For more information, contact Elmas Irmak at eirmak@umich.edu.

If you want to check the list of seminars held in previous terms, click on the appropriate link below.

Winter 2003,  Fall 2002, the year 2000-1.

Schedule of Talks

Abstracts

September 12

In the mid 1970's, Jaco and Shalen (JS), and independently Johannson (J) proved the existence of a canonical (JSJ) decomposition of a compact Haken 3-manifold. Starting in the mid-1980's, various authors have given purely algebraic analogues of this JSJ-decomposition. The first such result was by Kropholler for Poincare duality groups. Also Leeb and I gave a topological generalisation to higher dimensional manifolds, but only when the ambient manifold is closed and possesses a Riemannian metric of non-positive curvature. In this talk, I will introduce the topological results in three dimensions, discuss the generalisation due to Leeb and myself, and then discuss the results of Kropholler and recent work of Swarup and myself on Poincare duality groups and pairs.

September 26

In this talk I will survey my recent joint work with Bernardo Uribe and Mainak Poddar. I will explain how motivated by string theory, especially the work of Vafa and Witten, on one hand, and by Chen and Ruan's discovery of orbifold cohomology on the other; Uribe and I have developed a general localization principle in orbifold theories that encompasses diverse applications. In the last part of the talk I will explain how combining this principle with the ideas of Motivic Integration of Kontsevich, Denef, Loeser and Looijenga; Poddar and I have proved Ruan's conjecture on certain generalization of the McKay correspondence.

October 3

This will be an introduction to limit groups. We will give a few equivalent definitions and some of their properties. In particular, we will discuss relative hyperbolicity of limit groups and its consequences.

October 10

In this talk I will explain the "stringy" aspects of Chen-Ruan orbifold cohomology, its origins and its applications. I will define orbifold K-theory and I will state the known relations among the two. Finally I will elaborate on the different twistings that one can construct for both theories, namely discrete torsion, B-fields and gerbes with connection in general.

October 17

A construction of conformal field theories, in the sense of Segal and Kontsevich, from vertex operator algebras is essentially complete in genus zero and one. However, in higher genus, some basic analytic and geometric problems must be solved in order to even formulate some fundamental structures, such as modular functors and holomorphic weakly conformal field theories in the sense of Segal. The basic geometric objects are Riemann surfaces with analytically parametrized boundaries and their associated determinant lines. The formulations of these theories are based on the highly nontrivial assumptions that the moduli space of such surfaces is an infinite-dimensional complex manifold, the determinant lines form a holomorphic line bundle over this moduli space and that the sewing operation is holomorphic. I will outline proofs of these statements using classical results from Teichm\"uller theory and Schiffer variation.

October 24

In this talk I will discuss my work on lax algebraic structures, which is motivated by axiomatization of conformal field theory. In particular, I will define theories according to Lawvere, their generalization called 2-theories, and lax algebras over them with some relevant examples. My main work concerns constructing the right versions of lax limits, colimits and adjoints in the 2-categories of lax algebras and proving their existence.

October 31

This talk is on joint work with Elmas Irmak and Nikolai Ivanov. Let S be a closed, orientable, connected surface of genus g. Let n be a positive integer. The n-string surface braid group of S, denoted B_n(S), is the fundamental group of the space of unordered n-tuples of distinct points on S. Let {x(1), ..., x(n)} be the chosen basepoint for B_n(S). We prove that every automorphism of B_n(S) is induced by a self-homeomorphism of the pair (S, {x(1),...,x(n)}) provided that g > 1 and n > 2. This result establishes for the relevant surface braid groups what has been previously established for surface mapping class groups and Torelli groups.

November 21

I will explain the structure of a good category of equivariant spectra parametrized over a fixed base space together with base change functors relating such categories. Then I will describe some immediate consequences of the existence of such a good home to do parametrized stable homotopy theory. They include a conceptual understanding of the relation between the transfer constructions for bundles and fibrations as well as an easy conceptual proof of a duality theorem of Po Hu for equivariant bundles with closed smooth manifold fibers. That theorem includes the classical Wirthmuller and Adams isomorphisms as special cases. As time permits I will then discuss some of the challenges involved in obtaining the stable homotopy cateogory of equivariant parametrized spectra. This is joint work in progress with Peter May.

December 5

We will discuss recent work that aims to understand the topological relations of knots that are indistinguishable by their finite type invariants, using techniques from the theory of Dehn surgery.

This page last updated Wed Aug 27 09:03:42 EST 2003 .