Topology at IU

Seminars During Academic Year 2014-15

Fall 2014

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.

Spring 2015

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.

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