Counting matrix points via lattice zeta functions (Yifeng Huang, USC)

​I will introduce two general problems and explain how they surprisingly connect with each other and with other aspects of mathematics (for a glimpse, Sato—Tate, hypergeometric functions, moduli spaces of sheaves, Catalan numbers, Hall polynomials, etc.)​.

The first problem is to count finite-field points on so called “varieties of matrix points”. They are created from a simple and fully elementary recipe and can yet easily get very complicated. The second problem is analogous to counting full-rank sublattices of $\mathbb{Z}^d$ with index $n$, but with $\mathbb{Z}$ replaced by non-Dedekind rings, such as non-maximal orders in number fields. (Containing joint work with Ken Ono, Hasan Saad and joint work with Ruofan Jiang)