I will spend most of this talk on the well-known 1- and 2-dimensional situations.
Thursday April 3, 2014 in Rawles Hall 104: Matt Hedden, Floer homology and Fractional Dehn twists; There is a rational valued invariant of an automorphism of a surface with a single boundary component called the fractional Dehn twist coefficient. Roughly, it measures the twisting of the automorphism around the boundary. The fractional Dehn twist coefficient is related to the theory of taut foliations, essential laminations, and contact structures on 3-manifolds obtained by performing Dehn surgery on fibered knots. These connections arise by associating to a fibered knot in a 3-manifold the fractional Dehn twist coefficient of the monodromy of the fibration on its complement. I'll describe how the Heegaard Floer homology of a 3-manifold bounds the fractional Dehn twist coefficient of any of its fibered knots, and some consequences this has for contact structures on 3-manifolds. This is joint work with Tom Mark.
Wednesday April 9, 2014 in Rawles Hall 104: Paul Melvin, Stable isotopy in 4-manifolds; It is a well known principle in 4-dimensional topology that homeomorphic smooth simply-connected 4-manifolds become diffeomorphic after 'stabilizing' by connected summing with S^2 x S^2 sufficiently many times. In fact many explicit examples are known for which exactly one stabilization is required, and surprisingly, no examples have been shown to require more than one. Perhaps this exoticity dissolves quickly under stabilization. An analogous principle holds for 2-spheres embedded in a simply-connected 4-manifold, namely, any two that are topologically isotopic become smoothly isotopic after stabilizing the manifold sufficiently many times. Until now, however, bounds on the number of stabilizations needed have not been found for any examples. In this talk, we will describe families of topologically isotopic knots that are not smoothly isotopic, but become so after one stabilization. This is joint work with Auckly, Kim and Ruberman.
Wednesday April 16, 2014 in Rawles Hall 104: Will Kazez, Approximating C^0 foliations; Taut foliations and tight contact structures are important topological structures on 3-manifolds that are at opposite ends of the spectrum in the study of 2-plane distributions. One is integrable everywhere, the other is integrable nowhere. Eliashberg and Thurston have shown that they are closely related; in fact, every sufficiently smooth taut foliation can be perturbed to a tight contact structure. In joint work with Rachel Roberts, we show that the smoothness assumptions on the foliation can be removed. This allows the approximation theorem to be applied to a wide range of recent constructions of (non-smooth) foliations.
Tuesday April 22, 2014 in 104 Rawles Hall: Qayum Khan, Classification of free actions of Cp on S1xSn; Let p be an odd prime, and let n be a positive integer. We classify the set of equivariant homeomorphism classes of free C_p-actions on the product S^1 x S^n of spheres, up to indeterminacy bounded in p. The description is expressed in terms of number theory. The techniques are various applications of surgery theory and homotopy theory, and we perform a careful study of h-cobordisms. The p=2 case was completed by B Jahren and S Kwasik (2011). The new issues for the odd p case are the presence of nontrivial ideal class groups and a group of equivariant self-equivalences with quadratic growth in p. The latter is handled by the composition formula for structure groups of A Ranicki (2009).
Wednesday April 23, 2014 in Rawles Hall 104: Jonathan Hillman, Seifert fibered 2-knots; An n-knot manifold is the closed manifold obtained by elementary surgery on an n-knot K:S^n to S^{n+2}. (If n=1, we assume that the surgery is longitudinal, i.e., 0-framed.) Motivated by earlier work on geometries on 2-knot manifolds, we consider when such manifolds are Seifert fibred over 2-dimensional orbifolds. Consideration of the possible base orbifolds leads to a class of 2-knots with torsion-free solvable groups, hitherto overlooked, which completes the list of such groups. This is joint work with Jim Howie of Heriot-Watt University in Edinburgh.
Wednesday April 30, 2014 in Rawles Hall 104: Corrin Clarkson, Three-manifold mutations detected by Heegaard Floer homology; Given a self-diffeomorphism h of a closed, orientable surface S and an embedding f of S into a three-manifold M, we construct a mutant manifold N by cutting M along f(S) and regluing by h. We will consider whether there are any gluings such that for any embedding, the manifold and its mutant have isomorphic Heegaard Floer homology. In particular, we will demonstrate that if the gluing is not isotopic to the identity, then there exists an embedding of S into a three-manifold M such that the rank of the non-torsion summands of the Heegaard Floer homology of M differs from that of its mutant.
Wednesday August 27, 2014 in Rawles Hall 104: Elizabeth Housworth, Topology Faculty Meeting (tenure track only); "Small group" meeting with the chair. Tenure track faculty only.
Wednesday September 10, 2014 in Rawles Hall 104: Peter Lambert-Cole, Knot Floer homology, transverse knots and algorithmic computation; Knot Floer homology HFK(K) encodes much geometric data about knots in 3-manifolds and detects several classical invariants of knots, such as the 3-genus and fiberedness. However, efficient computation of this invariant has proved elusive. The first approach to combinatorial knot Floer homology, introduced by Manolescu, Ozsvath and Sarkar, computes HFK from a grid diagram of the knot. We develop an alternative approach to algorithmic computation of HFK(K), starting from a braid B whose closure is K. For many knots, the computational efficiency of our approach is significantly faster asymptotically and in practice. The algorithm is inspired by work of Baldwin, Vela-Vick and Vertesi on invariants of tranvserse knots in contact 3-manifolds and as a consequence, we can also compute the knot Floer invariant T(K) of transverse knots.
Wednesday October 1, 2014 in Rawles Hall 104: Prasit Bhattacharya, Higher Associativity of Moore Spectra; Practice talk for lectures at other universities.
Wednesday October 8, 2014 in Rawles Hall 104: Maxim Prasolov, Bypasses for rectangular diagrams; The central part of the talk is a criteria for a rectangular diagram to admit a simplification. This criteria is given in terms of Legendrian knots. We discuss several corollaries of this criteria: there are two types of simplifications which are mutually independent in a sense; Dynnikov's theorem about a monotonic simplification for the unknot is equivalent to the theorem of Eliashberg-Fraser about classification of topologically trivial Legendrian knots; a minimal rectangular diagram maximizes the Thurston-Bennequin number for the corresponding Legendrian links; a proof of Jones' conjecture about the invariance of the algebraic number of intersections of a minimal braid representing a fixed link type.
Thursday October 23, 2014 in Rawles Hall 104: Matt Hogancamp
Tuesday October 28, 2014 in Rawles Hall 104: Matt Hogancamp, exact triangle recognition lemma
Wednesday November 5, 2014 in Rawles Hall 104: Alexander Zupan, Bridge presentations for knotted surfaces in the 4-sphere; Recently, Gay and Kirby introduced a 4-dimensional analogue of a Heegaard splitting called a trisection. We adapt their approach to show that every knotted surface in the 4-sphere admits a bridge trisection; namely, there is a decomposition of the 4-sphere into three 4-balls which splits the surface into a collection of boundary parallel disks. This may be viewed as the 4-dimensional version of a bridge presentations for a knot in the 3-sphere, and as such, a bridge trisection has two complexity parameters akin to the bridge number of a knot. We will discuss a classification of bridge trisections with relatively low complexity. This talk is based on joint work with David Gay and Jeffrey Meier.
Tuesday November 11, 2014 in Rawles Hall 104: Andrei Vesnin, Hyperelliptic involutions of hyperbolic 3-manifolds; An involution of an n-dimensional manifold is said to be hyperelliptic if the quotient space by its action is homeomorphic to the n-sphere. A manifold having such an involution is said to be a hyperelliptic manifold. We will discuss two aspects of the theory of 3-dimensional hyperbolic hyperelliptic manifolds: (1) construction of hyperelliptic 3-manifolds uniformized by subgroups of Coxeter groups, and (2) number of hyperelliptic involutions of a hyperbolic manifold.
Wednesday November 12, 2014 in Rawles Hall 104: Evgeny Fominykh, On complexity of 3-manifolds; The most useful approach to a classication of 3-manifolds is the complexity theory introduced by S. Matveev. Exact values of complexity are known for few series of 3-manifold only. We present the results (joint with A. Vesnin) on the complexity of two infinite series of hyperbolic 3-manifolds with boundary.
Tuesday November 18, 2014 in Rawles Hall 104: Hans Boden, A classical approach to virtual knots; Given a virtual knot K, there are various groups naturally associated to K. We will discuss the invariants obtained from the elementary ideal theory of the virtual knot group VG_K, and we will show how to via Wirtinger_s method, and We will discuss groups that one can associate to virtual knots and invariants derived from these groups using elementary ideal theory. For instance, associated to the k=0 ideal is a polynomial H_K(s,t,q) in three variables, and we show how the q-width of H_K gives information about the virtual crossing number of K. The polynomial H_K(s,t,q) satisfies a skein formula, and one can define a twisted polynomial invariant of virtual knots for any representation from VG_K to GL_n(R).
Wednesday December 3, 2014 in Rawles Hall 104: Michael Mandell, Homotopy groups of A(*); Waldhausen's algebraic K-theory of spaces is related to the differential topology of high dimensional manifolds. This talk will describe joint work with Andrew Blumberg where we build on work of Rognes to compute the homotopy groups of A(*) in terms of the homotopy groups of the sphere spectrum, the homotopy groups of $\mathbb{C}P^\infty_{-1}$, and the homotopy groups of $K(\mathbb{Z})$.
Wednesday December 10, 2014 in Rawles Hall 104: Emily Riehl, The formal category theory of (∞,1)-categories; Theorems in abstract homotopy are occasionally stated in the language of _(∞,1)-categories,_ which can be modeled by quasi-categories ("∞-categories_) or by complete Segal spaces. In pioneering work of Joyal and Lurie, ordinary category theory has been extended to quasi-categories, making this model particularly convenient for proving abstract theorems. In joint work with Verity, we show that the category theory of quasi-categories is 2-categorical: new definitions of the basic notions _ (co)limits, adjunctions, fibrations _ equivalent to the Joyal-Lurie definitions, can be encoded in the homotopy 2-category of quasi-categories. From this new vantage point, the basic categorical theorems become easier to prove; the 2-categorical proofs in the homotopy 2-category mimic the classical ones in the 2-category CAT. Importantly, complete Segal spaces and more general categories of Rezk objects have an analogous homotopy 2-category, so this _formal_ approach to the development of the category theory of quasi-categories immediately extends to other models of higher homotopical categories.
Wednesday February 18, 2015 in Rawles Hall 104: David Gay, Trisections of 4-manifolds: the basics, some news, and speculations; Trisections are to 4-manifolds as Heegaard splittings are to 3-manifolds. This talk will serve as an introduction to the foundational results developed by myself and Kirby, will survey a few recent developments by others, and will hopefully devote a good amount of time to speculation (the analogy with Heegaard splittings poses far more questions than it answers, and most of these questions seem very interesting).
Wednesday February 25, 2015 in Rawles Hall 104: Jim Davis, Almost flat manifolds with cyclic holonomy are boundaries
Tuesday March 3, 2015 in Rawles Hall 104: Marithania Silvero, Kauffman's conjecture on alternative and pseudoalternating links; In 1983 Louis Kauffman introduced the family of alternative links, as a generalization of alternating links. It is known that alternative links are pseudoalternating. Kauffman conjectured the converse. In this talk we show that both families are equal in the particular case of knots of genus one. However, Kauffman's Conjecture does not hold in general, as we also show by finding two counterexamples. In the way we will deal with the intermediate family of homogeneous links, introduced by Peter Cromwell; the techniques used here allow us to give an explicit characterization of homogeneous links of genus 1.
Wednesday March 4, 2015 in Rawles Hall 104: Mark Bell, Conjugacy in the Mapping Class Group
Wednesday April 1, 2015 in Rawles Hall 104: Shida Wang, Connected Sums of Low Genus Knots in the Concordance Group; Fix any positive integer n. It is an interesting question that whether a knot is concordant to a connected sum of knots of genus not greater than n. For example, the Alexander polynomial can show the (p,q)-torus knot with q>2n+1 is not concordant to such a connected sum. We will explore more examples and give an obstruction using the invariants of epsilon equivalence classes defined by J. Hom. As a consequence, we will prove that there is an infinitely generated free summand lying in the group of topologically slice knots not concordant to connected sums of knots of genus not greater than n. If time permits, we will give an alternative proof of this fact using the Upsilon invariant recently defined by Ozsvath-Stipsicz-Szabo.
Wednesday April 8, 2015 in Rawles Hall 104: Danny Ruberman, Absolutely exotic 4-manifolds; Joint work with Selman Akbulut. We show the existence of exotic smooth structures on contractible 4-manifolds. These structures are absolute, in the sense that they do not depend on a specific marking of the boundary. This is in contrast to the phenomenon of corks, which are exotic relative to an automorphism of their boundaries. The technique is to modify a relatively exotic manifold to give an exotic one for which we have a good understanding of the automorphism group of the boundary.
Wednesday April 15, 2015 in Rawles Hall 104: Anna Marie Bohman, Constructing equivariant spectra; Equivariant spectra determine cohomology theories that incorporate a group action on spaces. Such spectra are increasingly important in algebraic topology but can be difficult to understand or construct. In recent work, Angelica Osorno and I have created a machine for building such spectra out of purely algebraic data based on symmetric monoidal categories. Our method is philosophically similar to classical work of Segal on building nonequivariant spectra. In this talk I will discuss an extension of our work to the more general world of Waldhausen categories. Our new construction is more flexible and is designed to be suitable for equivariant algebraic K-theory constructions.
Thursday April 16, 2015 in Rawles Hall 104: Kristen Hendricks, Involutive Heegaard Floer Homology; In joint work with C. Manolescu, we use the conjugation symmetry on the Heegaard Floer complexes to define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z_4-equivariant Seiberg-Witten Floer homology. From this we obtain two new invariants of homology cobordism, explicitly computable for surgeries on L-space knots and quasi-alternating knots, and two new concordance invariants of knots, one of which (unlike other invariants arising from Heegaard Floer homology) detects non-sliceness of the figure-eight knot.
Wednesday April 22, 2015 in Rawles Hall 104: Andrew Blumberg, Twists of K-theory and TMF
Tuesday April 28, 2015 in Rawles Hall 104: Amnon Neeman, Separable monoids come from etale covers; Given a monoidal category (a category with a tensor product), it is possible to define what it means for an object to be an "separable monoid". We will recall the definition. In the category of modules over a commutative ring the concept is classical and has been studied extensively, with literature going back to the 1960s. In the late 1980s and 90s Goodwillie looked at this construction in the category of modules over an E^infty ring spectrum. Recently Balmer and some collaborators started the study of separable monoids in any tensor triangulated category; we will recall some of the recent theorems. The main new result I will present says [among other things] that, in the derived category of modules over a noetherian commutative ring, any commutative separable monoid must come from an etale extension of the ring. We will explain this precisely - for the precise statement one cannot avoid generalizing from rings to schemes, the general statement is global and asserts that there is a simple description of all the separable monoids in the derived category of quasicoherent sheaves on an arbitrary noetherian scheme. This result raises many questions, the most basic being what remains true if we drop the noetherian hypothesis, or drop the hypothesis that the monoid is commutative. I have no idea - I will try to explain where the techniques of the proof depend essentially on the hypotheses, even though these hypotheses might well be redundant.
Wednesday June 17, 2015 in Rawles Hall 104: Diarmuid Crowley, New invariants in G_2 topology; 7-manifolds with G_2 holonomy are very difficult to construct, whereas every spin 7-manifold admits a G_2 structure. In this talk I report on a joint project with Sebastian Goette and Johannes Nordstr_m which develops new invariants for spin 7-manifolds, G_2 structures and G_2 manifolds. The new invariants are used to show the existence of homeomorphic but not diffeomorphic G_2 manifolds and that the moduli space of G_2 manifolds is in general disconnected.
Wednesday September 2, 2015 in Rawles Hall 104: Agnes Beaudry, Gluing data in chromatic homotopy theory; Understanding the stable homotopy groups of spheres is one of the great challenges of algebraic topology. They form a ring which, despite its simple definition, carries an amazing amount of structure. A famous theorem of Hopkins and Ravenel states that it is filtered by simpler rings called the chromatic layers. This point of view organizes the homotopy groups into periodic families and reveals patterns. There are many structural conjectures about the chromatic filtration. I will talk about one of these conjectures, the chromatic splitting conjecture, which concerns the gluing data between the different layers of the chromatic filtration.
Wednesday September 30, 2015 in Rawles Hall 104: Gwenael Massuyeau, Poisson brackets in representation algebras generalizing the Atiyah-Bott-Goldman structures; Goldman showed how to use intersection of curves in order to compute the Poisson bracket induced by the Atiyah-Bott symplectic structure on the moduli space of representations of a surface group in a Lie group. We will present some algebraic generalizations of Goldman's formulas where the surface group is replaced by a cocommutative Hopf algebra and the Lie group is replaced by a group scheme. Next, we will explain how to apply this general framework to higher-dimensional manifolds using intersection of cycles in loop spaces.
Wednesday October 7, 2015 in Rawles Hall 104: Danielle O'Donnol, Minimal embedding of Legendrian graphs and the total Thurston-Bennequin number.; A Legendrian graph is a graph embedded in such a way that its edges are everywhere tangent to the contact structure. We extend the classical invariants Thurston-Bennequin number and rotation number to Legendrian graphs. A minimal embedding is one where all of the minimal length cycles are unknots with $tb=-1$. I will talk about some new results about a new invariant, the total Thurston-Bennequin number, and discuss the consequences of these results for embeddings of $K_4$ and $K_{3,3}$.
Wednesday October 14, 2015 in Rawles Hall 104: Emmanuel Dror Farjoun, A new look at Postnikov sections; (with Wojtek C., Ramon F., Jerome S.) It turns out that for a nilpotent space X, the Postnikov sections P_nX have a strong "cellular" relation to X itself: For example if X is acyclic with respect to any homology theory so is P_nX. This allows strong statements about the value of certain functors F(Y)—–>Y (e.g higher covers) on these sections. The main tool is a modified Bousfield Kan completion tower which is easier to understand in certain respects.
Wednesday October 21, 2015 in Rawles Hall 104: Sarah Yeakel, A chain rule for Goodwillie calculus; In the homotopy calculus of functors, Goodwillie defines a way of assigning a Taylor tower of polynomial functors to a homotopy functor and identifies the homogeneous pieces as being classified by certain spectra, called the derivatives of the functor. Michael Ching showed that the derivatives of the identity functor of spaces form an operad, and Arone and Ching developed a chain rule for composable functors. We will review these results and show that through a slight modification to the definition of derivative, we have found a more straight forward chain rule for endofunctors of spaces.
Tuesday October 27, 2015 in SE 140: Inna Zakharevich, Spectral sequences associated to the Grothendieck ring of varieties; The Grothendieck ring of varieties is defined to be the free abelian group generated by varieties, modulo the relation that for a closed embedding $Y to X$, [X] = [Y]+[XY]. This ring comes with a filtration by dimension of variety, but the associated graded of this filtration is very difficult to compute, as higher-dimensional varieties can induce relations between lower-dimensional ones. To get around this difficulty we use algebraic K-theory to construct a spectral version of this ring, which will also have a dimension filtration. We show how to compute the associated graded of this filtration and discuss the resulting spectral sequence.
Wednesday November 4, 2015 in Rawles Hall 104: Palanivel Manoharan, Geometry of manifolds modeled on Hilbert modules
Wednesday November 11, 2015 in Rawles Hall 104: Cary Malkiewich, Duality in topological Hochschild homology (THH); Topological Hochschild homology (THH) is a beautiful and computable invariant of rings and ring spectra. In this talk, I will focus on the ring spectrum DX, and discuss a few different aspects of THH(DX). For example, it splits when X is a suspension, and we can use this for computations in topological cyclic homology. I will also recall the "Atiyah duality" between THH(DX) and the free loop space LX, and prove that this duality preserves the genuine S^1-structure. This uses the new "norm" model of THH, and a surprising technical result about orthogonal G-spectra. If there is time, I will apply these tools once more and describe an enrichment of the character map from representation theory.
Wednesday November 18, 2015 in Rawles Hall 104: Ralph Kaufmann, Around Feynman categories; We recall the notion of Feynman categories and then go on to discuss examples and constructions. We then turn to newer developments like Hopf algebras arising out of them, which are known from physics and number theory as well as newer decorated version.
Wednesday January 6, 2016 in Rawles Hall 104: Mark Bell
Wednesday January 20, 2016 in Rawles Hall 104: Peter Lambert-Cole, Classifying planar Legendrian graphs; A planar spatial graph in S^3 is a 1-complex embedded on a smooth sphere. The graph is Legendrian if it is everywhere tangent to the standard contact structure on the 3-sphere. Using convex surface theory, we solve several classification problems for planar Legendrian graphs, including Legendrian simplicity, the Legendrian mirror problem, and stabilization equivalence. This is joint with Danielle O'Donnol (Indiana).
Wednesday February 10, 2016 in Rawles Hall 104: Jim Davis, Topology of L^2-acyclic manifolds; A manifold is L^2-acyclic if its L^2-betti numbers vanish. An example is given by a manifold whose fundamental group is infinite cyclic where every homology class of its universal cover is annihilated by a nonzero Laurent polynomial in the deck transformation. After giving examples, we will discuss general fundamental groups and give analytic properties of such manifolds. This is likely to be the first in a series of talks.
Tuesday February 16, 2016 in Rawles Hall 104: Chuck Livingston, Survey of knot concordance
Wednesday March 9, 2016 in Rawles Hall 104: Birgit Richter, Ramified extensions of structured ring spectra; Well behaved multiplicative cohomology theories can be represented by ring spectra. Topological complex K-theory for instance is represented by a periodic ring spectrum $KU$ whose homotopy groups are a Laurent polynomial ring on a generator in degree two over the integers. Algebraic K-theory of ring spectra carries geometric information in some cases, but here the ring spectra involved are often connective, i. e., have homotopy groups concentrated in non-negative degrees, such as the sphere spectrum or connective complex K-theory. Even for Galois extensions of ring spectra, such as the complexification map from real to complex periodic K-theory, the associated map on connective covers behaves like a ramified extension of number rings. We discuss examples and present detection methods for ramification.
Wednesday March 23, 2016 in Rawles Hall 104: Nerses Aramian, The Integration Pairing and Extended Topological Field Theories; Given an hermitian line bundle with an hermitian connection on a manifold there is a straightforward way of producing 1-dimensional topological field theory over the manifold. One can generalize this procedure by replacing complex line bundles with $n$-gerbes bound by $U(1)$, and replacing connections with appropriate connective structures. The resulting topological field theories are $n$-dimensional and extended. In this talk, we provide an explicit construction of topological field theories using the higher categorical framework of Jacob Lurie, and show that the construction establishes equivalence between the smooth Deligne complex and the framed topological field theories.
Thursday March 24, 2016 in Rawles Hall 104: Qayum Khan, Stable existence of incompressible 3-manifolds in 4-manifolds; Given an injective amalgam at the level of fundamental groups and a specific 3-manifold, does there exist a corresponding geometric-topological decomposition of a given 4-manifold, in a stable sense? We find an algebraic-topological splitting criterion and express it via the associated orientation classes and universal covers. Along the way, we generalize the Lickorish–Wallace theorem to regular covers. This is joint work-in-progress with my PhD student, Gerrit Smith.
Thursday March 31, 2016 in RH104: Alexandra Kjuchukova, Branched covers of four-manifolds and related questions in knot theory; Let $f:Y\to X$ be a branched cover of closed oriented four-manifolds with an oriented, topologically flatly embedded branching set $B$. By an old formula of Viro, the signature of $Y$ can be computed in terms of data about $X$, $B$ and $[B]\in H_2(X; Z)$. I give a generalization of this formula to the case where $B$ is embedded in $X$ with an isolated cone singularity of type $K$ (here, $K$ is a knot in $S^3$) and $f$ is an irregular dihedral cover. The defect to signature of $Y$ resulting from the presence of a singularity can be expressed in terms of classical invariants of $K$. I discuss several natural questions that follow: which knot types can arise as singularities? what is the range of signatures of dihedral covers of $S^4$? can all possible signatures over a given base be realized using two-bridge knots? I'll report on some on-going work on these questions.
Wednesday April 6, 2016 in SE 140: Chris Herald, Traceless SU(2) character varieties of tangles and their symplectic properties; In this talk, we discuss the character variety of traceless fundamental group representations to SU(2) of a tangle complement. Here, traceless means meridians of all n strands are sent to traceless SU(2) elements. There is a restriction map from the traceless character variety of an n-strand tangle to the traceless character variety of the 2n-punctured sphere. Weâll describe the structure of these traceless character varieties and their symplectic properties, and outline a program to define a Lagrangian Floer homology counterpart to Kronheimer Mrowka reduced singular instanton homology, which uses these traceless character varieties. This is work by various combinations of Paul Kirk, Matt Hedden, and Matt Hogancamp and myself
Tuesday April 12, 2016 in Rawles Hall 104: Allan Edmonds, Geography of Aspherical 4-Manifolds; I will survey what is know about the pairs (signature, Euler characteristic) of closed, oriented, aspherical 4-manifolds and sketch a construction of new examples.
Thursday April 28, 2016 in Rawls Hall 104: Arunima Ray, Shake slice and shake concordant knots.; If K is a knot in the 3-sphere, then the 4-manifold W_K obtained by adding a single 2-handle to B^4 along K with zero framing has H_2 isomorphic to Z. If a generator of H_2(W_K) can be represented by an embedded sphere, K is said to be shake-slice. Any slice knot is shake-slice, but the converse is unknown. We define a relative version of this concept, known as shake concordance, and construct infinite families of knots that are pairwise shake-concordant but not concordant. We show that the concordance invariants tau, s, and slice genus are not invariants of shake-concordance. We also give a characterization of shake-concordant and shake-slice knots in terms of concordance. (This is joint work with Tim Cochran.)
Tuesday June 14, 2016 in Rawles 104: Brandy Doleshal, The search for distinct primitive/Seifert knots that admit the same Seifert fibered space surgery; Given a knot in a 3-manifold, one can obtain a new 3-manifold by removing a neighborhood of the knot and refilling the empty space created. This process, called Dehn surgery, is a great way of understanding 3-manifolds, which is a major goal of low dimensional topology. Primitive/Seifert knots are a class of knots that lie on the genus two surface in a way that guarantees that they admit a Seifert fibered space Dehn surgery along a particular slope. We would like to answer the following question: is it possible for two distinct primitive/Seifert knots to have the same Seifert fibered space surgery after surgery at the surface slope? In this talk, we will discuss some necessary definitions, the origin of this problem and the things that went wrong but turned out alright. This is joint work with Jeff Meier.
Wednesday September 7, 2016 in Rawles Hall 104: Carmen Rovi, Hirzebruch $\chi_y$-genera and the signature modulo $8$ of fibre bundles; In this talk we shall be concerned with the residues modulo $4$ and modulo $8$ of the signature $\sigma(M)$ in $\mathbb{Z}$ of an oriented 4k-dimensional geometric Poincare complex $M$. The precise relation between the signature modulo $8$ and the Brown-Kervaire invariant was worked out by Morita. We shall discuss how the relation between these invariants and the Arf invariant can be applied to the study of the signature modulo $8$ of a fibration. In particular it had been proved by Meyer in 1973 that a surface bundle has signature divisible by $4$. This was generalized to higher dimensions by Hambleton, Korzeniewski and Ranicki in 2007. I will explain two results from my thesis concerning the signature modulo $8$ of a fibration: firstly under what conditions can we guarantee divisibility of the signature by 8 and secondly what invariant detects non-divisibility by 8 in general. I will also talk about a joint project with S. Yokura where we investigate how multiplicativity properties of the signature modulo $8$ also hold for certain Hirzebruch $\chi_y$-genera.
Wednesday September 14, 2016 in Rawles Hall 104: Anthony Conway, Splitting numbers and signatures; The splitting number of a link is the minimal number of crossing changes between different components required to convert it into a split union of its components. Since 2012, this invariant has been studied using various tools such as Khovanov homology, covering link calculus, the Alexander polynomial and Heegaard-Floer homology. After briefly reviewing some of these methods, we will show how (multivariable) signatures give strong lower bounds on the splitting number. This is a joint work with David Cimasoni and Kleopatra Zacharova.
Wednesday September 21, 2016 in Rawles Hall 104: Henry Horton, A functorial symplectic instanton homology via traceless character varieties
Wednesday October 5, 2016 in Rawles Hall 104: Peter Lambert-Cole, Conway Mutation and knot Floer homology; Mutant knots are notoriously hard to distinguish. Many, but not all, knot invariants take the same value on mutant pairs. Khovanov homology with coefficients in Z/2Z is known to be mutation-invariant, while the bigraded knot Floer homology groups can distinguish mutants such as the famous Kinoshita-Terasaka and Conway pair. However, Baldwin and Levine conjectured that delta-graded knot Floer homology, a singly-graded reduction of the full invariant, is preserved by mutation. In this talk, I will give a new proof that Khovanov homology mod 2 is mutation-invariant. The same strategy can be applied to delta-graded knot Floer homology and proves theBaldwin-Levine conjecture for mutations on a large class of tangles.
Wednesday October 26, 2016 in Rawles Hall 104: Michael Mandell, The algebraic K-theory of the sphere spectrum; We've learned a lot about the algebraic K-theory of the sphere spectrum since I spoke on this topic 2 years ago. I'll say some things about this.
Wednesday November 2, 2016 in Rawles Hall 104: Gabe Angelini-Knoll, Red-Shift Phenomena in Iterated Algebraic K-theory of Finite Fields; In the early 2000âs, Ausoni and Rognes initiated a program to study the arithmetic of structured ring spectra. In particular, they conjectured that algebraic K-theory should take ring spectra of chromatic telescopic complexity n to ring spectra of chromatic telescopic complexity n + 1, generalizing the number theoretic Quillen-Lichtenbaum conjecture. To verify this conjecture, it is necessary to do computations. However, only a few algebraic K-theory computations of ring spectra which are not Eilenberg-Mac Lane, have been completed. The iterated algebraic K-theory of a finite field of order q, where q is a prime power generator of the p-adic units, is of special interest because, after p- completion algebraic K-theory of these finite fields is equivalent to the connective cover of the K(1)-local sphere, when p is odd. This computation therefore fits into an old program of Waldhausenâs, where he suggested computing algebraic K-theory of suitable connective covers of certain localizations of the sphere spectrum.In my talk, I will describe a method for computing topological Hochschild homology, a linear approximation to algebraic K-theory, using a multiplicative Whitehead tower of a connective commutative ring spectrum as a filtration. I will also show how certain height two classes in the homotopy groups of the Smith-Toda complex V(1) map nontrivially to the circle homotopy fixed points of mod (p,v_1) THH of p-complete algebraic K-theory of fields of order q. As a consequence, I can show that the Î²-family, a periodic family of height two in the homotopy groups of spheres, is detected in mod (p,v_1)-homotopy of the iterated algebraic K-theory a finite field of order q. I will also provide some evidence indicating that it may be infinitely generated as a P(v_2)-module, which would imply that iterated algebraic K-theory of finite fields does not have telescopic complexity two. This means that the red-shift conjecture may need to be reformulated for it to hold in this case, even though it certainly holds in spirit.
Thursday November 3, 2016 in Rawles Hall 104: Gabriel C. Drummond-Cole, Homotopy probability theory on a Riemannian manifold; Homotopy probability theory is a homological enrichment of algebraic probability theory, a toy model for the algebra of observables in a quantum field theory. I will introduce the basics of the theory and use it to describe a reformulation of fluid flow equations on a compact Riemannian manifold.
Monday November 7, 2016 in Swain West 218: Dror Bar-Natan, A Poly-Time Knot Polynomial Via Solvable Approximation; I will construct the first poly-time-computable knot polynomial since Alexander's (1928) by using some new commutator-calculus techniques and a Lie algebra $g_1$ which is at the same time solvable and an approximation of the simple Lie algebra $sl_2$
Monday November 7, 2016 in Swain West 218: Dror Bar-Natan, (cont) A Poly-Time Knot Polynomial Via Solvable Approximation; I will construct the first poly-time-computable knot polynomial since Alexander's (1928) by using some new commutator-calculus techniques and a Lie algebra $g_1$ which is at the same time solvable and an approximation of the simple Lie algebra $sl_2$
Wednesday November 30, 2016 in Rawles Hall 104: Wai-Kit Yeung, Homotopy braid closure and knot contact homology; In this talk, I will present a universal construction, called homotopy braid closure, that produces invariants of links in R^3 starting with a braid group action on objects of a (model) category C. Applying this construction to the natural action of the braid group B_n on the category of perverse sheaves on the two-dimensional disk with singularities at n marked points, we obtain a differential graded (DG) category that gives knot contact homology in the sense of L. Ng. As an application, we show that the category of finite-dimensional modules over the 0-th homology of this DG category is equivalent to the category of perverse sheaves on R^3 with singularities at most along the link. [This is joint work with Yu. Berest and A. Eshmatov.]
Wednesday December 7, 2016 in Swain West 220: Wenzhao Chen, Heegaard Floer inviants of cable knots; The Upsilon invariant is a knot concordance invariant introduced by Ozsvath, Stipsicz, and Szabo in 2014, using knot Floer homology. It can be viewed as powerful generalization of an earlier known concordance invariant, the Ozsvath-Szabo tau-invariant. In this talk, we will discuss how the Upsilon invariant behaves correspondingly under the cabling operation on knots. I will present a partial answer, which is an inequality relating the Upsilon invariant of a knot and that of its cable. This is a natural generalization of results of Hedden and Van Cott on the tau-invariant of cable knots. As an application, I will also show how one can use this inequality to easily see the existence of an infinite-rank summand of topologically slice knots.
Wednesday March 1, 2017 in Rawles Hall 104: Birgit Richter, Inhabitants of interesting subsets of the Bousfield lattice; Bousfield introduced an equivalence relation on the stable homotopy category that sorts spectra according to their acyclics. The set of Bousfield classes has some important subsets such as the distributive lattice, DL, of all classes which are smash idempotent and the complete Boolean algebra, cBA, cBA of closed classes. Hovey and Palmieri stated in 1999 that no explicit example of a class in DL \ cBA is known. In joint work with Brooke-Taylor and Loewe we provide uncountably many examples of spectra that are in DL but not in cBA; in particular, for every prime p, the Bousfield class of the Eilenberg-MacLane spectrum of Z/pZ is in DL but not in cBA.
Wednesday March 29, 2017: Matthias Nagel (UQEM)
Thursday March 30, 2017 in Rawles Hall 104: Inbar Klang, Factorization homology and topological Hochschild cohomology of Thom spectra; By a theorem of Lewis, the Thom spectrum of an n-fold loop map to BO is an E_n-ring spectrum. I will discuss a project studying the factorization homology and the higher topological Hochschild cohomology of such Thom spectra, and talk about some applications, such as computations, and a duality between higher topological Hochschild homology and cohomology of such Thom spectra. This talk will include an exposition to factorization homology and higher Hochschild cohomology.
Wednesday April 5, 2017 in RH 104: Kun Wang, The L-theoretic Farrell-Jones conjecture under group extensions and its applications; In this talk, I will introduce what the Farrell-Jones conjecture is and present a general result on its L-theoretic version under group extensions. I will then discuss its applications to the s-cobordism and h-cobordism Borel conjectures and to the Novikov conjecture.
Wednesday April 12, 2017 in RH 104: Peter Feller, "Concordance with a view toward embedding problems in affine algebraic geometry"; Abstract: We first discuss classical questions about polynomial embeddings of the complex line C into complex spaces such as C^m and affine algebraic groups. Next, we consider torus knots and discuss questions related to their non-sliceness motivated by singularity theory. Finally, we use a knot theory perspective to indicate proofs for the embedding questions discussed first.
Wednesday July 26, 2017 in RH 104: Matthias Nagel, Milnor's triple linking numbers and C-complexes; We relate fillability of two link exteriors, and the question when two links admit equivalent C-complexes to (a refinement) of Milnor's triple linking numbers. This extends a theorem of Davis-Roth to include also links with nonvanishing linking numbers. This is joint work with C. Davis, P. Orson and M. Powell.
Monday August 7, 2017 in SE 140: Midwest Topology Summer School: Teena Gerhardt, An Overview of Computations in Algebraic K-theory Via Trace Methods I
Monday August 7, 2017 in SE 140: Midwest Topology Summer School: Michael Mandell, Introduction to Equivariant Stable Homotopy Theory
Monday August 7, 2017 in SE 140: Midwest Topology Summer School: Teena Gerhardt, An Overview of Computations in Algebraic K-theory Via Trace Methods II
Monday August 7, 2017 in SE 140: Midwest Topology Summer School: Thomas Nikolaus, On Topological Cyclic Homology and Cyclotomic Spectra: THH and the Frobenius
Tuesday August 8, 2017 in SE 140: Midwest Topology Summer School: Lars Hesselholt, Topological Hochschild Homology and Arithmetic: Bökstedt Periodicity
Tuesday August 8, 2017 in SE 140: Midwest Topology Summer School: Gijs Heuts, Tate Diagonals
Tuesday August 8, 2017 in SE 140: Midwest Topology Summer School: Thomas Nikolaus, On Topological Cyclic Homology and Cyclotomic Spectra: Cyclotomic Spectra and TC
Tuesday August 8, 2017 in SE 140: Midwest Topology Summer School: Teena Gerhardt, An Overview of Computations in Algebraic K-theory Via Trace Methods III
Wednesday August 9, 2017 in SE 140: Midwest Topology Summer School: Lars Hesselholt, Topological Hochschild Homology and Arithmetic: Smooth Algebras
Wednesday August 9, 2017 in SE 140: Midwest Topology Summer School: Thomas Nikolaus, On Topological Cyclic Homology and Cyclotomic Spectra: Examples
Wednesday August 9, 2017 in SE 140: Midwest Topology Summer School: Aaron Royer and Sarah Yeakel, The Dundas McCarthy Theorem
Wednesday August 9, 2017 in SE 140: Midwest Topology Summer School: Saul Glasman, Symmetric Power Structures on Algebraic K-Theory
Thursday August 10, 2017 in SE 140: Midwest Topology Summer School: Lars Hesselhot, Topological Hochschild Homology and Arithmetic: De Rham Cohomology over S
Thursday August 10, 2017 in SE 140: Midwest Topology Summer School: Michael Mandell, Introduction to Non-commutative Derived Algebraic Geometry
Thursday August 10, 2017 in SE 140: Midwest Topology Summer School: Aaron Royer and Sarah Yeakel, The Dundas McCarthy Theorem
Thursday August 10, 2017 in SE 140: Midwest Topology Summer School: Nick Rozenblyum, A Geometric Approach to the Cyclotomic Trace
Friday August 11, 2017 in SE 140: Midwest Topology Summer School: Aaron Royer and Sarah Yeakel, The Dundas McCarthy Theorem
Friday August 11, 2017 in SE 140: Midwest Topology Summer School: Michael Mandell, The Strong Künneth Theorem for TP
Wednesday September 6, 2017 in RH 104: Guillem Cazzasus, "Towards extended Floer field theories"; Abstract: Donaldson polynomials are powerful invariants associated to smooth four-manifolds. The introduction by Floer of Instanton homology groups, associated to some 3-manifolds, allowed to define analogs of such polynomials for (some) four-manifolds with boundary, that have a structure similar with a TQFT.
Wehrheim and Woodward developed a framework called "Floer field theory" which, according to the Atiyah-Floer conjecture, should permit to recover Donaldson invariants from a 2-functor from the 2-category Cob_{2+1+1} to a 2-category Symp they defined, which is an enrichment of Weinstein's symplectic category.
I will describe a framework that should permit to extend such a 2-functor to lower dimensions. This framework should permit to define new invariants in Manolescu and Woodward's symplectic instanton homology (sutured theory, equivariant version). This is work in progress.
Wednesday September 13, 2017 in RH104: Kenji Kozai, "Hyperbolic Structures from Sol"; Abstract: Thurston's hyperbolization theorem states that a three-manifold fibering over a circle admits a hyperbolic structure if and only if the monodromy is a pseudo-Anosov diffeomorphism of a hyperbolic surface. Although his construction does not give an explicit way of describing the geometric structure, a pseudo-Anosov diffeomorphism has a "stretch-squeeze" dynamic that can be encoded in combinatorial objects called train tracks. This leads to a singular Sol metric on the manifold. We show that the singular Sol structure on the manifold can be locally deformed to cone hyperbolic structures by viewing the eight Thurston geometries as specializations of real projective geometry.
Wednesday September 20, 2017 in RH104: Samantha Allen, "The nonorientable four-genus of knots"; The 4–genus of a knot K is the minimal genus of a surface in B^4 whose boundary is K. Similarly, we can define the nonorientable 4–genus of a knot K as the minimal “nonorientable genus” of a surface in B^4 whose boundary is K. Finding the nonorientable 4–genus of a knot can be quite intractable; existing methods exploit the relationship between nonorientable genus and normal Euler number of the nonorientable surface. In this talk, I will give an overview of the interplay between the nonorientable genus and normal Euler number of nonorientable surfaces in B^4. I will define both of these invariants and discuss their computation for closed surfaces and then for surfaces with boundary a knot. In particular, when fixing a knot K, we can ask what pairs of nonorientable genus and normal Euler number are realizable for a surface whose boundary is K. We will see that both classical invariants and Heegaard–Floer invariants can be used towards answering this question.
Wednesday October 11, 2017 in RH 104: Pat Gilmer, "On the Kauffman bracket skein module of the 3-torus"; Carrega has shown that the Kauffman bracket skein module of the 3-torus over the field of rational functions in the variable A can be generated by 9 skein elements. We show this set of generators is linearly independent.
Thursday October 26, 2017 in RH104: Alejandro Adem, (University of British Columbia) "Homotopy Group Actions and Group Cohomology"; Abstract: Understanding the symmetries of a topological space is a classical problem in mathematics. In this talk we will consider the somewhat more flexible notion of a group action up to homotopy. This leads to interesting interactions between topology, group theory and representation theory. This is joint work with Jesper Grodal.
Wednesday November 1, 2017 in RH 104: Bernardo Villarreal Herrera (IUPUI), "A classifying space for commutativity in low dimensional Lie groups"; In this talk I will define the space BcomG arising from commuting tuples in G originally defined by A. Adem, F. Cohen and E. Torres. This space sits inside the classifying space BG and I will focus on describing the space BcomG for G=SU(2), U(2) and O(2), via its integral and mod 2 cohomology ring. If time permits, for the Lie groups above, I'll describe the homotopy type of the homotopy fiber of the inclusion BcomG into BG, denoted EcomG. This is joint work with O. Antolín and S. Gritschacher.
Wednesday November 8, 2017 in RH 104: Peter Bonventre (UKY), Genuine Equivariant Operads; A classic result by Elmendorf states that G-spaces and G-coefficient systems are Quillen equivalent. Moreover, this result remains true if one replaces spaces with other well-behaved categories (including simplicial sets and categories). However, when considering G-operads, this equivalence fails to capture certain subtleties of equivariant commutative monoids. In this talk, I will introduce (G-)operads and review this failure. I will then define a new algebraic gadget called genuine G-operads (playing the role of coefficient systems), and state an Elmendorf-type result in this context. This is joint work with L. Pereira.
Wednesday November 15, 2017 in RH104: Diana Hubbard, "The Braid Index of Highly Twisted Braid Closures"; Abstract: The braid index of a knot is the least number of strands necessary to represent it as the closure of a braid. If we view a braid as an element of the mapping class group of the punctured disk, its fractional Dehn twist coefficient measures, informally, the amount of twisting it exerts about the boundary. In this talk I will discuss joint work with Peter Feller showing that if an n-braid has fractional Dehn twist coefficient greater than n-1, then its closure is of minimal braid index, which draws a connection between braids as topological and geometric objects.
Monday November 27, 2017 in RH104: Rafael Zentner (U. Regensburg), "Irreducible SL(2,C)-representations of homology 3-spheres "; Abstract: We prove that the splicing of any two non-trivial knots in the 3-sphere admits an irreducible SU(2)-representation of its fundamental group. This uses instanton gauge theory and in particular holonomy perturbations of the flatness equation in an essential way. Using a result of Boileau, Rubinstein and Wang, it follows that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C).
Wednesday January 17, 2018 in RH104: Artem Kotelskiy, "Bordered Theory for Pillowcase Homology"; Abstract: Pillowcase homology is a geometric construction, which was developed in order to better understand and compute a knot invariant called singular instanton knot homology. We will introduce an algebraic extension of pillowcase homology. First we will associate an algebra A to the pillowcase. Then to an immersed curve L inside the pillowcase we will associate an A∞ module M(L) over A. Then we will show how, using these modules, one can recover and compute Lagrangian Floer homology for immersed curves.
Wednesday January 24, 2018 in RH104: Katherine Raoux (MSU), "Tau-invariants for knots in rational homology spheres"; Abstract: Using the knot filtration on the Heegaard Floer chain complex, Ozsvath and Szabo defined an invariant of knots in the three sphere called tau(K), which they also showed is a lower bound for the 4-ball genus. Generalizing their construction, I will show that for a (not necessarily null-homologous) knot, K, in a rational homology sphere, Y, we obtain a collection of tau-invariants, one for each spin-c structure on Y. In addition, these invariants can be used to obtain a lower bound on the genus of a surface with boundary K properly embedded in a negative definite 4-manifold with boundary Y.
Wednesday February 28, 2018 in RH104: Birgit Richter, "A strictly commutative model for the cochain algebra of a space"; One of the key tools in rational homotopy theory is the strictly commutative model for the cochains on a space, the Sullivan algebra of polynomial forms. Over general commutative rings such a model cannot exist in the category of chain complexes – that contradicts for instance the presence of cohomology operations that exist because of the non-commutativity on chain level. However, using diagram categories, one can construct a variant of Sullivan cochains that model the cochains of a space and that are strictly commutative. This is joint work with Steffen Sagave.
Wednesday April 25, 2018 in RH 104: Kursat Sozer, Extended HQFTs in dimension 2; Topological quantum field theories (TQFTs), inspired by theoretical physics, produce manifold invariants behaving well under gluing. For every discrete group G, homotopy quantum field theories (HQFTs) are G-equivariant versions of TQFTs. In this talk, we define and classify extended 2-dimensional HQFTs by generalizing methods introduced for TQFTs by Chris Schommer-Pries in 2009. We list generators and relations for the extended G-equivariant bordism bicategory and use them to classify extended 2-dimensional HQFTs.
Saturday April 28, 2018 in SE 105: Midwest Topology Seminar: Charles Weibel (Rutgers), "The Real Topological Analogue of the Witt Group"
Saturday April 28, 2018 in SE 105: Midwest Topology Seminar: Guillem Cazassus (IU), "Towards extended Floer field theories"; Donaldson polynomials are powerful invariants associated to smooth four-manifolds. The introduction by Floer of Instanton homology groups, associated to some 3-manifolds, allowed to define analogs of such polynomials for (some) four-manifolds with boundary, that have a structure similar with a TQFT.
Wehrheim and Woodward developed a framework called "Floer field theory" which, according to the Atiyah-Floer conjecture, should permit to recover Donaldson invariants from a 2-functor from the 2-category Cob_{2+1+1} to a 2-category Symp they defined, which is an enrichment of Weinstein's symplectic category.
I will outline a possible way of extending such a 2-functor to lower dimensions. This should permit to define new invariants in Manolescu and Woodward's symplectic instanton homology (sutured theory, equivariant version). This is work in progress.
Saturday April 28, 2018 in SE 105: Midwest Topology Seminar: Jonathan Campbell (Vanderbilt), "Combinatorial K-theory, Devissage and Localization"; What sorts of categories can K-theory be defined for? We know that exact categories and Waldhausen categories can be used as appropriate input. However, there are geometric categories where we would like to define K-theory where we are only allowed to cut and paste rather than quotient — examples of these include the category of varieties, and the category of polytopes. I'll define a more general context where one may talk about the algebraic K-theory of these categories, and outline a proof of a geometric version of Quillen's devissage and localization. I'll outline applications to studying derived motivic measures and the scissors congruence problem. This is joint work with Inna Zakharevich.
Saturday April 28, 2018 in SE 105: Midwest Topology Seminar: Julie Bergner (UVA), "2-Segal sets arising from graphs and their associated Hall algebras"; The notion of 2-Segal set allows us to consider structures which behave like categories but for which composition may not always exist or may be multiply-defined. While many examples only have partially defined composition, the 2-Segal set associated to a graph gives a nice example where maps can be composed in different ways. Thus, it gives a good way to explore the general properties of 2-Segal structures. In particular, following a definition of Dyckerhoff and Kapranov, this 2-Segal set has an associated Hall algebra which is much smaller than most natural examples of such algebras and has a curious description as a cohomology ring.
Tuesday May 15, 2018 in RH104: Swatee Naik (University of Nevada, Reno) " Periodic knots and Heegaard Floer correction terms"; Description: Periodic knots, such as the trefoil knot, are invariant under a rotation, and this symmetry can be easily illustrated in a knot diagram drawn in the plane. It so happens that the orbit space is a three-sphere in which the image of a periodic knot is called a quotient knot. Many properties of periodic knots are direct consequences of the branched covering set up that occurs between various three-manifolds that are naturally associated with the periodic knot and the quotient knot, respectively. In this talk we will begin with definitions and examples, introduce the basics of the theory, and demonstrate how properties of periodic knots can be used to detect knots that are not periodic. Our tools will include knot polynomials, homology of branched covers, and Heegaard-Floer correction terms.
Wednesday May 16, 2018 in RH104: Kate Ponto ( University of Kentucky ) , "Connections between K-theory and fixed point invarients"; Abstract: A major reason the Euler characteristic is a fantastic topological invariant is because it is additive on subcomplexes. It is also a fixed point invariant - it is a signed count of the number of fixed points of the identity map. (Replace the map by a homotopic map with isolated fixed points.) These two observations raise questions about how to think about fixed point invariants. We'll describe an approach to fixed point invariants that, while initially motivated by classical invariants, is reaching toward prioritizing additivity.
Wednesday May 23, 2018 in RH104: Peter Lambert-Cole, "Thom Conjecture"
Wednesday May 30, 2018 in RH 104: Mark Powell (Durham University), Non slice knots with vanishing Casson-Gordon invariants, revisited; Abstract: Around 2000, Cochran, Orr, and Teichner found the first example of a non- topologically slice knots with vanishing Casson-Gordon slice obstructions. With Allison Miller, we revisited these examples and provided an easier proof of non-sliceness that can easily be applied to more examples. The key ingredient in our obstruction theory is a twisted Blanchfield form, an extension of the Kirk-Livingston twisted Alexander polynomial slice obstruction. We showed how to compute the twisted Blanchfield form by means of an explicit description of the symmetric chain complex of the knot exterior. A computer then replaces the "25 foot extension cord" used by Cochran-Orr-Teichner.
Thursday July 19, 2018 in RH 104: Taehee Kim (Konkuk University, S. Korea), The 4-genus of knots and links; abstract: The 4-genus of a knot measures the complexity of the knot via the oriented compact surfaces in the 4-ball bounded by the knot. In particular, a knot is slice if the 4-genus is zero. In this talk, I will give lower bounds on the 4-genus of a knot using the infinite cyclic cover of the knot and metabelian Cheeger-Gromov-von Neumann rho invariants, refining the work of Jae Choon Cha, and give the first examples of knots with nontrivial stable 4-genus for which the Casson-Gordon invariants vanish. I will also discuss lower bounds on the 4-genus of links which are highly solvable in the sense of Cochran-Orr-Teichner. This is joint work with Jae Choon Cha and Min Hoon Kim.
Wednesday September 5, 2018 in RH104: Jonathan Campbell (Vanderbilt University), Fixed Point Theory and the Cyclotomic Trace; Fixed point theory has been the motivation for many of the most celebrated results of 20th century mathematics: the Lefschetz fixed point theorem, the Atiyah-Singer index theorem, and the development of etale cohomology. In this talk I'll describe work, joint with Kate Ponto, that relates classical fixed point theory to algebraic K-theory, topological Hochschild homology and the cyclotomic trace. The relationship seems to clarify both domains, and readily suggests generalizations, via machinery of Lindenstrauss-McCarthy, that relate to dynamical zeta functions. The link turns on careful considerations of the bicategorical structure of THH. Prerequisites: knowledge of the stable homotopy category, and an appetite for category theory; I will strive for the talk to be otherwise self-contained.
Wednesday September 19, 2018: G. Cazassus (IU), Equivariant Floer theory via cotangent bundles
Wednesday September 26, 2018 in RH 104: Min Hoon Kim (KIAS), A family of freely slice good boundary links.; Title: A family of freely slice good boundary links.
Abstract: The still open topological surgery conjecture for 4-manifolds is equivalent to the statement that all good boundary links are freely slice. In this talk, I will show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links. This is joint work with Jae Choon Cha and Mark Powell.
Wednesday October 10, 2018: Juanita Pinzon-Caicedo (NCSU): Satellites of infinite rank in the smooth concordance group; Many winding number zero satellite operations induce an infinite rank function on the smooth concordance group
Wednesday October 17, 2018 in RH 104: Stephan Stolz (Notre Dame), "Invertible topological field theories are SKK manifold invariants"; Topological field theories in the sense of Atiyah-Segal are symmetric monoidal functors from a bordism category to the category of complex (super) vector spaces. A field theory E of dimension d associates vector spaces to closed (d-1)-manifolds and linear maps to manifolds of dimension d. It turns out that if E is invertible, i.e., if the vector spaces associated to (d-1)-manifolds have dimension one, then the complex number E(M) that E associates to a closed d-manifold M, is an SKK manifold invariant. Here these letters stand for schneiden=cut, kleben=glue and kontrolliert=controlled, meaning that E(M) does not change when modifying the manifold by cutting and gluing along hypersurfaces in a controlled way. The main result of this joint work with Matthias Kreck and Peter Teichner is that the map described above gives a bijection between topological field theories and SKK manifold invariants.
Wednesday October 24, 2018 in RH 104: Siddhi Krishna (BC), Taut Foliations, Positive 3-Braids, and the L-Space Conjecture; The L-Space Conjecture is taking the low-dimensional topology community by storm. It aims to relate seemingly distinct Floer homological, algebraic, and geometric properties of a closed 3-manifold Y. In particular, it predicts a 3-manifold Y isn't "simple" from the perspective of Heegaard-Floer homology if and only if Y admits a taut foliation. The reverse implication was proved by Ozsvath and Szabo. In this talk, we'll present a new theorem supporting the forward implication. Namely, we'll discuss how to build taut foliations for manifolds obtained by surgery on positive 3-braid closures. No background in Heegaard-Floer or foliation theories will be assumed.
Wednesday October 31, 2018 in RH 104: Lisa Piccirillo (Texas), The Conway knot is not slice; Surgery-theoretic classifications fail for 4-manifolds because many 4-manifolds have second homology classes not representable by smoothly embedded spheres. Knot traces are the prototypical example of 4-manifolds with such classes. I’ll give a flexible technique for constructing pairs of distinct knots with diffeomorphic traces. Using this construction, I will show that there are knot traces where the minimal genus smooth surface generating second homology is not the obvious one, resolving question 1.41 on the Kirby problem list. I will also use this construction to show that Conway knot does not bound a smooth disk in the four ball, which completes the classification of slice knots under 13 crossings and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.
Wednesday November 7, 2018: Enrico Toffoli, Rho invariants for 3-manifolds with toral boundary
Tuesday November 13, 2018 in RH 104: Junghwan Park (Georgia Tech), Rational cobordisms and integral homology; We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology embeds in any other element in the same class. As an application, we show that the natural map from the Z/pZ homology cobordism group to the rational homology cobordism group has large cokernel, for each prime p. This is joint work with Paolo Aceto and Daniele Celoria.
Wednesday November 14, 2018 in RH 104: Nathan Dowlin, A spectral sequence from Khovanov homology to knot Floer homology; Khovanov homology and knot Floer homology are two knot invariants which are defined using very different techniques, with Khovanov homology having its roots in representation theory and knot Floer homology in symplectic geometry. However, they seem to contain a lot of the same topological data about knots. Rasmussen conjectured that this similarity stems from a spectral sequence from Khovanov homology to knot Floer homology. In this talk I will give a construction of this spectral sequence. The construction utilizes a recently defined knot homology theory HFK_2 which provides a framework in which the two theories can be related.
Tuesday January 29, 2019 in RH 104: Lee Kennard (Syracuse)
Wednesday February 13, 2019 in RH 104: Birgit Richter (Hamburg), "Splitting results for higher THH"; For a commutative ring (spectrum) R and a (pointed) finite simplicial set X one can build the tensor product X \otimes R. Important special cases are (topological) Hochschild homology for X=S^1 and torus homology for X=S^1 x...xS^1. In joint work with Bobkova, Hoening, Lindenstrauss, Poirier and Zakharevich we investigate how we can play off the algebraic variable R against the space variable and use this to show several splitting results.
Tuesday February 19, 2019 in RH 104: Dror Bar-Natan (University of Toronto), Computation without Representation; A major part of "quantum topology" (you don't have to know what's that) is the definition and computation of various knot invariants by carrying out computations in quantum groups (you don't have to know what are these). Traditionally these computations are carried out "in a representation", but this is very slow: one has to use tensor powers of these representations, and the dimensions of powers grow exponentially fast.
In my talk I will describe a direct-participation method for carrying out these computations without having to choose a representation and explain why in many ways the results are better and faster. The two key points we use are a technique for composing infinite-order "perturbed Gaussian" differential operators, and the little-known fact that every semi-simple Lie algebra can be approximated by solvable Lie algebras, where computations are easier.
This is joint work with Roland van der Veen and continues work by Rozansky and Overbay.
Wednesday February 20, 2019: Claudius Zibrowius (University of British Columbia), "Khovanov homology and the Fukaya category of the 3-punctured disc"; This talk will focus on a classification result for complexes over a certain quiver algebra and its consequences for Khovanov homology of 4-ended tangles. In particular, I will introduce a family of immersed curve invariants for pointed 4-ended tangles, whose intersection theory computes reduced Khovanov homology. This is joint work in progress with Artem Kotelskiy and Liam Watson, which was inspired by recent work of Matthew Hedden, Christopher Herald, Matthew Hogancamp and Paul Kirk.
Wednesday March 20, 2019 in RH 104: Maggie Miller (Princeton), Concordance of Light Bulbs; I will describe a concordance version of Gabai's 4-dimensional light bulb theorem. The light bulb theorem says that two 2-spheres $R$ and $R'$ that are homotopically embedded in a 4-manifold and mutually intersect another 2-sphere $G$ (with trivial normal bundle) in one point are isotopic, modulo a condition on 2-torsion in the fundamental group of the 4-manifold. I'll show that if we relax the condition on $R'$ so that $R'$ intersects $G$ in many points, then we may still conclude that $R$ and $R'$ are concordant (modulo the same 2-torsion obstruction). The proof is constructive, and will entail explicitly constructing a 3-manifold handle by handle in a movie of 5-dimensional space.
Wednesday March 27, 2019: Nate Bottman, The symplectic (A-infinity,2)-category; title: The symplectic (A-infinity,2)-category
abstract: I will describe ongoing work, partly joint with Katrin Wehrheim and Shachar Carmeli, which aims to implement a complete version of functoriality for the Fukaya category by constructing Symp, an (A-infinity,2)-category whose objects are symplectic manifolds. Symp associates operations to chains in 2-associahedra, which are configuration spaces of marked vertical lines in the plane. There are three aspects to this project: algebra (define notions of 2-associahedra and (A-infinity,2)-categories), analysis (regularize moduli spaces of pseudoholomorphic quilts involving the strip-shrinking degeneration), and computational (how can we make computations in Symp?). I will touch on each of these aspects, focusing on the algebraic one, and, time permitting, the computational one.
Wednesday April 3, 2019 in RH 104: Arunima Ray (MPI, Bonn), "The 4-dimensional sphere embedding theorem"; The disc embedding theorem for simply connected 4-manifolds was proved by Freedman in 1982 and forms the basis for his proofs of the h-cobordism theorem, the 4-dimensional Poincaré conjecture, 4-dimensional surgery, and the classification of simply connected 4-manifolds, all in the topological category. The disc embedding theorem for more general manifolds is proved in the book of Freedman and Quinn. However, the geometrically transverse spheres claimed in the outcome of the theorem are not constructed. We close this gap by constructing the desired transverse spheres. We also outline where transverse spheres appear in surgery and the classification of 4-manifolds and give a general 4-dimensional sphere embedding theorem. This is a joint project with Mark Powell and Peter Teichner.
Wednesday April 10, 2019: Vardan Oganesyan (Stony Brook), Monotone Lagrangian submanifolds and toric topology; Let N be the total space of a bundle over some k-dimensional torus with fibre Z, where Z is a connected sum of sphere products. It turns out that N can be embedded into C^n as a monotone Lagrangian submanifold. Also, it is possible to construct embeddings of N with different minimal Maslov number and get submanifolds which are not Lagrangian isotopic. In addition, we can show that some of our embeddings are smoothly isotopic but not isotopic through Lagrangians.
Wednesday May 1, 2019 in RH 104: Ben Knudsen (Harvard), Higher enveloping algebras and configuration spaces of manifolds; We provide Lie algebras with enveloping algebras over the little n-disks operads. These algebras enjoy a combination of good formal properties and computability, the latter afforded by a Poincare–Birkhoff–Witt type result. The main application pairs this theory with the theory of factorization homology in a study of configuration spaces, leading to a number of computations and qualitative results.
Tuesday June 4, 2019 in RH 104: Allison Moore (University of California, Davis), Obstructions to band surgery; Band surgery is a topological operation that transforms a knot into a new knot or link. When the operation respects orientations on the links involved, it is called coherent band surgery, otherwise it is called non-coherent. We will classify all band surgery operations from the trefoil knot to the T(2, n) torus knots and links, and prove some related obstructions to non-coherent band surgery. This is done by analyzing the behavior of the Heegaard Floer d-invariants under integral surgery along knots in the branched double cover of the links involved in the banding. We’ll also mention some newer results on the structure of the band-Gordian graph.
Tuesday July 16, 2019: Taehee Kim (Konkukk University, S. Korea), "Polynomial splittings of knot concordance invariants"; In this talk we will address the following question: if two nonslice knots have coprime Alexander polynomials, then are they not concordant? I will talk about the results on this question regarding various concordance invariants in both smooth and topological settings. Parts of this talk are based on joint work with Min Hoon Kim and Se-Goo Kim.
Thursday September 19, 2019 in SE 245: Tye Lidman (North Carolina State University), "Ribbon homology cobordisms"; Homology cobordisms are an important class of manifolds which show up in the theory of triangulations. We study some properties of 4-dimensional homology cobordisms which have particularly nice handle decompositions using Floer homology, character varieties, and other techniques. This is joint work with Aliakbar Daemi, Shea Vela-Vick, and Mike Wong.
Wednesday September 25, 2019 in RH 104: Paolo Ghiggini, "Generators of the wrapped Fukaya category of a Weinstein manifold"; A Weinstein manifold is an open symplectic manifold admitting a Morse function adapted to the symplectic structure. It turns out that the indices of such a Morse function have index at most half of the dimension. When the index is half the dimension, the unstable manifolds are Lagrangian planes called "Lagrangian cocore planes" In a joint work with Baptiste Chantraine, Georgios Dimitroglou Rizell and Roman Golovko, we decompose any object in the wrapped Fukaya category of a Weinstein manifold as an iterated cone of Lagrangian cocores planes.
Wednesday October 23, 2019 in RH 104: Colleen Delaney (IU), "Quantum link invariants and topological order"; We give a tour of the mathematical landscape of “topologically ordered” quantum phases of matter by way of the framed link invariants that encode certain spacetime trajectories of its quasiparticles.
Guided by two humble framed links – the twisted unknot and the Hopf link – we discuss applications of low-dimensional topology to the classification of braided fusion categories, characterization of phase transitions in theoretical condensed matter physics, and quantum information.
Wednesday October 30, 2019 in RH 104: Yu Zhang (Ohio State), "Topological Quillen localization of structured ring spectra"; Homotopy groups and stable homotopy groups of spaces are main invariants in algebraic topology. Homotopy groups are very powerful but difficult to compute in practice. Stable homotopy groups, on the other hand, are easier to work with, at the expense of losing unstable information.
Structured ring spectra are spectra with certain algebraic structure encoded by the action of an operad O. For such O-algebras, the analog of stable homotopy groups is played by Topological Quillen (TQ) homology groups. In this talk, we will discuss the following question: What information can be seen by TQ-homology? In particular, we will discuss TQ-localization of O-algebras and show the TQ-Whitehead theorem for homotopy pro-nilpotent O-algebras.
Wednesday November 13, 2019 in RH104: Noah Snyder, "Homotopy group calculations in synthetic homotopy theory"; There are two rather different approaches to classical plane geometry: Euclid's synthetic approach and Descartes analytic approach. In Descartes approach circles and lines are defined in terms of simpler notions as a set of points, while in Euclid's approach circles and lines are fundamental objects satisfying certain axioms and allowing certain constructions (like intersecting lines to get a point). In Euclid's approach, you can ask whether a point lies on a line, but you can't say that a line consists of a collection of points. Homotopy Type Theory was inspired by computer science, but it gives a synthetic approach to homotopy theory. That is, we think of spaces as basic notions which satisfy certain axioms and allow certain constructions, rather than defining them in terms of simpler notions (like certain topological spaces or certain simplicial sets). I will discuss some calculations of small homotopy groups using this synthetic approach. The rough idea is that CW complexes are \infty-groupoids with a given presentation in terms of generators and relations, and computing their homotopy groups is roughly like solving the word problem for a group. The usual way we solve the word problem for nice ordinary groups is to write down a free and transitive group action on something well understood. I will explain how to compute several examples of homotopy groups by writing down free and transitive actions of higher groups. This talk will be a mix of explaining results of Buchholtz, Brunerie, van Doorn, Rijke, Licata, Lumsdaine, and Shulman, and some new ideas from joint work with my student Nachiket Karnick and joint in part with our REU advisee Jacob Prinz. No previous knowledge of Homotopy Type Theory will be assumed (and I will mostly avoid using the language of type theory at all), and my hope is that this is a relaxing fun talk mostly about a different way to think about very old math beautiful mathematics of Pontryagin and others.
Wednesday November 20, 2019 in RH 104: Foling Zou (University of Chicago), "Nonabelian Poincare duality theorem in equivariant factorization homology"; The factorization homology groups are invariants of $n$-dimensional manifolds with some fixed tangential structures that take coefficients in suitable $E_n$-algebras. In this talk, I will give a definition for the equivariant factorization homology of a framed manifold for a finite group $G$ by monadic bar construction following Miller-Kupers. Then I will prove the equivariant nonabelian Poincare duality theorem in this case. As an application, in joint work with Asaf Horev and Inbar Klang, we compute the equivariant factorization homology on equivariant spheres for certain Thom spectra.
Wednesday December 11, 2019 in RH104: Deniz Kutluay (IU), "Winding Homology of Knotoids"
Wednesday January 29, 2020 in RH 104: Antonio Alfieri (Univ. of British Columbia), "Connected Floer homology of covering involutions"; I will outline theconstruction of some knot concordance invariants based on the Heegaard Floerhomology of double branched coverings. The construction builds on some ideasdeveloped by Hendricks, Manolescu, Hom and Lidman. This is joint work withAndras Stipsicz, and Sungkyung Kang.
Tuesday March 24, 2020 in RH 104: Allison Miller (Rice University), "Satellite operations and knot genera"; The satellite construction, which associates to a patternknot P in a solid torus and a companion knot K in the 3-sphere the so-calledsatellite knot P(K), features prominently in knot theory and low-dimensionaltopology. Besides the intuition thatP(K) is “more complicated” than either P or K, one can attempt to quantify howthe complexity of a knot changes under the satellite operation. In this talk,I’ll discuss how several notions of complexity based on the minimal genus of an embedded surfacewhose boundary is the given knot and which satisfies various additionalconstraints change under satelliting. In the case of the classical Seifertgenus of a knot, Schubert gives an exact formula. In the 4-dimensional contextthe situation is more complicated, and depends on whether we work in the smoothor topological category: the smooth category is asymptotically similar to theclassical setting, but our main results demonstrate that the topologicalcategory is much weirder. This talk isbased on joint work with Peter Feller and Juanita Pinzón-Caicedo.
Thursday April 2, 2020 in RH 104: Anthony Conway (Max Planck Institute), "Non-slice linear combinations of iterated torus knots"
Wednesday November 10, 2021 in Zoom Only: Michael Mandell (IU), "Virtual Topology Seminar: Toward HHR norms and equivariant factorization homology for compact Lie groups"; This seminar isonline only. Contact Prof. Mandell for a zoom link/password.
The Hill-Hopkins-Ravenel norm for finite groups is a powerful technical tooland is now an intrinsic part of the foundations of equivariant stable homotopytheory. Beyond finite groups, we currently only know a construction of the normfor 1-dimensional compact Lie groups, and the relationship to THH hints thatthe norm for higher dimensional compact Lie groups could open up exciting newdirections in the subject. The norm is also the starting point for a genuinetheory of equivariant factorization homology, and vice versa. This talk will bemostly non-equivariant and discuss a preliminary project (joint with AndrewBlumberg) to adapt the proofs of foundational theorems in factorizationhomology to better accommodate equivariant generalizations for positivedimensional compact Lie groups.
Thursday April 28, 2022 in RH 368 & Zoom: John Nicholson (Imperial College, London), "4-manifolds and the homotopy type of a finite 2-complex"; Abstract: It has been known since the 1930s that two finite2-dimensional CW-complexes with the same fundamental group and Euler characteristic become homotopy equivalent after wedging with a number of copies of S^2. However, it took until the 1970s to find examples which were not themselves homotopy equivalent. A long-standing problem has since been to determine for which k there exists examples with Euler characteristic k above the minimal value for a fixed fundamental group G. In the first part of this talk, I will discuss my complete resolution to this problem using techniques from integral representation theory.
A large part of the motivation for this work is a close analogy between the homotopy type of a finite 2-complex and the homeomorphism type of a smooth closed 4-manifold. I will explain this analogy and then discuss one aspect of the classification of 4-manifolds on which I hope my work on 2-complexes will eventually have an impact.
Zoom Id: https://iu.zoom.us/j/88198565168
Wednesday September 7, 2022 in SE 105: Daniel Ramras (IUPUI), "Flat connections and the topological Atiyah-Segal map"; Given a closed, aspherical manifold M, the topological Atiyah–Segal is a map of ring spectra relating the topological K-theory of M to a spectrum built from representations of the fundamental group of M. The homotopy fiber of this map can be described in terms of spaces of flat connections on vector bundles over M. Combined with work of Tyler Lawson, this allows one to extract information about spaces of flat connections from homological data regarding spaces of irreducible representations. I’ll explain this set-up, and describe some applications.
Wednesday October 19, 2022 in SE 105: Agniva Roy (GA Tech), "Small caps and embedding rational homology balls in CP^2"; Given a symplectic 4 manifold and a contact 3 manifold, it is natural to ask whether the latter embeds in the former as a contact type hypersurface. We explore this question for CP^2 and lens spaces. We will discuss a construction of small symplectic caps, using ideas first laid out by Gay in 2002, for rational homology balls bounded by lens spaces. This allows us to explicitly understand embeddings of these rational balls in CP^2 that were earlier understood only through almost toric fibrations. This is joint work with John Etnyre, Hyunki Min, and Lisa Piccirillo.
Wednesday October 26, 2022 in SE 105: Seppo Niemi-Colvin, "Invariance of Knot Lattice Homology and Homotopy"; Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from algebraic complex surfaces and curves inside them. Némethi created lattice homology as an invariant for links of normal surface singularities which developed out of computations for Heegaard Floer homology. Later Ozsváth, Stipsicz, and Szabó defined knot lattice homology for generalized algebraic knots, which is known to play a similar role to knot Floer homology and is known to compute knot Floer in some cases. I discuss a proof that knot lattice is an invariant of the smooth knot type, which had been previously suspected but not confirmed.
Wednesday November 2, 2022: Kai Smith (IU), "Traceless SU(2) Character Varieties of Tangles"; Hedden, Herald, and Kirk initiated a program to study knots by decomposing them into tangles and studying the interaction between certain perturbed moduli spaces associated to each tangle. I will explain this program and how it is related to gauge theory, in particular Kronheimer and Mrowka's singular instanton homology. I will present my own results which lead to computations of these perturbed moduli spaces for many new tangles. These computations shed some light on a proposed knot invariant called pillowcase homology.
Wednesday November 9, 2022 in Zoom Only: Sally Collins (Georgia Tech), "Homology cobordism and knot concordance"; Array
Wednesday November 16, 2022 in SE 105: Taehee Kim (Konkuk University), "Iterated satellite operators and knot concordance"; Hedden and Pinz\'{o}n-Caicedo conjecture that the image of a non-constant satellite operator generates an infinite rank subgroup in the knot concordance group. For a satellite operator P, they also ask a similar question on the P-filtration of the knot concordance group which is obtained by applying P iteratively. In this talk, we will address the conjecture/question and related problems using L^2-signatures. In particular, we will show that for a winding number zero satellite operator P, if the axis has nontrivial Blanchfield self-linking form, then each of the successive quotient groups of the P-filtration has infinite rank. This is joint work with Jae Choon Cha.
Wednesday January 18, 2023 in Zoom Only: 609 643 9799 Password: derivative: David Boozer (Princeton University), "Khovanov homology and the Fukaya category of the traceless character variety for the twice-punctured torus"; We describe a strategy for constructing reduced Khovanov homology for links in lens spaces by generalizing a symplectic interpretation of reduced Khovanov homology for links in $S^3$ due to Hedden, Herald, Hogancamp, and Kirk. The strategy relies on a partly conjectural description of the Fukaya category of the traceless $SU(2)$ character variety of the 2-torus with two punctures. From a diagram of a 1-tangle in a solid torus, we construct a corresponding object $(X,\delta)$ in the $A_\infty$ category of twisted complexes over this Fukaya category. The homotopy type of $(X,\delta)$ is an isotopy invariant of the tangle diagram. We use $(X,\delta)$ to construct cochain complexes for links in $S^3$ and some links in $S^2 \times S^1$. For links in $S^3$, the cohomology of our cochain complex reproduces reduced Khovanov homology, though the cochain complex itself is not the usual one. For links in $S^2 \times S^1$, we present results that suggest the cohomology of our cochain complex may be a link invariant.
Wednesday April 19, 2023 in SE 105: Taehee Kim (Konkuk Univ.), "The double slice genus of a knot"; The double slice genus of a knot in the 3-sphere is the minimum genus among unknotted surfaces in the 4-sphere whose intersection with the 3-sphere is the knot. In this talk, we relate the double slice genus of a knot to the second Betti number of a closed 4-manifold into which the zero-surgery on the knot can be embedded preserving first homology. Then, using the relationship and the von-Neumann rho invariants, we give examples of knots with vanishing Casson-Gordon invariants whose double slice genera are arbitrarily large. This joint work with Se-Goo Kim.
Wednesday April 26, 2023 in SE 105: Shawn Williams (IU), "A Higher Order Fox-Milnor Theorem"; Abstract: In a celebrated theorem, Fox and Milnor showed the Alexander polynomial of a slice knot factors as $f(t)f(t^{-1})$. We introduce a family of Alexander modules $\mathcal{A}_n^\mathcal{P}$ that peer more deeply into the knot group than the classical Alexander module. Our main result is a new sequence of Fox-Milnor Theorems for (n+1)-solvable knots.
Monday August 28, 2023 in SE 010: William Balderrama (Univ. of Virginia), "The equivariant J-homomorphism and RO(G)-graded periodic phenomena"; I will describe how the G-equivariant J-homomorphism can be "desuspended" in a way that produces nontrivial RO(G)-graded periodicities in equivariant stable homotopy theory, such as in the G-equivariant stable stems. When G = C_2, these are essentially versions of James periodicity, and I will explain how this recovers and unifies theorems of Bredon, Araki and Iriye, and Behrens and Shah.
Wednesday September 13, 2023 in SE 140: Manuel Rivera (Purdue), "Coalgebras, loop spaces, and homotopy theory"; A central problem in algebraic topology is to determine how much information about topological spaces considered up to a specified notion of equivalence is preserved by a particular functorial invariant. In this talk, I will discuss a version of this problem for spaces up to “$\pi_1$-R-equivalence” by means of R-coalgebras, for a fixed commutative ring R. A continuous map of spaces is called a $\pi_1$-R-equivalence if it induces an isomorphism on fundamental groups and an R-homology equivalence between universal covers. I will describe a corresponding notion of weak equivalence for simplicial R-coalgebras using ideas from differential graded Koszul duality. When R is an algebraically closed field, the R-chains functor embeds fully and faithfully the homotopy theory of spaces up to $\pi_1$-R-equivalence in the corresponding homotopy theory of simplicial R-coalgebras. One of the key ingredients is the new (somewhat surprising) observation that the coalgebra of chains on a space, appropriately considered, completely determines the fundamental group (and much more). This is based on a series of joint projects with F. Wierstra, M. Zeinalian, and G. Raptis.
Wednesday September 20, 2023 in SE 140: Ningchuan Zhang (IU), "Equivariant algebraic K-theory and Weil L-functions of finite fields"; In 1972, Quillen computed higher algebraic K-groups of finite fields. The orders of these groups are very closely related to their Weil L-functions. In this talk, I will generalize this connection to Weil L-functions of abelian characters of Galois groups of finite fields. On the homotopy theory side, we compute Galois equivariant algebraic K-groups of finite fields with coefficients in those characters. If time allows, I will explain how this generalization works for function fields and number fields. This is joint work in progress with Elden Elmanto.
Wednesday September 27, 2023 in SE 140: Michael Mandell (IU), "Factorization Homology and HHR Norms"; This talk is nominally about a project with Blumberg and Hill to generalize the Hill-Hopkins-Ravenel norm to compact Lie groups, but it will mostly consist of an introduction to factorization homology and some of its properties as background, motivation, and introduction to the HHR norm problem.
Wednesday October 4, 2023 in SE 140: Yun Liu (IU), "Relative Join Construction and Towers of Borel Fibrations"; In a recent work, Yu. Berest and A. C. Ramadoss formulated and studied the realization problem for rings of quasi-invariants of finite reflection groups in terms of classifying spaces of compact Lie groups The main tool used in their work is the fiber-cofiber construction introduced in topology by T. Ganea.
In the first part of this talk, which is based on joint work with Yu. Berest and A. C. Ramadoss, we will describe a natural generalization of the fiber-cofiber construction in the context of abstract homotopy theory, which we call relative join construction. Then, in the second part, we will show how to apply this relative join construction to obtain analogues of spaces of quasi-invariants which can be identified with homotopy quotient of certain $G$-spaces for a compact Lie group $G$. The rational cohomology rings of these generalized spaces of quasi-invariants have nice algebraic properties that are similar to those of classical quasi-invariants. Moreover, the equivariant K-theory of those $G$-spaces exhibits similar properties.
Wednesday October 11, 2023 in SE 140: Dylan Thurston (IU), “Floer Homology beyond Borders” -- postponed
Wednesday October 18, 2023 in SE 140: Daniel Tolosa (Purdue), "An algebraic model for the free loop space as an S^1-space"; The free loop space of a topological space has a canonical circle action given by rotating loops, making it an S^1-space. The work of Jones, Goodwillie, and others, relates the equivariant homology of the free loop space to the cyclic homology of the algebra of singular chains on the topological monoid of based loops. One can model the free loop space in terms of the chains on the underlying space considered as a categorical coalgebra, a notion Koszul dual to a non-negatively graded dg category. This construction is ”as small as possible”, has no hypotheses on the underlying space and is suitable for computations in (non-simply connected) string topology. I will present a cyclic theory for categorical coalgebras and dg-categories extending the theory of cyclic homology for (dg) algebras and coalgebras. In particular, the cyclic chains of the categorical coalgebra of chains on a simplicial set provides a model for the S^1-equivariant chains on the free loop space that is suitable for computations. The proofs of these results can be understood in terms of a combinatorial model for the unit of the Bar-Cobar adjunction.
Wednesday October 25, 2023 in SE 140: Ishan Levy (MIT), “Telescopic stable homotopy theory”; Chromatic homotopy theory attempts to study the stable homotopy category by breaking it into v_n-periodic layers corresponding to height n formal groups. There are two natural ways to do this, via either the K(n)-localizations which are computationally accessible, or via the T(n)-localizations, which detect the v_n-periodic parts of the stable homotopy groups of spheres. Ravenel's telescope conjecture asks that these two localizations agree. For n at least 2 and all primes, I will discuss counterexamples to Ravenel’s telescope conjecture. Our counterexamples come from using trace methods to compute the T(n) and K(n)-localizations of the algebraic K-theory of a family of ring spectra, which in the case n=2 are certain finite Galois extensions of the K(1)-local sphere. I will then explain that this can be used to obtain an infinite family of elements in the v_n-periodic stable homotopy groups of spheres, giving the best known lower bound on the asymptotic average ranks of the stable stems. Finally, I will explain that the Galois group of the T(n)-local category agrees with that of the K(n)-local category, and how the failure of the telescope conjecture comes entirely from the failure of Galois hyperdescent. This talk comes from projects that are joint with Burklund, Carmeli, Clausen, Hahn, Schlank, and Yanovski.
Wednesday November 1, 2023 in SE 140: Hana Kong (Harvard), “A deformation of Borel equivariant homotopy”; The real motivic stable homotopy category has a close connection to the $C_2$-equivariant stable homotopy category. From a computational perspective, the real motivic computation can be viewed as a simpler version which “removes the negative cone” in the $C_2$-equivariant stable homotopy groups. On the other hand, by work of Burklund–Hahn–Senger, one can build the completed Artin–Tate real motivic category from the completed $C_2$-equivariant category using the deformation construction associated to the $C_2$-effective filtration.
In work with Gabriel Angelini-Knoll, Mark Behrens, and Eva Belmont, we try to build an analog of this deformation story for a general finite group $G$. We give a new interpretation of the $C_2$-effective filtration in the Borel equivariant category which generalizes for $G$. Using this new interpretation, the deformation construction gives a deformation of the Borel equivariant stable homotopy category for general finite groups.
Wednesday November 8, 2023 in SE 140: David Mehrle (UKY), “Hochschild homology of polynomial Tambara functors for cyclic p-groups”; Hochschild homology is a ubiquitous invariant that carries both geometric and algebraic information. One of the first calculations in Hochschild homology is the Hochschild homology of $k[x]$ as a $k$-algebra. Inspired by recent progress in equivariant trace methods, we carry out an equivariant version of this calculation, where the polynomial ring $k[x]$ is replaced by a polynomial Tambara functor for a cyclic $p$-group. The key tool in this calculation is the Koszul resolution of a polynomial Tambara functor, which becomes a multicomplex in the equivariant case. This is joint work with J.D. Quigley and Michael Stahlhauer.
Wednesday November 15, 2023 in SE 140: Dylan Thurston (IU), “Floer Homology Beyond Borders”; Heegaard Floer Homology is an invariant of 3-manifolds and 4-manifolds, providing striking topological information. Bordered Floer Homology extends that theory to apply to 3-manifolds with boundary, allowing computation for 3-manifolds from elementary pieces, developed by Robert Lipshitz, Peter Ozsvath, and the speaker.
In this talk we will survey the construction, structure, applications, and future directions of the subject.
Wednesday November 29, 2023 in SE 140: Eva Belmont (Case Western), “Equivariant homological stability for unordered configuration spaces”; In foundational work, McDuff and Segal proved "homological stability" in the setting of unordered configuration spaces $C_n(M)$ of $n$ points in an open manifold $M$; that is, $H_d(C_n(M)) \cong H_d(C_{n+1}(M))$ for $n\gg d$. In joint work with J.D. Quigley and Chase Vogeli, we prove an equivariant generalization of McDuff and Segal's result which applies to Bredon homology of unordered configuration spaces on G-manifolds for a finite abelian group G.
Wednesday December 6, 2023 in SE 140: Hari Rau-Murthy (Notre Dame), “An HKR theorem for factorization homology”; This talk will be on a generalization of the so-called “HKR theorem”. The classical HKR theorem concerns two different ways of generalizing the notion of differential forms on a ring – the theorem asserts that they agree. I will give a generalization of this theorem for ring spectra which will allow us to recalculate the factorization homology of rational KU. A written version of this work is available on arXiv 2305.00658.
Wednesday January 17, 2024 in SE 140: Tanushree Shah (Vienna), “Tight contact structures on Seifert fibered 3-manifolds”; I will start by introducing contact structures. They come in two flavors: tight and overtwisted. Classification of overtwisted contact structures is well understood as opposed to tight contact structures. Tight contact structures have been classified on some 3 manifolds like S^3, R^3, Lens spaces, toric annuli, and almost all Seifert fibered manifolds with 3 exceptional fibers. We look at classification on one example of the Seifert fibered manifold with 4 exceptional fibers. I will explain the Legendrian surgery and convex surface theory, which help us calculate the lower bound and upper bound of a number of tight contact structures. We will look at what more classification results can we hope to get using the same techniques and what is far-fetched.
Wednesday January 24, 2024 in SE 140: Daniel Lopez Neumann (IU), “Genus bounds from quantum groups at roots of unity”; Quantum invariants of knots and 3-manifolds form a large class of topological invariants built from the theory of quantum groups, and more generally, Hopf algebras and tensor categories. A big open problem in the field is to understand the geometric content of such invariants. This is at most the subject of several conjectures for the Jones polynomial, which is obtained through quantum SL2 at a generic parameter q. In this talk I’ll explain that, if the parameter q of a quantum group is specialized to a root of unity, a different family of link polynomials arises and the degrees of these give lower bounds to the Seifert genus of the link. This is joint work with Roland van der Veen.
Wednesday January 31, 2024 in SE 140: Danika Van Niel (MSU), “Computational approaches to twisted topological Hochschild homology”; Topological Hochschild homology (THH) is an invariant of ring spectra and is a key component of the trace method approach to algebraic K-theory. One of the main computational tools for THH is the Bökstedt spectral sequence. The study of the algebraic structure of THH and the Bökstedt spectral sequence have advanced our computational ability. In recent years, a generalization of THH for equivariant ring spectra called twisted THH has been developed along with an equivariant version of the Bökstedt spectral sequence. In this talk I’ll introduce THH, twisted THH, and discuss computations of twisted THH using the equivariant Bökstedt spectral sequence as well as work on the algebraic structure of twisted THH. I will also talk about Mackey functor fields, which are an equivariant analog of classical fields, and how they can be used in computations of twisted THH.
Wednesday February 7, 2024 in SE 140: Dror Bar-Natan (Univ. of Toronto), "Cars, Interchanges, Traffic Counters, and some Pretty Darned Good Knot Invariants"
Wednesday February 14, 2024 in SE 140: Jim Davis (IU), “The Borel Conjecture and aspherical 4-manifolds”; The Borel Conjecture states that the homeomorphism type of a closed aspherical manifold is determined by its fundamental group.
Freedman and Quinn proved that the disc embedding conjecture (and hence surgery and the s-cobordism theorem) holds for 4-manifolds with EA (= elementary amenable) fundamental group.
In joint work with Jonathan Hillman, we formulate versions (uniqueness and existence) of the Borel Conjecture for compact aspherical manifolds with boundary, and prove that they hold for 4-manifolds with EA fundamental group. Furthermore, we characterize all EA groups which are the fundamental group of a compact aspherical 4-manifold.
Wednesday February 21, 2024 in SE 140: Vladimir Turaev (IU), “Knots and links in 2-complexes”
Wednesday February 28, 2024 in SE 140: Liam Keenan (Minnesota), “Chromatic vanishing results for topological restriction homology”; Over the past several years, there has been an enormous development in our understanding of how algebraic K-theory, an invariant of ring spectra, interacts with chromatic homotopy theory, a powerful tool for organizing computations of stable homotopy groups. These developments are part of Rognes' chromatic redshift philosophy, which, roughly speaking, asserts that K-theory should increase chromatic height by one. In this talk, I will give an introduction to this circle of ideas and explain some recent work, joint with Jonas McCandless, regarding how topological restriction homology fits into this picture.
Wednesday March 6, 2024 in SE 140: Birgit Richter (Hamburg), “Loday constructions for Tambara functors”
Wednesday March 20, 2024 in SE 140: Jonathan Campbell (CCR-L), “The Many Uses of the K-theory of Endomorphisms”; In this talk I'll discuss work with Kate Ponto and Inna Zakharevich on some properties of the K-theory of endomorphisms and a new approach to the cyclotomic trace in K-theory. To us, the definition of the cyclotomic trace has always seemed ad hoc, and so we try to put it on a cleaner conceptual footing via the K-theory of endomorphisms, K(End). I'll try to convince you that K(End) is a somewhat more natural invariant than K-theory for some purposes, prove some interesting properties of K(End) (for example, that it is a "shadow"), and show how these properties point towards a natural definition of the cyclotomic trace. For folks who aren't homotopy theorists: don't fret! I'll explain all of the terms in this abstract.
Wednesday March 27, 2024 in SE 140: Maxine Calle (Penn), “Nested cobordisms and TQFTs”; The folklore theorem identifying 2-dimensional topological quantum field theories (TFQTs) with Frobenius algebras is a starting point for a lot of interesting mathematics, from mathematical physics to homotopy theory to higher category theory. In this talk, we will explore what happens if we replace the cobordism category with a category of nested cobordisms, where 2-dimensional surfaces may have embedded 1-dimensional submanifolds, and what kind of algebraic structure the corresponding nested TQFTs pick out. This talk is based on joint work with R. Hoekzema, L. Murray, N. Pacheco-Tallaj, C. Rovi, and S. Sridhar.
Wednesday April 3, 2024 in SE 140: Bob Oliver (Paris XIII), “The loop space homology of a small category”; In an article published in 2009, Dave Benson described, for a finite group $G$, the mod p homology of the space $\Omega (BG^{\wedge}_p)$ (the loop space of the p-completion of $BG$) in purely algebraic terms. In joint work with Carles Broto and Ran Levi, we tried to better understand Benson’s result by generalizing it. Among other things, we showed that when $\mathcal{C}$ is a small category, $|\mathcal{C}|$ is its geometric realization, $R$ is a commutative ring, and $|\mathcal{C}|^+_R$ is a “plus construction” of $|\mathcal{C}|$ with respect to homology with coefficients in $R$, then $H_*(\Omega(|\mathcal{C}|^+_R); R)$ is the homology of any chain complex of projective $R\mathcal{C}$-modules that satisfies certain conditions. Benson’s theorem is then the special case where $\mathcal{C}$ is the category associated to a finite group $G$ and $R = \mathbb{F}_p$ , so that p-completion appears as a special case of the plus construction.
Wednesday April 10, 2024 in SE 140: David Chan (MSU), “Equivariant Algebraic K theory and the stable parametrized h-cobordism theorem”; The stable parametrized h-cobordism theorem is a result which connects the study of cobordisms of manifolds to stable homotopy theory. It is formulated in terms of Waldhausen’s algebraic K-theory of spaces and helps to explain the connection between algebraic K-theory and geometric topology through Wall’s finiteness obstruction and Whitehead torsion. In this talk we will discuss a refinement of Waldhausen’s construction which accounts for symmetries of the underlying manifold, implemented as the algebraic K theory of an equivariant ring spectrum. As an application, we will explain how this framework can be used as a step to proving an equivariant stable parametrized h-cobordism theorem.
Wednesday April 17, 2024 in SE 140: J.D. Quigley (UVA), “Detecting exotic spheres and their symmetries with stable homotopy theory”; An exotic n-sphere is a smooth n-manifold which is homeomorphic, but not diffeomorphic, to the n-sphere with its standard smooth structure. Even after decades of intense study, there are many simple-sounding questions about exotic spheres which are unanswered. In which dimensions do exotic spheres exist? Can you rotate an exotic sphere as you would a standard sphere? Are exotic spheres "round" like the standard sphere? In this talk, I will survey what is known about exotic spheres and their geometry. I will then report on work with Mark Behrens and Mark Mahowald, Prasit Bhattacharya and Irina Bobkova, and Boris Bovtinnik on detecting exotic spheres and their smooth symmetries.
Wednesday April 24, 2024 in SE 140: Noah Snyder (IU), “Towards the Quantum Exceptional Series”; Many Lie algebras fit into discrete families like GL_n, O_n, Sp_n. By work of Brauer, Deligne and others, the corresponding symmetric tensor categories fit into continuous families GL_t and OSp_t. A similar story holds for quantum groups, so we can speak of two parameter families (GL_t)_q and (OSp_t)_q. These are the tensor categories attached to the HOMFLY and Kauffman polynomials. There are a few remaining Lie algebras which don’t fit into any of the classical families: G2, F4, E6, E7 and E8. By work of Deligne, Vogel, and Cvitanovic, there is a conjectural 1-parameter continuous family of planar algebras which interpolates between these exceptional Lie algebras. Similarly to the classical families, there ought to be a 2-parameter family of planar algebras which introduces a variable q, and yields a new exceptional knot polynomial. In work joint with Dylan Thurston and joint in part with Scott Morrison arXiv:2402.03637, we give a skein theoretic description of what this knot polynomial would have to look like. In particular, we show that any braided tensor category whose box spaces have the appropriate dimension and which satisfies some mild assumptions must satisfy these exceptional skein relations.
Saturday April 27, 2024: Midwest Topology Seminar; https://mts.math.indiana.edu/2024/
Wednesday May 22, 2024 in SW 220: Jonathan Hillman (Univ. of Sydney), "Hypersurfaces in S^4"; Abstract: I shall outline the contents of a book manuscript that I have been writing, with the title "Locally flat embeddings of 3-manifolds in S^4"
The emphases in this book are on embeddings of Seifert fibred 3-manifolds and on properties of the complementary regions.
The choice of "locally flat" rather than "smooth" reflects Freedman's result on homology spheres, which all have locally flat embeddings, whereas the smooth problem is almost intractable, and on the hope of applying topological surgery when the complementary regions have good fundamental groups.
Wednesday September 11, 2024 in SE 105: Jeremy Miller (Purdue), “Scissors congruence automorphism groups”; Two polytopes are called scissors congruent if they can be cut up into isometric pieces. The question of classifying polytopes up to scissors congruence dates back to Hilbert. In joint work with Lemann, Malkiewich, and Sroka, we study an orthogonal question, investigating the group of scissors congruence automorphisms of a fixed polytope. We prove a homological stability result for these groups. This yields a new model of Zakharevich’s scissors congruence spectrum. Combining our result with work of Malkiewich on Tits building models of the scissors congruence spectra yields an approach to calculating the homology of these groups and their variants.
Wednesday September 18, 2024 in SE 105: David Boozer (IU), “The combinatorial and gauge-theoretic foam evaluation functors are not the same”; Kronheimer and Mrowka have outlined a new approach that could potentially lead to the first non-computer based proof of the four-color theorem. Their approach relies on a functor J-sharp, which they define using gauge theory, from a category of webs in R^3 to the category of finite-dimensional vector spaces over the field of two elements. They have also suggested a possible combinatorial replacement J-flat for J-sharp, which Khovanov and Robert proved is well-defined on a subcategory of planar webs. We exhibit a counterexample that shows the restriction of the functor J-sharp to the subcategory of planar webs is not the same as J-flat.
Wednesday September 25, 2024 in SE 105: Myungsin Cho (IU), “Fiber of the cyclotomic trace of the sphere spectrum and K-theoretic Tate-Poitou duality at the prime 2”; Understanding the algebraic K-theory of the sphere spectrum has long been recognized as a fundamental problem in algebraic and differential topology. Since the homotopy fiber of its p-completed cyclotomic trace depends only on the zeroth homotopy group, we can apply algebraic methods to study it. Blumberg and Mandell’s work demonstrates that, for odd primes, Tate-Poitou duality can be enhanced to an Anderson duality between the homotopy fiber and the K(1)-local K-theory of the integers. In this talk, I will present this connection and extend the result to the case where p=2.
Wednesday October 2, 2024 in SE 105: Paul Kirk (IU), “A lagrangian correspondence on traceless SU(2) character varieties induced by a marking”; I’ll describe a non-trivial endofunctor on the Fukaya A_infty category of the SU(2) character variety of a Conway sphere (S^2,4) induced by a cobordism obtained by adding a marking to the product (S^2,4) x I. (You don’t need to know what any of these words mean.)
Wednesday October 9, 2024 in SE 105: Juan Moreno (CU Boulder), “Duals of higher real K-theories at p=2”; Morava E-theories are higher chromatic analogs of complex K-theory, KU. They come equipped with cyclic group actions generalizing the complex conjugation action on KU. Taking fixed-points in a suitable sense gives rise to generalizations of KO called higher real K-theories. The higher KO’s form a large class of interesting cohomology theories that are great at detecting elements in the stable homotopy groups of spheres. Interestingly, one often finds that the higher real K-theories are self-dual up to a suspension. To the speaker’s knowledge, whether this is the case in general is an open question.
In this talk, we will explore a connection between the self-duality of higher real K-theories and equivariant periodicities of E-theories generalizing the classical Bott periodicity. In particular, we establish the self-duality of a particularly interesting higher real K-theory known to detect the Kervaire invariant one elements.
Wednesday October 16, 2024 in SE 105: Ben Spitz (UVA), “Algebraically Closed Tambara Functors”; In equivariant stable homotopy theory, objects called "Tambara functors" play the role of commutative rings. Tambara functors are abstract algebraic objects: they consist of sets with certain operations satisfying certain axioms; however, the theory of Tambara functors is much less developed than the theory of commutative rings, in part because it is not clear exactly how to define the "Tambara analogues" of many classical notions.
In their proof of the Redshift Conjecture, Burklund-Schlank-Yuan introduced the notion of a "Nullstellensatzian object" in a category C. When C is the category of commutative rings, this recovers precisely the notion of an algebraically closed field, so we therefore define an "algebraically closed Tambara functor" to be a Nullstellensatzian object in the category of Tambara functors.
In this talk, I will present a classification of algebraically closed G-Tambara functors (for any finite group G). As a corollary, we reduce the K-theory of algebraically closed Tambara functors to the ordinary K-theory of algebraically closed fields. This is joint work with Jason Schuchardt and Noah Wisdom.
Wednesday October 23, 2024 in SE 105: Nikolai Konovalov (Chicago), “Algebraic Goodwillie spectral sequence”; I will discuss two tools for computing homotopy groups of a space: Goodwillie and Adams spectral sequences. The first one is more fundamental, however less known and less understood. Vice versa, the second one is less fundamental, but far more investigated. I will also talk about how these two spectral sequences interact with each other.
Wednesday October 30, 2024 in SE 105: Hana Jia Kong (Zhejiang/Harvard), “Motivic Milnor formulas”; In the latest research, Lin, Wang, and Xu have advanced the Adams computation of stable homotopy groups using computer programs. Developing these algorithms requires a deep understanding of the Steenrod algebra. While Milnor's work provides a foundation in the classical case, the motivic case is much more complex. In this talk, I'll share my work with Weinan Lin on the motivic formulas. This is part of a larger project with Lin, Wang, and Xu, aiming to compute real motivic stable stems over a large range with the assistance of computers.
Wednesday November 6, 2024 in SE 105: Zhonghui Sun (MSU), “Twisted bicategorical shadows and traces”; Bicategorical shadows, defined by Ponto, provide a framework which generalizes (topological) Hochschild homology. Bicategorical shadows have important properties, such as Morita invariance, and allow one to generalize the symmetric monoidal trace to a bicategorical trace. Topological Hochschild homology (THH), which is an essential component to the trace methods approach for algebraic K-theory, is a key example of a bicategorical shadow.
In recent years, equivariant versions of topological Hochschild homology have emerged. In particular, for a C_n-ring spectrum there is a theory of C_n-twisted THH, constructed via equivariant norms. However, twisted THH fails to be a bicategorical shadow. In this talk, we will explain a new framework of equivariant bicategorical shadows and explain why twisted THH is a g-twisted shadow. We also explore g-twisted bicategorical traces.
Wednesday November 13, 2024 in SE 105: Yunze Lu (UCSD), “RO(G)-graded homotopy fixed point spectral sequence for height 2 Morava E-theory”; The homotopy fixed points of Lubin-Tate theories are central objects in chromatic homotopy theory, as they are building blocks of K(n)-local spheres. This talk will describe the use of equivariant structure and vanishing line result to compute the RO(Q_8)-graded homotopy fixed points spectral sequence at height 2 and prime 2. This is joint work with Zhipeng Duan, Hana Jia Kong, Guchuan Li, and Guozhen Wang.
Wednesday November 20, 2024 in SE 105: Daniel Lopez Neumann (Universidad de Concepción, Chile), “Obstructions to knot fibering from quantum groups”; The quantum invariants of knots and 3-manifolds are a wide family of topological invariants coming from the theory of braided monoidal categories, and in particular, quantum groups. The Jones and HOMFLY polynomials of knots are the most studied special cases of this theory. However, it is not known how these invariants are related to classical geometric properties of knots, such as the Seifert genus or fibredness.
In this talk, I’ll propose a partial solution to this problem: quantum invariants coming from "non-semisimple" quantum groups do provide genus bounds and obstructions to fibredness. More precisely, I’ll state and show a q-deformation of the classical fibering obstruction for the Alexander polynomial. The talk doesn't assumes familiarity with quantum groups. This is joint work with Roland van der Veen.
Thursday November 21, 2024 in SE 105: Seymour Sherman Memorial Lecture: Adam Levine (Duke), “New Constructions of Exotic 4-Manifolds”; The search for exotic manifolds — pairs or families of smooth manifolds that are homeomorphic but not diffeomorphic — has been a motivating theme in topology for decades, particularly in dimension 4. In this talk, we will discuss some new constructions of exotic 4-manifolds and new methods for distinguishing between them. We exhibit several novel phenomena, including the first known examples of exotic 4-manifolds with definite intersection form. This is joint work with Tye Lidman and Lisa Piccirillo.
Wednesday January 29, 2025 in SE 105: Maximillien Peroux (MSU)
Wednesday February 5, 2025 in SE 105: J. D. Quigley (UVA), “On the Tambara affine line”; Tambara functors are the analogue of commutative rings in equivariant algebra. A little over a decade ago, Nakaoka defined the Zariski spectrum of prime ideals of a Tambara functor, and Nakaoka and Calle-Ginnett computed the analogue of $Spec(\mathbb{Z})$ over finite cyclic groups. In this talk, I will survey some new techniques for studying the spectra of Tambara functors. The main application is a description of the Tambara affine line, the equivariant analogue of $Spec(\mathbb{Z}[x])$, over cyclic groups of prime order. This is joint work with David Chan, David Mehrle, Ben Spitz, and Danika Van Niel.
Wednesday February 12, 2025 in SE 105: Noah Wisdom (Northwestern)
Wednesday February 19, 2025 in SE 105: Noah Riggenbach (Northwestern)
Wednesday February 26, 2025 in SE 105: Maru Sarazola (Minnesota), “Squares K-theory and 2-Segal spaces”; It's a well-known result that a simplicial set satisfies the Segal condition precisely if it's the nerve of a category; then, categories represent all possible 1-Segal sets. But what about 2-Segal? This question was studied by Bergner, Osorno, Ozornova, Rovelli and Scheimbauer, who proved that 2-Segal sets are in turn related to a certain class of double categories via a "K-theory construction". In this talk, I will discuss recent work joint with Maxine Calle where we explore the connection between 2-Segal spaces and a recently introduced K-theory framework based on double categories called "Squares K-theory".
Wednesday March 12, 2025 in SE 105: Matthew Kahle (OSU)
Wednesday March 26, 2025 in SE 105: Cary Malkiewich (Binghamton)
Wednesday April 2, 2025 in SE 105: Hyungseop Kim (Université Paris-Saclay)

IU/PU/IUPUI Joint Topology Seminar

Fall 2014: Saturday November 8, 2014: IU/PU/IUPUI Joint Topology Seminar: Ralph Kaufman / Kate Ponto, Fall 2014 Seminar
Winter 2015: Saturday January 31, 2015: IU/PU/IUPUI Joint Topology Seminar: Jim Davis / Bert Guillou, IU/PU/IUPUI Joint Topology Seminar
Spring 2015: Saturday April 25, 2015: IU/PU/IUPUI Joint Topology Seminar: Carl McTeague / Jim McClure, IU/PU/IUPUI Joint Topology Seminar
Fall 2015: Saturday December 5, 2015: IU/PU/IUPUI Joint Topology Seminar: Charmaine Sia / Marco Varisco, Fall Seminar
Spring 2016: Saturday April 16, 2016: IU/PU/IUPUI Joint Topology Seminar: Jeremy Miller / Cary Malkievich, Spring 2016 Seminar
Fall 2016: Saturday November 12, 2016: IU/PU/IUPUI Joint Topology Seminar: Mentor Stafa / Andrew Putman, Fall 2016 Meeting
Fall 2017: Saturday November 4, 2017: IU/PU/IUPUI Joint Topology Seminar: Peter Patzt (Purdue) and Christopher Schommer-Pries (Notre Dame)
Fall 2018: Saturday October 13, 2018: IU/PU/IUPUI Joint Topology Seminar: Paul VanKoughnett (Purdue) and Ernie Fontes (Ohio State)
Fall 2022: Saturday September 24, 2022: IU/PU/IUPUI Joint Topology Seminar, Sanjana Agarwal (IU) and Sam Nariman (Purdue), TBA
Fall 2023: Saturday December 9, 2023: IU/PU/IUPUI Joint Topology Seminar, Ningchuan Zhang (IU) / Manuel Rivera (Purdue)
Fall 2024: Saturday September 28, 2024: IU/PU/IUPUI Joint Topology Seminar, Kate Ponto (UKY) / David Boozer (IU)

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