09/03:
Ralph Kaufmann (Purdue):
Open/Closed Hochschild actions: a universal formula and a few surprises
Abstact:
We discuss the generalization of our Hochschild actions to the open
closed case. There is a generalization of our universal formula in this
situation. However the construction of the action is not as
straightforward as one might expect. This leads to new somewhat
unexpected restrictions.
09/10: Peter Albers (Purdue): Symplectic topology of cotangent bundles
and its connection to loop spaces.
Abstract:
We discuss two symplectic invariants of cotangent bundles T*B:
symplectic homology and Rabinowitz Floer homology. Both are closely
related to the homology of the free loop space of B. Then we explain
old and new applications of this to symplectic topology and dynamical
systems. The new applications are joint work with Urs Frauenfelder.
09/14: Andrei Gabrielov
(Purdue): Triangulation of monotone families of compact sets.
Abstract:
Let $S_t$, be a one-parametric family of compact sets in a compact
$K\subset R^n$, such that $S_u\subset S_t$ for $0<t<u$.
If the family $S_t$ and the compact $K$ are ``tame'' (for example, real
semialgebraic) we construct a triangulation of $K$ such that, for each
open simplex
$\sigma$, the family $\sigma\cap S_t$ is homeomorphic to one of the
``standard'' families.
10/08: Javier Zuniga
(Purdue): Bordered Ribbon Graphs.
Abstract:
I will sketch the construction of a cellular
decomposition for the space of decorated bordered Riemann surfaces and
for one of its compactifications. This extends the definition of stable
ribbon graphs to the bordered case and gives a chain model computing
the homology of the moduli spaces of bordered Riemann surfaces
10/22: Dan Edidin (U. of
Missouri): Logarithmic Restriction and Orbifold Products
Abstract:
A variant on a classical question about eigenvalues of sums of
Hermitian matrices is the following: What are the possible eigenvalues
for a collection of unitary matrices whose product is the identity? I
will explain how a fundamental inequality (proved by Falbel and
Wentworth) for the logarithms of eigenvalues of unitary matrices can be
used to define an operation on vector bundles
called the logarithmic restriction. This allows us to show, without
reference to orbicurves, that a certain K-theory class defined by
Jarvis, Kaufmann and Kimura is canonically represented by a vector
bundle. In turn this allows us a to give a purely equivariant
definition of orbifold products for quotient Deligne-Mumford stacks.
This talk is based on joint work with Tyler Jarvis and Takashi Kimura.
10/29:
Mohammed Abouzaid (MIT):
Topological models for Fukaya categories.
Abstract:
Extending ideas of Floer and Donaldon, Fukaya assigned to each
symplectic manifold a category whose objects are Lagrangian
submanifolds. Unfortunately, there is no procedure in general for
computing such categories. In this talk, I will explain how, in
the case of the cotangent bundle of a smooth manifold, understanding
the Fukaya category reduces to understanding the Pontryagin product on
the based loop space. Time permitting, I will explain some
generalisations of this idea beyond the case of cotangents.
11/12:
Philip Hackney (Purdue)
: Homology
operations in the spectral sequence of a cosimplicial space
Abstract:
Given a cosimplicial space $X$, the spectral sequence of the title
abuts to the mod 2 homology of $\operatorname{Tot}(X)$. If $X$ is a
cosimplicial infinite loop space, then $\operatorname{Tot} (X)$ is an
infinite loop space, so the target of the spectral sequence admits
Araki-Kudo operations; our natural impulse is to lift these operations
to the spectral sequence. This was accomplished by Jim Turner in 1998.
We take a different approach and define, for any cosimplicial space
$X$, external operations which land in the spectral sequence associated
to
$E C_2 \times_{C_2} X \times X$. In the case when $X$ is a cosimplicial
infinite loop space there is a cosimplicial map
$E C_2 \times_{C_2} X \times X \to X$ which induces the internal
operations.
