Notre Dame Topology Seminar

Felix Klein Seminar (on geometric things)
Notre Dame Department of Mathematics

Questions? Contact Mark Behrens

Fall Semester 2014 Schedule

3:15–4:15 (Eastern Time) Thursdays @ 258 Hurley Hall

30 October: Michael Mandell (University of Indiana); The homotopy groups of K(S)
6 November: Cary Malkiewich (UIUC); Coassembly in algebraic K-theory
MONDAY,10 November, 3:30-4:30: Andrew Ranicki (University of Edinburgh); The total surgery obstruction
13 November: Rune Haugseng (MPIM); The higher Morita category of E_n-algebras
4 December: Charles Rezk (UIUC); Power operations and elliptic cohomology

Spring Semester 2015 Schedule

3:00–4:00 (Eastern Time) Tuesdays @ 258 Hurley Hall, unless otherwise noted

24 February: Ezra Getzler (Northwestern); Towards a symplectic form on the derived stack of perfect complexes
11:00-12:00, 3 March, DEBARTOLO 213: Theo Johnson-Freyd (Northwestern); Twisted field theories and higher-categorical (op)lax transfors
4:00-5:00, MONDAY, 20 April, 258 Hurley: Agnes Beaudry (Chicago); The Chromatic Splitting Conjecture at n=p=2
3:45-4:45, WEDNESDAY, 29 April, 229 HAYES-HEALY: Inna Zakharevich (Chicago); Analyzing the Grothendieck ring of varieties using K-theory

Abstracts of Invited Talks

30 October 2014: Michael Mandell

The homotopy groups of K(S)

This talk will describe joint work with Andrew Blumberg where we build on work of Rognes to compute the homotopy groups of K(S) in terms of the homotopy groups of the sphere spectrum, the homotopy groups of CP_-1^oo, and the homotopy groups of K(Z). If time permits, I will say something about nilpotence of the multiplicative structure.

6 November 2014: Cary Malkiewich

Coassembly in algebraic K-theory

The coassembly map allows us to approximate any contravariant homotopy-invariant functor by an excisive functor, i.e. one that behaves like a cohomology theory. We apply this construction to a contravariant form of Waldhausen's algebraic K-theory of spaces, and its corresponding THH functor. The results are somewhat surprising: a certain dual form of the A-theory Novikov conjecture is false, but when the space in question is the classifying space BG of a finite p-group, coassembly on THH is split surjective after p-completion. The method of proof suggests new conjectures about both the assembly and coassembly maps for the A-theory of BG. If there is time, we will also discuss related work on the equivariant structure of THH.

10 November 2014: Andrew Ranicki

The total surgery obstruction

The total surgery obtruction invariant was introduced 35 years ago to unify the two stages of the classical Browder-Novikov-Sullivan-Wall surgery theory of n-dimensional topological manifold types in the homotopy types of spaces with n-dimensional global Poincare duality, with n>4. The invariant quantifies the failure of local Poincare duality, vanishing if and only if the space is homotopy equivalent to a topological manifold. Moreover, the various vanishings quantify the topological manifold types. The talk will review progress in the total surgery obstruction theory.

13 November 2014: Rune Haugseng

The higher Morita category of E_n-algebras

I will discuss a construction of a higher category of E_n-algebras and iterated bimodules, generalizing the classical bicategory of algebras and bimodules. This leads to generalizations of the Picard and Brauer groups, which have been studied in stable homotopy theory as interesting invariants of ring spectra, and should also lead to an "algebraic" construction of factorization homology as an extended topological quantum field theory.

4 December 2014: Charles Rezk

Power operations and elliptic cohomology

24 February 2015: Ezra Getzler

Towards a symplectic form on the derived stack of perfect complexes

Shulman (following ideas of Bott) constructed closed differential forms in the total de Rham complex of the nerve of a compact Lie group which realize the Chern-Weil correspondence. (In the case of U(n), these classes lift to differential characters by the work of Cheeger and Simons: it would be very interesting to extend this to all groups. This has been done for classes in H^4 when G is simple by Gawedzki and collaborators, where this problem is seen as corresponding to quantization of the so-called WZW model.) Shulman's main theorem is that the differential form associated to an invariant polynomial of degree k is, in the terminology of Pantev, Toën, Vacquié and Vezzosi, a form of degree 2k in the complex of closed k-forms. The second Chern class defines a "shifted symplectic form" on this stack. Toën and Vezzosi have extended this class to the derived stack where one considers not the groupoid of isomorphisms between vector spaces, but rather the higher groupoid of quasi-isomorphisms between perfect complexes. Their proof of its existence invokes Lurie’s proof of the cobordism hypothesis. In this talk, I describe partial results in constructing this form explicitly.

11:00-12:00, 3 March 2015, DEBARTOLO 213: Theo Johnson-Freyd

Twisted field theories and higher-categorical (op)lax transfors

A "Schrodinger picture" (extended) quantum field theory is a functor from some (higher) category of "spacetimes" to some (higher) category of "Hilbert spaces". This framework is powerful and well-studied. Unfortunately, it does not capture many important examples. Instead, most interesting quantum field theories are best described as "morphisms" of some sort between functors from the category of spacetimes --- these are called "twisted" or "relative" or "Heisenberg picture" quantum field theories. The most natural notion of "morphisms of functors" is "natural transformation." Unfortunately, plain (i.e. "strong") natural transformations still fail to accommodate most examples. Instead, what is needed are "lax" or "oplax" natural transformations. In this talk, based on joint work with Claudia Scheimbauer, I will describe the definition of "(op)lax natural transformation" between functors of higher categories, and discuss qualitative differences between "lax" and "oplax" twisted quantum field theories.

4:00-5:00, MONDAY, 20 April 2015: Agnes Beaudry

The Chromatic Splitting Conjecture at n=p=2

In its strongest form, the chromatic splitting conjecture gives a precise description of the homotopy type of L_1L_K(2)S, which has been shown to hold for p ≥ 5 by Hopkins and for p=3 by Goerss, Henn and Mahowald. In this talk, I will explain why this description cannot hold at the prime p=2. More precisely, let V(0) be the mod 2 Moore spectrum. I will give a summary of how one uses the duality resolution techniques to show that π_kL_1L_K(2)V(0) is not zero when k is congruent to 5 modulo 8. I will explain how this contradicts the decomposition of L_1L_K(2)S predicted by the chromatic splitting conjecture.

3:45-4:45, WEDNESDAY, 29 April 2015, HAYES-HEALY 229: Inna Zakharevich

Analyzing the Grothendieck ring of varieties using K-theory

The Grothendieck ring of varieties, K_0[V_k] was introduced by Grothendieck in a 1964 letter to Serre. It is defined to be the free abelian group generated by varieties (over a fixed field k), under the relation that if Y is a closed subvariety of X then [X] = [Y] + [X \ Y]. Multiplication is defined via the Cartesian product. Thus, in particular, if X and Y are varieties which are piecewise isomorphic then [X] = [Y] in the ring. There are two important structural questions about this ring: 1. If two varieties X and Y satisfy [X] = [Y], are they piecewise isomorphic? 2. Is the class of the affine line a zero divisor? The answers to these questions are "no" and "yes", respectively; this was shown in a recent paper of Borisov, where he constructed an example to resolve question (2), which also coincidentally answered question (1). In this talk we will analyze the structure of this ring using techniques of algebraic K-theory. In particular, we will construct a spectrum K(V_k) whose pi_0 is the Grothendieck ring of varieties, and whose higher homotopy groups contain further geometric information about piecewise isomorphisms of varieties. We will then analyze the geometry of this spectrum to show that the kernel of multiplication by the affine line is generated by varieties X and Y such that [X] = [Y] but X and Y are not piecewise isomorphic, thus showing that Borisov's coincidence is not a coincidence at all.