The ACO Seminar (2022–2023)

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 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 (in Langland's program). Arthur's trace formula concerns the trace of the left action of a reductive group G on the space L^2(G/Γ) where Γ is a discrete (arithmetic) subgroup. The combinatorics involved is closely related to compactifications of ''locally symmetric spaces''. 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). If time permits I will briefly touch upon connections with toric varieties. The talk is aimed at combinatorics crowd and mostly involves geometry and combinatorics of convex polytopes. This is a joint work with Mahdi Asgari (OkSU).