# Daniel Rostamloo

## 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

**Major:**Pure Math**Sponsor:**Kwatinetz Fund**Mentor:**David Eisenbud