Purdue University

Department of Mathematics
Topology Seminar


Fall 2009
Time and place: Thursdays 4:30-5:30 in MATH 211

Organizer: Ralph Kaufmann. Contact organizer.


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.