Topology at IU

Seminars During Academic Year 2017-18

Fall 2017

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).

Spring 2018

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.

IU/PU/IUPUI Joint Topology Seminar

Fall 2017: Saturday November 4, 2017: IU/PU/IUPUI Joint Topology Seminar: Peter Patzt (Purdue) and Christopher Schommer-Pries (Notre Dame)