Evaluations of the Tutte polynomial

The Tutte polynomial is a fundamental invariant associated to a matroid, which includes graphs, vector arrangements, or hyperplane arrangements. Due to its computational complexity, it is often specialized at specific values which give us enumerative, geometric, algebraic information. In the case of the complete graph, one particular evaluation connects the number of spanning trees to the Eulerian numbers, which count the number of up-down permutations. In this talk, we will explore some specific combinatorial and geometric settings for the Tutte polynomial and discuss ongoing investigations on how to better understand the Tutte polynomial in these settings.