Seminar

Jack Jeffries: "Local cohomology of determinantal nullcones": The coordinate ring  $R$ of the variety of $m\times n$ matrices with rank at most $t$ can be realized in a natural way as an invariant ring of a polynomial ring $S$; the same is true if one replaces "$m\times n$ matrices" by "$n\times n$ symmetric matrices" or "$n\times n$ alternating matrices". This realization as a ring of invariants explains many of the nice algebraic properties enjoyed by determinantal rings, at least in characteristic zero, since $R$ is a retract of $S$ via the Reynolds operator. Motivated by understanding the relationship between $R$ and $S$ in arbitrary characteristic, we consider the ideal generated by positive degree invariants $R_+$ inside of the ambient polynomial ring $S$. For example, certain varieties of complexes as introduced by Buchsbaum and Eisenbud occur like so. In this talk, we will discuss some aspects of the behavior of local cohomology with support in these ideals in different characteristics and some applications. This is based on work in progress with Pandey, Singh, and Walther, and earlier joint work with Hochster, Pandey, and Singh.

Uli Walther: "Matroidal Polynomials and their Singularities": (joint with Dan Bath) We introduce a class of polynomials attached to matroids (or flags of matroids) and the choice of a base field that includes various other polynomials that have appeared in the literature, including Kirchhoff polynomials, configuration polynomials, matroid basis polynomials, and multivariable Tutte polynomials. The common property of matroidal polynomials is a Deletion-Contraction formula. In the talk I will explain various geometric properties of matroidal polynomials; in particular, for connected matroids they have rational singularities over the complex numbers, and (if homogeneous coming from a single matroid) are �-regular in positive characteristic. The idea of the proof in characteristic zero can be extended to polynomials called Feynman integrands, and which show up in certain integrals in scattering theory. In particular, these do have rational singularities as well under certain genericity hypotheses on masses and momenta. The plan is to explain all these words, and hint at the proofs, which involve jets and the Frobenius.