University of Michigan
|
|---|
For more information, contact Valentina Joukhovitski.
If you want to check the list of seminars held in previous terms, click on the appropriate link below.
Fall 2006, Winter 2006, Fall 2005, Winter 2005, Fall 2004, Winter 2004, Fall 2003, Winter 2003, Fall 2002, the year 2000-1.
| Date | ||
| 01/11 | ||
| 01/18 | ||
| 01/25 | ||
| 02/01 | ||
| 02/08 | ||
| 02/15 | ||
| 02/22 | ||
| 03/01 | ||
| 03/08 | ||
| 03/15 | ||
| 03/22 | ||
| 03/29 | ||
| 04/05 | ||
| 04/12 | Special time: 2-3 p.m. Different room: 3096 EH |
Abstract: Several years ago Mike Hopkins and collaborators have introduced topological modular forms, tmf for short. After a while Mark Behrens has described how to construct tmf with level-N structures when N is prime to the characteristic of the base field p.
I will briefly introduce topological modular forms as understood by Mike Hopkins and collaborators along with work of Mark Behrens on level-N structures on tmf. As a conclusion of the talk I will outline my own work on the construction of tmf with level-pn structures.
Abstract: In these talks I will discuss cohomological and K-theoretic invariants of orbifolds. After developing the basic machinery of orbifolds I will describe how "stringy" products can be introduced in this context, in particular we will discuss the recently defined stringy product in twisted K-theory, which is joint work with Y.Ruan and B.Zhang. Abstract: The group of invertible 2x2 matrices with integer coefficients has three different interpretations: it is the group of automorphisms of Z^2, the group of isotopy classes of diffeomorphisms of the 2-torus, and the group of outer automorphisms of the free group F_2 of rank 2. These descriptions lead to three much studies families of discrete groups: GL_n(Z), mapping class groups, and Out(F_n). Even though the techniques differ, there is a kind of a dictionary between the basic results about these families and often a theorem in one class motivates the study in another. I will try to describe the basic spaces on which these groups act and parts of the dictionary. In the second half of the talk I will report on recent progress in the study of the cohomology of Torelli subgroups of mapping class groups and of Out(F_n). These are the subgroups consisting of all elements that act trivially in homology. For example, the cohomological dimension was computed in the latter case, and in the former an alternative proof was found of a theorem of Mess which states that the genus 2 Torelli group is free. The techniques seem likely to generalize to higher genus. This is joint work with Kai-Uwe Bux and Dan Margalit. Abstract: Since Thomason put a model structure on the category Cat of small categories in 1980, model categories have become increasingly important in category theory. Various simplicial definitions of higher category were recently proved to be Quillen equivalent by Bergner, Joyal, Tierney and others. In another development, the homotopy theory of 2-categories was seen to resemble the homotopy theory of categories (Hess, Worytkiewicz, et al, Lack). This talk will report on recent progress concerning model structures in ``wider'' category theory.Higher categories are obtained by weakening iterated enrichment. Wider categories, on the other hand, are obtained by weakening iterated internalization. Ehresmann's double categories (internal categories in Cat) are the n=2 case and are the subject of this talk. A familiar example of a weak double category has objects rings, horizontal morphisms bimodules, vertical morphisms ring homomorphisms, and squares twisted equivariant maps of bimodules. Double categories contain 2-categories, but just how different are they? Our initial conclusions are that the homotopy theory of double categories is much richer than the homotopy theory of 2-categories. In this talk I will recall the notion of double category and its topological examples, and then describe the various model structures. This is joint work with Simona Paoli and Dorette Pronk.
Abstract: I will talk about my joint work with J. D. McCarthy. We prove that each injective simplicial map of the arc complex of a compact, connected, orientable surface with nonempty boundary is induced by a homeomorphism of the surface. We deduce, from this result, that the group of automorphisms of the arc complex is naturally isomorphic to the extended mapping class group of the surface, provided the surface is not a disc, an annulus, a pair of pants, or a torus with one hole. We also show, for each of these special exceptions, that the group of automorphisms of the arc complex is naturally isomorphic to the quotient of the extended mapping class group of the surface by its center. Abstract: There is a forgetful map from the mapping class group of a punctured surface to that of the surface with one fewer puncture. We relate the action of the kernel on the curve complex to a family of actions on trees. Together with a geometric compactness argument we further prove that finitely generated purely pseudo-Anosov subgroups of the kernel are convex cocompact in the sense of Farb and Mosher. As a corollary, this answers their question of local convex cocompactness for Whittlesey's group. This is joint work with R. Kent and S. Schleimer. Abstract: Working over an arbitrary field k, I will construct a map from the algebraic K-theory of k into the motivic stable homotopy groups over k. This map is analogous to the complex J-homomorphism of classical homotopy theory. It allows us to construct a non-zero element of the motivic stable homotopy group in dimension (4k-1,2k) for every k > 1. I will also discuss some other ideas for lifting classical stable homotopy computations to motivic stable homotopy theory. Abstract: Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first $3g-3$ geodesics. We use Brooks and Makover's construction of random Riemann surfaces to investigate the distribution of short ($< \log (g)$) geodesics on random Riemann surfaces. We calculate the expected value of the shortest geodesic, and show that if one orders prime non-intersecting geodesics by length $\gamma_1\le \gamma_2\le \dots \le \gamma_i ,\dots$, then for fixed $k$, if one allows the genus to go to infinity, the length of $\gamma_{k}$ is independent of the genus.This page last updated on September 4, 200720 .