Current Series

[View Past Series]

Mon Oct 01

Topology Seminar

3:30pm - Vincent Hall 301
Topology Seminar: Grothendieck-Teichmuller group and braided Hopf algebras
Craig Westerland, University of Minnesota

The primitives in a classical Hopf algebra form a Lie algebra (in fact, a Lie subalgebra of the Hopf algebra).  For a braided Hopf algebra, this is no longer the case.  Consequently, all of the structure theorems for Hopf algebras (e.g., the Milnor-Moore and Poincaré-Birkhoff-Witt theorems) break down in this setting. This is a report on ongoing work in which we construct an operad (a braided form of the Lie operad) which governs the algebraic structure of the primitives in a braided Hopf algebra.  We can interpret this operad in terms of the homology of covering spaces of the 2-dimensional little disks operad.  This gives rise to an action of Drinfeld's Grothendieck-Teichmuller group on this operad which may be related to Drinfeld's original definition of this group.

Mon Oct 08

Topology Seminar

3:30pm - Vincent Hall 301
Topology Seminar: A chromatic approach to tmf cooperations
Paul van Koughnett, Purdue

The topological modular forms spectrum, tmf, is a cohomology theory constructed from elliptic curves that captures information about stable homotopy at chromatic heights less than or equal to 2. We present a description of the height 1 part of the algebra of tmf cooperations, and interpret it in terms of modular forms. This is joint work with Dominic Culver.

Mon Oct 15

Topology Seminar

3:30pm - Vincent Hall 301
Topology Seminar: Rigidity of warped cones and coarse geometry of expanders
Wouter van Limbeek, UIC

Finitely generated subgroups of compact Lie groups give rise to expander graphs via a warped cone construction. We study the dependence of the coarse geometry of such expander graphs on the original subgroup and establish a dynamical analogue of quasi-isometric rigidity theorems in geometric group theory: Namely, the coarse geometry of the warped cone determines the subgroup up to commensurability, unless the group has abelian factors. This is joint work with David Fisher and Thang Nguyen.

Mon Oct 22

Topology Seminar

3:30pm - Vincent Hall 301
Topology Seminar: Finitely generated sequences of linear subspace arrangements
Nir Gadish, University of Chicago

Hyperplane arrangements are a classical meeting point of topology, combinatorics and representation theory. Generalizing to arrangements of linear subspaces of arbitrary codimension, the theory becomes much more complicated. However, a crucial observation is that many natural sequences of arrangements seem to be defined using a finite amount of data.

In this talk I will describe a notion of 'finitely generation' for collections of arrangements, unifying the treatment of known examples. Such collections turn out to exhibit strong forms of stability, both in their combinatorics and in their cohomology representation. This structure makes the appearance of representation stability transparent and opens the door to generalizations.

Mon Oct 29

Topology Seminar

3:30pm - Vincent Hall 301
Topology Seminar: Symmetric monoidal categories and \Gamma-categories
Amit Sharma, Kent State University

A \Gamma-category is a functor from the category of finite based sets and basepoint preserving functions \Gamma^op to Cat. We construct a model category structure on the category of \Gamma-categories, which is symmetric monoidal closed to the Day convolution product. The fibrant objects in this model category structure are those
\Gamma-categories which are often called special \Gamma-categories. The main objective of this research is to establish a Quillen equivalence between a natural model category structure on the category of (small) permutative categories and strict symmetric monoidal functors Perm and our model category structure on \GammaCat. The weak equivalences of the natural model category structure are equivalences of underlying categories. In the paper [1], Segal defined a functor from the category of (small) symmetric monoidal categories into \GammaCat which can be described as a nerve functor for symmetric monoidal categories. The right adjoint \bar{K} of our Quillen equivalence is a thickening of Segal's nerve functor. We construct a permutative category L called Leinster's category, having the universal property that each \Gamma-category extends uniquely to a symmetric monoidal functor along an inclusion functor \Gamma^op —> L. The left adjoint L of our Quillen equivalence is a composite functor composed of the symmetric monoidal extension functor indicated above followed by a homtopy colimit functor. In the paper [2], Mandell had shown that Segal's nerve functor (followed by the ordinary nerve functor) induces an equivalence between a homotopy category of Perm, obtained by inverting those strict symmetric monoidal functors which induce a weak homotopy equivalence of simplicial sets upon applying the nerve functor, and a homotopy category of \Gamma-spaces \GammaS obtained by inverting pre-stable equivalences which are those maps of \Gamma-categories which induce a degreewise weak homotopy equivalence of simplicial sets upon applying an E_\infty-completion functor. The objective of Mandell's work is to understand the relation between connective spectra and \Gamma-spaces obtained by applying the Segal's nerve functor to symmetric monoidal categories whereas our objective is to construct a model category of symmetric monoidal categories which is
symmetric monoidal closed.

References:
[1] G. Segal, Categories and cohomology theories, Topology 13 (1974) 293-312.
[2] M. A. Mandell, An Inverse K-theory functor, Doc. Math 15 (2010) 765-791.

Mon Nov 05

Topology Seminar

3:30pm - Vincent Hall 301
Topology Seminar: An isovariant Elmendorf's theorem
Sarah Yeakel, University of Maryland

An isovariant map is an equivariant map which preserves isotropy groups. Isovariant maps show up in equivariant surgery theory and in other settings when homotopy theory is applied to geometry. For a finite group G, we consider the category of G-spaces with morphisms given by isovariant maps. We will discuss a cofibrantly generated model structure on this category, along with isovariant versions of Elmendorf's theorem and a theorem of Klein and Williams about homotoping a map off a submanifold.

Mon Nov 12

Topology Seminar

3:30pm - Vincent Hall 301
Topology Seminar: Localizing the E_2 page of the Adams spectral sequence
Eva Belmont, Northwestern

The Adams spectral sequence is one of the central tools for calculating the stable homotopy groups of spheres, one of the motivating problems in stable homotopy theory. In this talk, I will discuss an approach for computing the Adams E_2 page for the sphere at p = 3 in an infinite region, by computing its localization by the non-nilpotent element b_{10}. This approach relies on computing an analogue of the Adams spectral sequence in Palmieri's stable category of comodules, which can be regarded as an algebraic analogue of stable homotopy theory. This computation fits in the framework of chromatic homotopy theory in the stable category of comodules.

Mon Nov 19

Topology Seminar

3:30pm - Vincent Hall 301
Topology Seminar: The spectrum of units of a height 2 theory
Jeremy Hahn, Harvard University

The space BSU admits two infinite loop space structures, one arising from addition of vector bundles and the other from tensor product. A surprising fact, due to Adams and Priddy, is that these two infinite loop spaces become equivalent after p-completion. I will explain how the Adams-Priddy theorem allows for an identification of sl_1(ku_p), the spectrum of units of p-complete complex K-theory. I will then describe work, joint with Andrew Senger, that attempts to similarly understand the spectrum of units of the 2-completion of tmf_1(3).

Mon Nov 26

Topology Seminar

3:30pm - Vincent Hall 301
Topology Seminar: Cohomology of arithmetic groups and characteristic classes of manifold bundles
Bena Tshishiku, Harvard University

A basic problem in the study of fiber bundles is to compute the ring H*(BDiff(M)) of characteristic classes of bundles with fiber a smooth manifold M. When M is a surface, this problem has ties to algebraic topology, geometric group theory, and algebraic geometry. We have a good understanding of the cohomology in the "stable range", but this accounts for a small percentage of the total cohomology, and little is known beyond that. In this talk I describe some new characteristic classes (in the case dim M >>0) that come from the unstable cohomology of arithmetic groups.

Fri Dec 07

Topology Seminar

3:30pm - 301 Vincent Hall
Topology Seminar: Power operations in normed motivic spectra
Tom Bachmann, MIT

In joint work with M. Hoyois, we established (the beginnings of) a theory of "normed motivic spectra". These are motivic spectra with some extra structure, enhancing the standard notion of a motivic E_oo-ring spectrum (this is similar to the notion of G-commutative ring spectra in equivariant stable homotopy theory). It was clear from the beginning that the homotopy groups of such normed spectra afford interesting *power operations*. In ongoing joint work with E. Elmanto and J. Heller, we attempt to establish a theory of these operations and exploit them calculationally. I will report on this, and more specifically on our proof of a weak motivic analog of the following classical result of Würgler: any (homotopy) ring spectrum with 2=0 is generalized Eilenberg-MacLane.

Mon Dec 10

Topology Seminar

3:30pm - Vincent Hall 301
Topology Seminar: The geometric average size of Selmer groups over function fields
Aaron Landesman, Stanford

We use topological methods to investigate the average size of n-Selmer groups of elliptic curves over F_q(t).
Loosely speaking, the n-Selmer group of an elliptic curve measures objects which look like the n-torsion of the elliptic curve.
We relate the question of computing the average size of the n-Selmer group to demonstrating
homological stability for a sequence of moduli spaces
of these n-Selmer elements.
Via monodromy arguments, we show the number of components of these moduli spaces stabilizes, which determines the
average size after taking a large q limit.

Mon Sep 09

Topology Seminar

2:30pm - Ford Hall 110
Cohomology of the space of complex irreducible polynomials in several variables
Weiyan Chen, University of Minnesota

We will show that the space of complex irreducible polynomials of degree d in n variables satisfies two forms of homological stability: first, its cohomology stabilizes as d increases, and second, its compactly supported cohomology stabilizes as n increases. Our topological results are inspired by counting results over finite fields due to Carlitz and Hyde.

Mon Sep 16

Topology Seminar

2:30pm - Ford Hall 110
Transfer in the homology and cohomology of categories
Peter Webb, University of Minnesota

The cohomology of a category has many properties that extend those that are familiar when the category is a group. Second cohomology classifies equivalence classes of category extensions, first cohomology parametrizes conjugacy classes of splittings, first homology is the abelianization of the fundamental group, and second homology has a theory that extends that of the Schur multiplier. Defining restriction and corestriction maps on the homology of categories is problematic: most attempts to do this require induction and restriction functors to be adjoint on both sides, and this typically does not happen with categories. We describe an approach to defining these maps that includes all the situations where they can be defined in group cohomology, at least when the coefficient ring is a field. The approach uses bisets for categories, the construction by Bouc and Keller of a map on Hochschild homology associated to a bimodule, and the realization by Xu of category cohomology as a summand of Hochschild cohomology.

Mon Sep 23

Topology Seminar

2:30pm - Ford Hall 110
Homology class of Deligne-Lusztig varieties
Dongkwan Kim, University of Minnesota

Since first defined by Deligne and Lusztig, a Deligne-Lusztig variety has become unavoidable when studying the representation theory of finite groups of Lie type. This is a certain subvariety of the flag variety of the corresponding reductive group, and its cohomology groups are naturally endowed with the action such finite groups, which in turn gives a decomposition of irreducible representations called Lusztig series. In this talk, I will briefly discuss the background of Deligne-Lusztig theory and provide a formula to calculate the homology class of Deligne-Lusztig varieties in the Chow group of the flag variety. If time permits, I will also discuss their analogues.

Mon Sep 30

Topology Seminar

2:30pm - Ford Hall 110
Mysterious Duality
Sasha Voronov, University of Minnesota

“Mysterious Duality” was discovered by Iqbal, Neitzke, and Vafa in 2001. They noticed that toroidal compactifications of M-theory lead to the same series of combinatorial objects as del Pezzo surfaces do, along with numerous mysterious coincidences: both toroidal compactifications and del Pezzo surfaces give rise to the exceptional series E_k; the U-duality group corresponds to the Weyl group W(E_k), arising also as a group of automorphisms of the del Pezzo surface; a collection of various M- and D-branes corresponds to a set of divisors; the brane tension is related to the “area” of the corresponding divisor, etc. The mystery is that it is not at all clear where this duality comes from. In the talk, I will present another series of mathematical objects: certain versions of multiple loop spaces of the sphere S^4, which are, on the one hand, directly connected to M-theory and its combinatorics, and, on the hand, possess the same combinatorics as the del Pezzo surfaces. This is a report on an ongoing work with Hisham Sati.

Mon Oct 07

Topology Seminar

2:30pm - Ford Hall 110
Topology Seminar
Craig Westerland, University of Minnesota

TBA

Mon Oct 14

Topology Seminar

2:30pm - Ford Hall 110
Topology Seminar
TBA
Mon Oct 21

Topology Seminar

2:30pm - Ford Hall 110
Topology Seminar
TBA
Mon Oct 28

Topology Seminar

2:30pm - Ford Hall 110
Topology Seminar
Elden Elmanto, Harvard University

TBA

Mon Nov 04

Topology Seminar

2:30pm - Ford Hall 110
Topology Seminar
TBA
Mon Nov 11

Topology Seminar

2:30pm - Ford Hall 110
Topology Seminar
TBA
Mon Nov 18

Topology Seminar

2:30pm - Ford Hall 110
Topology Seminar
TBA
Mon Nov 25

Topology Seminar

2:30pm - Ford Hall 110
Topology Seminar
TBA
Mon Dec 02

Topology Seminar

2:30pm - Ford Hall 110
Topology Seminar
TBA
Mon Dec 09

Topology Seminar

2:30pm - Ford Hall 110
Topology Seminar
TBA