Convex polytopes, toric varieties and combinatorics of Arthur's trace formula Kiumars Kaveh, University of Pittsburgh Host: Mahdi Asgari Contact Mahdi Asgari for Zoom info.
Abstract: I start by discussing two beautiful well-known theorems about decomposing a convex polytope into a signed sum of cones, namely the classical Brianchon-Gram theorem and the Lawrence-Varchenko theorem. I will explain the connection with toric varieties and how Brianchon-Gram theorem can be thought of as a computation of Euler characteristic using Cech cohomology. I will then explain a generalization of the Brianchon-Gram which can be summarized as ``truncating a function on the Euclidean space with respect to a polytope". This is an extraction of the combinatorial ingredients of Arthur's ``convergence'' and ``polynomiality'' results in his famous trace formula. Arthur's trace formula concerns the trace of left action of a reductive group G on the space $L^2(G/\Gamma)$ where $\Gamma$ is a discrete (arithmetic) subgroup. The combinatorics involved is closely related to compactifications of ''locally symmetric spaces'' (which by the way are hyperbolic manifolds). Our ''combinatorial truncation'' can be thought of as an analogue of Arthur's truncation over a toric variety (in place of a compactification of a locally symmetric space). This is a joint work (in progress) with Mahdi Asgari (OSU).
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