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.

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

Schedule of Talks

Abstracts

January 16

Let R be a compact, connected, orientable surface of genus g with p boundary components. Let C(R) be the complex of curves on R and Mod_R^* be the extended mapping class group of R. Suppose that either g = 2 and p > 1 or g > 2 and p >= 0. We prove that a simplicial map lambda from C(R) to C(R) is superinjective if and only if it is induced by a homeomorphism of R. As a corollary, we prove that if K is a finite index subgroup of Mod_R^* and f is an injective homomorphism from K to Mod_R^*, then f is induced by a homeomorphism of R and f has a unique extension to an automorphism of Mod_R^*. This extends the previous results about closed connected orientable surfaces of genus at least 3, to the surface R.

Januray 23

If a group G splits as an amalgamated free product or HNN extension, there is an associated action of G on a tree. There is also a converse to this. This is what Bass-Serre theory is about. This is related to the theory of ends of groups, because if G splits over a subgroup H then the pair (G,H) has more than one end. However, there is no simple converse to this fact. About 10 years ago, Sageev showed that if a group G has a subgroup H such that the pair (G,H) has more than one end, then there is an associated action of G on a Cat-0 cubed complex (a cubing). In this talk, I will start by describing Sageev's construction motivated by pictures involving curves on surfaces. Then I will describe recent work of Niblo, Sageev, Swarup and myself in which we modify Sageev's construction to obtain a smaller and more canonical cubing, and give some applications.

February 6

In this talk, I will discuss a generalized analogue of the string topology results of Chas-Sullivan and of Cohen-Jones to general finite simplicial complexes and to higher-dimensional unframed little disk operads. I will also discuss the connection of string topology to Koszul duality. In particular, I will show that the $k$-dimensional little disk operad $C_k$ is Koszul-dual to itself for all $k$, as well as the result that the Hoshschild cohomologies of Koszul-dual $C_k$-algebras are naturally equivalent.

February 13

I will explain the work of Betz-Cohen on Morse field theory. I will review the beautiful classical results in Morse theory from a different perspective, I will explain how a 0-1 TQFT theory can be constructed using Morse functions and to finish I will elaborate how these methods can be used on manifolds with finite group actions.

March 5

Following Thurston, we study subgroups of the mapping class group generated by two positive multi-twists. We determine the configurations of curves for which the corresponding groups exhibit certain exceptional behaviors. In particular, we identify a pseudo-Anosov automorphism whose dilatation is Lehmer's number and show that this is uniquely minimal for all the groups under consideration.

March 12

Let QF(S) denote the space of isometry classes of marked quasi-Fuchsian structures on the once-punctured torus, S. The deformation space AH(S) is the closure of QF(S). A marked hyperbolic manifold N in AH(S) is a self-bumping point provided that for any sufficiently small neighbourhood V of N in AH(S), the intersection of V with QF(S) is disconnected. We say that a marked hyperbolic 3-manifold N in AH(S) does not wrap provided that for any geometrical limit of any sequence of structures in AH(S) converging algebraically to N, there is a compact core for N which embeds in the geometric limit under the covering map. We show that if N does not wrap then it is not a self-bumping point.

March 18

For each closed orientable surface we introduce a simplical complex with some additional structure which is a version of the complex of curves of this surface adjusted to investigation of its Torelli group. We call this complex the Torelli geometry of our surface and prove that every automorphism of the Torelli geometry is induced by a diffeomorphism of the surface in question. We also provide an intrinsic algebraic characterization of some natural elements of the Torelli group and of some geometric relations between them. When combined, these results allow us to compute the automorphism group, the outer automorphism group and the abstract commensurator of the Torelli group for surfaces of genus at least 5.

April 2

I will describe the construction, by Hopkins, Rezk, and myself, of an E_\infty orientation from the spectrum MO<8> to the spectrum of topological modular forms, which refines the Witten genus and the sigma orientation.

This page last updated Jan 12 09:03:42 EST 2004 .