## Daniel Rostamloo L&S Sciences

### Explicit Non-Vanishing of Asymptotic Syzygies

Algebraic geometry is a rich area of mathematics that investigates the properties of geometric objects (like a variety the solution set of a system of polynomial equations) using their underlying algebraic structure. The closely related field of homological algebra studies how mappings between algebraic spaces (e.g., collections of polynomials) can be understood in terms of more concrete representations with tools from topology and algebra combined to understand the geometric structure of varieties. One homological invariant is a table of numbers called the Betti table, which captures nuanced geometric information about the variety. Despite being an active area of research since the 1980s, the Betti tables of higher dimensional varieties (i.e., varieties having dimension greater than 1) remain poorly understood. This research seeks to extend the understanding of Betti tables by investigating interesting cases in which Betti numbers are nonzero, namely for projective varieties where each point represents a line through the origin and products of spaces like these.

#### Message To Sponsor

I give my sincerest thanks to the SURF program and the generous donor who makes funding for aspiring undergraduate researchers like myself possible. The opportunity to collaborate with experienced researchers in math in the past has been an invaluable catalyst for my own development as a mathematician and a learner, and I'm incredibly grateful to the program for affording me this opportunity which will undoubtedly be a formative experience in my undergraduate career. I look forward to joining this year's SURF cohort with the goal of contributing to the math community and developing myself as a researcher along with my peers.**Major:**Pure Math

**Mentor:**David Eisenbud

**Sponsor:**Kwatinetz Fund