Notre Dame Topology Seminar

Spring 2008 Schedule

Tuesdays from 9:45-10:45 a.m.
Hurley 258
(unless otherwise noted)

Date	Speaker/Affiliation	Title
January 29	Working Seminar on Equivariant Topology
February 5	Mike Shulman (University of Chicago)	Parametrized monoidal structures and framed bicategories
February 12	Working Seminar on Equivariant Topology
February 19	Working Seminar on Equivariant Topology
February 26	No Seminar
March 4	Spring Break -- No Seminar
March 11	Working Seminar on Equivariant Topology
March 18	Working Seminar on Equivariant Topology
March 25	Working Seminar on Equivariant Topology
April 1	No Seminar
April 8 (9 am intro talk, followed by 10 am talk)	Ralph Kaufmann (Purdue University)	Operations on Hochschild co-chains: solving and generalizing Deligne's conjecture using surfaces
April 15 (9 am intro talk, followed by 10 am talk)	Urs Schreiber (University of Hamburg, Germany)	On nonabelian differential cohomology
April 22
April 29

Spring 2008 Abstracts

February 5, 2008: Mike Shulman

Parametrized monoidal structures and framed bicategories Frequently in mathematics we see a collection of categories parametrized by objects of some "base" category, such as modules parametrized by rings, manifolds by their boundaries, or ex-spaces by their base spaces. Many examples clearly have monoidal structures and/or are enriched in some sense; the problem is to make this precise. It turns out that there are different kinds of parametrized monoidal structure. One particularly useful kind, exemplified by the tensor product of bimodules and the gluing of cobordisms, can be described using "framed bicategories". Classical notions such as monoidal functors, enriched categories, and duality generalize naturally from monoidal categories to framed bicategories. I will describe this theory, its relationship to other parametrized monoidal structures, and applications to parametrized enriched category theory and a definition of parametrized ring spectra.

April 8, 2008: Ralph Kaufmann

Operations on Hochschild co-chains: solving and generalizing Deligne's conjecture using surfaces We will discuss how to build complexes from surfaces with extra structures and use these complexes to give algebraic operations. The first such example endows a vector space with the structure of a unital commutative associative algebra with a non-degenerate pairing. This is commonly known as a Frobenius algebra. Allowing different structures and applying this to multilinear functions, we obtain a Gerstenhaber Bracket and other operators appearing in Deligne's conjecture and String Topolology. In the most general setting we obtain operations corresponding to moduli spaces of surfaces.

April 15, 2008: Urs Schreiber

On nonabelian differential cohomology Nonabelian cohomology classifies higher bundles (higher gerbes). These higher bundles may be equipped with connection structure, thus yielding nonabelian differential cohomology. We discuss explicit realization of these structures in terms of parallel transport n-functors and L-infinity-valued differential forms.

Fall 2007 Schedule

Date

Speaker/Affiliation

Title

September 18 (**12:30 pm in Hurley 258**)

Nathan Habegger (University of Nantes-France)

On the work of Xiao-Song Lin

September 25

Liang Kong (Max-Planck-Institute for Mathematics, Bonn, Germany)

An introduction to open-closed conformal field theory

September 27 (**3:30 pm in Hayes-Healy 229**)

Liang Kong (Max-Planck-Institute for Mathematics, Bonn, Germany)

An introduction to open-closed conformal field theory, Part II

October 4 (**3:30 pm in Hayes-Healy 229**)

Chris Schommer-Pries (Berkeley)

Two-Dimensional Topological Quantum Field Theories

October 11 (**3:00 pm in Hayes-Healy 231**)

John Harper (University of Notre Dame)

(Co)operations on Quillen homology

Fall 2007 Abstracts

September 25, 2007: Liang Kong

An introduction to open-closed conformal field theory Open-closed conformal field theory describes perturbative open-closed string theory and some critical phenomena in condensed matter physics. It provides a tool to study the still mysterious object called "D-brane". In this talk, I will outline a mathematical study of open-closed conformal field theory based on the theory of vertex operator algebra. In particular, I will give a tensor-categorical formulation of open-closed conformal field theory with boundary condition preserving a rational vertex operator algebra. I will also briefly discuss what D-branes are in this framework. This talk will be accessible to those who know nothing about conformal field theory and vertex operator algebra.

October 4, 2007: Chris Schommer-Pries

Two-Dimensional Topological Quantum Field Theories This talk will describe several manner of 2-D TQFTs, including some recent work classifying extended 2-D TQFTs. An extended 2-D TQFT is a (symmetric monoidal) functor from the topological bordism bicategory of (zero-manifolds, 1-manifolds, 2- manifolds) to the bicategory of (algebras, bimodules, intertwiners) and, up to natural isomorphism, these are in bijection with semi- simple Frobenius algebras, up to Morita equivalence. This talk will begin with the classical theorems that 1+1 TQFTs are the same as commutative Frobenius algebras and that another version of "extended" TQFTs, the so called open-closed 1+1 TQFTs, are the same as "knowledgeable Frobenius algebras". After reviewing these classical 1- categorical theorems we will sketch a proof of the above theorem.

October 11, 2007: John Harper

(Co)operations on Quillen homology In this talk I will give an overview of my current research direction, present the motivating ideas behind various constructions and approaches, and indicate partial results for my two primary examples of interest: symmetric spectra and unbounded chain complexes. Along the way, I will introduce Quillen's derived functor version of homology and indicate why symmetric sequences equipped with operad actions naturally appear on the scene. I will also discuss some homotopical analogies useful for studying (co)operations on Quillen homology, along with some interesting connections to Koszul cooperads/duality.

Notre Dame Department of Mathematics

Questions? Contact Stacy Hoehn