The Arthur Trace Formula (ATF), in its various incarnations, has played a very important role in Number Theory and Automorphic Forms in the last 50 years. The non-invariant ATF, the first incarnation, is an equality of two distributions, the so-called geometric and spectral sides, on suitably chosen test functions an adelic reductive group. However, the usual trace diverges. Arthur proved, using complicated geometric/combinatorial and analytical techniques, that a truncated version of the trace, depending on a truncation parameter, is indeed convergent when the parameter is sufficiently regular, and indeed gives a polynomial in the parameter. These facts form the basis of the development of his theory of the ATF.

The combinatorial aspects of Arthur’s proof, when mixed with the analytic techniques, appear somewhat mysterious. My goal in this talk is to explicate the geometric/combinatrial aspects of Arthur’s proof by recasting them in the language of convex polytopes and fans, making the geometric and combinatorial aspects more transparent and natural. Apart from the motivation and background, the talk can be considered as being purely about combinatorics of polytopes. Of course, here there are connections to other areas, such as toric varieties, that we are currently exploring as well. This is joint work with Kiumars Kaveh (University of Pittsburgh).