Events of the Week

The semicircle law is one of the fundamental results in random matrix theory. It describes a remarkable phenomenon: the empirical distribution of eigenvalues of large random symmetric matrices converges to a universal limiting distribution.
In this talk, we will derive the semicircle law for Gaussian random matrices using the method of moments. By applying Isserlis' theorem, we will express the expected spectral moments in terms of pairings and show how their asymptotic behavior is governed by Catalan numbers. This approach reveals an elegant connection between combinatorics and random matrix theory, providing insight into the emergence of the semicircle law and its universality.

Depending on the base field, the cohomology of an algebraic variety may be equipped with either a Hodge structure or the structure of a Galois representation. Motives are an attempt to unify these structures; in a sense, they are a "universal structure" on the cohomology of a variety over any fielld. In this talk, we study the motives of Enriqeus surfaces, where the Hodge/Galois structure doesn't carry any interesting information. This is joint work with Jake Huryn.

Morse Theory - continued

A matrix representing a matroid over a field records which sets of columns are dependent, and its maximal minors assign a determinant to each basis. Can a tropical matrix do both for a valuated matroid, whose bases already carry weights?
I’ll show how to construct such a matrix using special valuated cocircuits as its rows. The resulting columns are tropically dependent exactly on the dependent sets of the matroid, and valuated circuits give explicit coefficients for those relations. To recover the basis weights, I’ll introduce a tropical volume motivated by the Cauchy–Binet formula.