Gunnar Carlson (Stanford): Representations and K-theory
Recent years have seen striking developments in the calculation
of the algebraic K-theory of fields. Specifically, there is a
spectral sequence converging to the algebraic K-theory of a
field \(F\), whose \(E_2\)-term is describable in terms of
cohomology groups of the absolute Galois group \(G_F\). In this
talk, we will describe a construction which constructs the
K-theory spectrum of F itself from the representation theory of
\(G_F\) for a wide class of fields. Byproducts are
expressions of completed Milnor K-groups in terms of the
homotopy groups of a derived version of completion of the
representation ring of \(G_F\).
Fred Cohen
(U. of Rochester):
Moment-angle complexes, Polyhedral products and their
applications
This talk
will give descriptions of the polyhedral product functor, and
applications in joint work. with T. Bahri, M. Bendersky, S.
Gitler. Further evolving applications with L. Taylor, F.
Callegaro,
and M. Salvetti will also be given .
Dan Dugger (U. of
Oregon): Characteristic
classes for Z/2-equivariant bundles
I
will describe a recent computation of the \(RO(G)\)-graded,
Eilenberg-MacLane cohomology of real, equivariant Grassmannians
for the group \(G={\mathbf Z}/2\). The answer exhibits
several interesting patterns related to the combinatorics of
partitions, and the proof involves some new
techniques. I will explain how this work ties in to
some questions about characteristic classes for
quadratic bundles with values in mod 2 motivic cohomology.
Qayum Khan
(Notre Dame): Rigidity of
pseudo-free group actions on contractible manifolds
We prove
the following equivariant topological rigidity theorem.
Let \(X\) be a contractible Riemannian \(n\)-manifold of
nonpositive sectional curvature. Let \(\Gamma\) be a
discrete cocompact group of isometries of \(X\). Suppose
\(\Gamma\) is virtually torsionfree and each nontrivial finite
subgroup \(H\) of \(\Gamma\) has finite normalizer
\(N_{\Gamma}(H)\).
Assume \(n \geq 5\). Then there is a
unique \(\Gamma\)-homeomorphism class of topological manifolds
\(M\) equipped with a cocompact proper \(\Gamma\)-action if
\(\Gamma\) is 2-torsionfree, or if \(n \equiv 0, 1 \pmod{4}\).
For the other cases, where \(\Gamma\) has elements of
order two and \(n \equiv 2, 3 \pmod{4}\), we show there are
infinitely many such classes, and we completely classify them
using quadratic forms and quadratic linking forms over the
function field \(\mathbb{F}_2(t)\).
The motivating special case comes from
lifting the involutions on the \(n\)-torus \(T^n\) that induce
negative the identity map on the first integral homology
group.
Matilde Marcolli
(Caltech). Noncommutative
numerical motives and the Tannakian formalism
I will
describe recent work with Goncalo Tabuada, where we consider
analogs of Grothendieck's standard conjectures C and D for a
suitable category of noncommutative numerical motives and we
show that, assuming these conjectures, one can make this
category into a Tannakian category. The motivic Galois group
of this category surjects onto the kernel of the homomorphism
from the motivic Galois group of the category of (commutative)
numerical motives to the multiplicative group, determined by
the inclusion of the subcategory of Tate motives.
Lizhen Qin
(Purdue): A Bona Fide
CW Decomposition of Compact Manifolds Resulting from Morse
Theory
Consider a
Morse function on a compact Riemannian manifold. This manifold
is a disjoint union of unstable manifolds of the negative
gradient of the Morse function. These unstable manifolds are
open cells. Does this give a CW complex which is homeomorphic
(not only homotopy equivalent) to the compact manifold? This
talk will describe the history of the above problem and give
an affirmative answer to it.
Sasha Voronov (U. of
Minnesota): A higher category
of cobordisms and TQFT
I will describe a categorical formalism for (Extended)
Topological Quantum Field Theories (TQFTs) and present them as
functors from a suitable category of cobordisms with corners to
a (linear) category, generalizing open-closed string theory to
higher dimensions. The approach is based on the notion of an
n-fold category by C. Ehresmann, weakened in the spirit of
monoidal categories (associators, interchangers, Mac Lane's
pentagons and hexagons), in contrast with the simplicial (weak
Kan and complete Segal) approach of Jacob Lurie.