Midwest Topology Seminar

April 2-3, 2011
University of Chicago


All the talks will be held in Ryerson 251.

Saturday April 2

8:45-9:30 Coffee

9:30-10:30 Benson Farb (University of Chicago)
Representation Theory and Homological Stability

In this talk I will explain some recent work with Tom Church. The story begins when we were working out some cohomology computations for a problem in geometric topology. Hints of a pattern emerged, but we struggled to find a way to describe it. We finally developed a language to do this, and called the phenomenon "representation stability".

As we began to look more broadly we started to see representation stability in many different areas of mathematics: from group cohomology to the topology of configuration spaces to classical representation theory to flag varieties to Lie algebras to algebraic combinatorics. We have been able to apply representation stability to prove theorems and make new predictions in these directions (some since proved by others). Many conjectures remain.

My goal in this talk will be to explain representation stability through examples and applications. I will also explain how Church, Jordan Ellenberg and I are applying this theory in order to compute and explain various combinatorial statistics in number theory.

11:00-12:00 Bjørn Dundas (University of Bergen)
Fixed points of smash powers of commutative ring spectra

Let A be a connective commutative ring spectrum. The Redshift Conjecture suggests that the algebraic K-theory K(A) sees more periodic phenomena than A does. The conjecture is supported by calculations by Ausoni, Bokstedt, Hesselholt, Madsen and Rognes, but there seems to be no K-theoretic reason for believing the conjecture.

However, this sort of redshift would follow if a certain interplay beween finite group actions and periodic phenomena of the fixed points of smash powers of A is a piece of a more general picture. Ideally one would hope to see a higher chromatic continuation of the Burnside-Witt-construction which lifts from finite to infinite characteristic. In the talk I will try to isolate key questions and test situations as well as review what is known.

12:00-1:30 Lunch break

1:30-2:30 Gonçalo Tabuada (University of Lisbon)
The fundamental theorem via derived Morita invariance, localization, and A1-homotopy invariance

In this talk I will prove that every functor defined on dg categories, which is derived Morita invariant, localizing, and A1-homotopy invariant, satisfies the fundamental theorem. As an application, we recover in a unified and conceptual way, Weibel and Kassel’s fundamental theorems in homotopy algebraic K-theory, and periodic cyclic homology, respectively.

2:30-3:00 Coffee

3:00-4:00 Jim McClure (Purdue University)
The Ranicki orientation of STop bundles is an E ring map

(Joint with Gerd Laures). In his MIT notes on Geometric Topology, Sullivan showed that STop bundles have a canonical KO orientation away from 2. Later, Ranicki observed that STop bundles have a canonical orientation with respect to the spectrum L(Z) (that is, the symmetric L-theory of Z); the latter spectrum is equivalent to KO away from 2. In the lecture, I will outline a proof that the Ranicki orientation gives an E ring map from MSTop to L(Z).

4:30-5:30 Chris Schommer-Pries (Massachusetts Institute of Technology)
Dualizability and Locality in 3D Topological Field Theory

Higher category theory provides a rich background in which algebraic notions, such as the dualizability of objects in a category, are on a equal footing with topological notions, such as homotopy actions and homotopy fixed point spaces. As such, it also provides an ideal context in which to understand the interplay between these types of structures.

In this talk I will report on recent work which shows that fusion categories are fully-dualizable objects in a certain natural 3-category and identifies the induced O(3)-action on the moduli space of fusion categories. Fusion categories themselves are well-known and arise in several areas of mathematics and physics - conformal field theory, operator algebras, representation theory of quantum groups, and others.

In light of Hopkins' and Lurie's work on the cobordism hypothesis, this provides a fully local 3D TQFT for arbitrary fusion categories. Moreover by understanding various homotopy fixed point spaces, we will uncover how many familiar structures from the theory of fusion categories are given a natural explanation from the point of view of 3D TQFTs. This is joint work with Christopher Douglas and Noah Snyder.

Sunday April 3

8:45-9:30 Coffee

9:30-10:30 Vesna Stojanoska (Northwestern University)
Duality and Topological Modular Forms

It has been observed that certain localizations of the spectrum of topological modular forms tmf are self-dual (Mahowald-Rezk, Gross-Hopkins). We provide an integral explanation of these results that is internal to the geometry of the (compactified) moduli stack of elliptic curves Mell, yet is only true in the derived setting. When p is inverted, choice of level-p-structure for an elliptic curve provides a geometrically well-behaved cover of Mell, which allows one to consider tmf as the homotopy fixed points of tmf(p), topological modular forms with level-p -structure, under a natural action by GL2(Z/p). Specializing to p=2 or p=3 we obtain that as a result of Grothendieck-Serre duality, tmf(p) is self dual. The vanishing of the associated Tate spectra then makes tmf itself Anderson self-dual.

,br> 11:00-12:00 Stefan Stolz(Notre Dame)
Equivariant deRham cohomology via gauged field theories

Motivated by physics, Graeme Segal defined the notion of a d-dimensional field theory over a manifold X. In joint work with Peter Teichner, we showed that concordance classes of supersymmetric field theories over X correspond to DeRham cohomology for d=0 (joint with Hohnhold and Kreck), K-theory for d=1, and conjectured that topological modular form theory is obtained for d=2. In this talk, we show that for d=0 this can be extended to the equivariant context by showing that supersymmetric gauged field theories over a G-manifold X lead to the equivariant deRham cohomology of X. This is joint work with Fei Han, Chris Schommer-Pries and Peter Teichner.

12:00-1:30 Lunch break

1:30-2:30 Paul Goerss (Northwestern University)
On the chromatic splitting conjecture

In the chromatic picture of stable homotopy theory, the homotopy type of a finite p-local spectrum X is reassembled from its various monochromatic layers, which isolate very specific sorts of periodic phenomena in the homotopy groups. In the early 1990s, Hopkins proposed an ingenious conjecture for how the reassembly process works. I'll review the conjecture and the state of the art - including a verfication of the conjecture at p=3 and chromatic level 2, where the question is not simply algebraic, and where we might have looked for a counterexample. This is joint work with Hans-Werner Henn.

2:45-3:45 Bertrand Guillou (University of Illinois)
G-spectra are spectral Mackey functors