Past Topology Seminar

3 November 2014
John Lind

 Parametrized spectra are topological objects that represent
twisted forms of cohomology theories.  In this talk I will describe a theory
of parametrized spectra as highly structured bundle-like objects.  In
particular, we can make sense of the structure "group" of a bundle of
spectra.  This point of view leads to new examples and a good framework for
twisted equivariant cohomology theories.  


27 October 2014
Andre Henriques

Given a 3-dimensional TQFT, the "conformal blocks" are the
values of that TQFT on closed Riemann surfaces.
The construction that we'll present (joint work with Douglas &
Bartels) takes as only input the value of the TQFT on discs. Towards
the end, I will explain to what extent the conformal blocks that we
construct agree with the conformal blocks constructed e.g. from the
theory of vertex operator algebras.


20 October 2014
Andras Juhasz

 We describe a framework for defining and classifying TQFTs via
surgery. Given a functor 
from the category of smooth manifolds and diffeomorphisms to
finite-dimensional vector spaces, 
and maps induced by surgery along framed spheres, we give a set of axioms
that allows one to assemble functorial coboridsm maps. 
Using this, we can reprove the correspondence between (1+1)-dimensional
TQFTs and commutative Frobenius algebras, 
and classify (2+1)-dimensional TQFTs in terms of a new structure, namely
split graded involutive nearly Frobenius algebras 
endowed with a certain mapping class group representation. The latter has
not appeared in the literature even in conjectural form. 
This framework is also well-suited to defining natural cobordism maps in
Heegaard Floer homology.


13 October 2014
Ulrike Tillmann

Vector bundles over a compact manifold can be defined via transition 
functions to a linear group. Often one imposes 
conditions on this structure group. For example for real vector bundles on 
may  ask that all 
transition functions lie in the special orthogonal group to encode 
orientability. Commutative K-theory arises when we impose the condition 
that the transition functions commute with each other whenever they are 
simultaneously defined.

We will introduce commutative K-theory and some natural variants of it, 
and will show that they give rise to  new generalised 
cohomology theories.

This is joint work with Adem, Gomez and Lind building on previous work by 
Adem, F. Cohen, and Gomez.

16 June 2014
Piotr Nowak
In this talk I will discuss a deformation principle for cohomology with coefficients in representations on Banach spaces. The main idea is that small, metric perturbations of representations do not change the vanishing of cohomology in degree n, provided that we have additional information about the cohomology in degree n+1. The perturbations considered here happen only on the generators of a group and even for isometric representations give rise to unbounded representations. Applications include fixed point properties for affine actions and strengthening of Kazhdan’s property (T). This is joint work with Uri Bader.
2 June 2014
Stefan Schwede
The filtration on the infinite symmetric product of spheres by number of factors provides a sequence of spectra between the sphere spectrum and the integral Eilenberg-Mac Lane spectrum. This filtration has received a lot of attention and the subquotients are interesting stable homotopy types. In this talk I will discuss the equivariant stable homotopy types, for finite groups, obtained from this filtration for the infinite symmetric product of representation spheres. The filtration is more complicated than in the non-equivariant case, and already on the zeroth homotopy groups an interesting filtration of the augmentation ideal of the Burnside rings arises. Our method is by `global' homotopy theory, i.e., we study the simultaneous behaviour for all finite groups at once. In this context, the equivariant subquotients are no longer rationally trivial, nor even concentrated in dimension 0.
26 May 2014
Andras Stipsicz
Knot Floer homology (introduced by Ozsvath-Szabo and independently by Rasmussen) is a powerful tool for studying knots and links in the 3-sphere. In particular, it gives rise to a numerical invariant, which provides a nontrivial lower bound on the 4-dimensional genus of the knot. By deforming the definition of knot Floer homology by a real number t from [0,2], we define a family of homologies, and derive a family of numerical invariants with similar properties. The resulting invariants provide a family of homomorphisms on the concordance group. One of these homomorphisms can be used to estimate the unoriented 4-dimensional genus of the knot. We will review the basic constructions for knot Floer homology and the deformed theories and discuss some of the applications. This is joint work with P. Ozsvath and Z. Szabo.
19 May 2014
Tatiana Smirnova-Nagnibeda
An invariant random subgroup in a (finitely generated) group is a probability measure on the space of subgroups of the group invariant under the inner automorphisms of the group. It is a natural generalization of the the notion of normal subgroup. I’ll give an introduction into this actively developing subject and then discuss in more detail examples of invariant random subgrous in groups of intermediate growth. The last part of the talk will be based on a recent joint work with Mustafa Benli and Rostislav Grigorchuk.