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Pierre Albin, University of Illinois, Urbana-Champaign
Stratified spaces arise naturally even when studying smooth objects, e.g., as algebraic varieties, orbit spaces of smooth group actions, and many moduli spaces. There has recently been a lot of activity developing analysis on these spaces and studying topological invariants such as the signature. I will report on joint work with Jesse Gell-Redman in which we study families of Dirac-type operators on stratified spaces and establish a formula for the Chern character of their index bundle.
Florian Pop, University of Pennsylvania
Grothendieck's anabelian geometry originates from his famous "Esquisse d'un programme" and "Letter to Faltings". Among the topics of this program, Grothendieck proposed to give a non-tautological description of absolute Galois groups, especially of the absolute Galois group G_Q of the rational numbers. After intensive work by many -- starting with Deligne, Ihara, Drinfel'd -- this development led to the so called Ihara/Oda-Matsumoto conjecture, for short I/OM, which gave (conjecturally) a topological combinatorial description of G_Q. In the talk I will review/explain the question and present the state of the art, in particular recent refinements of I/OM, based on the so called Bogomolov (birational anabelian) Program.
Helen Wong, Institute for Advanced Study
Abstract: The definition of the Kauffman bracket skein algebra of an oriented surface was originally motivated by the Jones polynomial invariant of knots and links in space, and more precisely by Witten's topological quantum field theory interpretation of the Jones invariant. But the skein algebra is also closely related to the \( SL_2 \mathbb C\) -character variety of the surface. We'll describe two seemingly different methods for constructing finite-dimensional representations of the skein algebra --- one uses combinatorial skein theory whereas the other comes from the quantum Teichmuller space. Very recently, Frohman, Le and Kania-Bartoszynska show that for generic representations, the two methods yield exactly the same representations. We'll discuss implications of this result and some of the many questions that remain.
Philip Dames, Department of Mechanical Engineering, Temple University
There are no conferences next week.