All talks will be held in Chemistry-Physics Building Room 139
12:30pm-1pm Check-in; coffee / snacks available
1pm-1:30pm Pre-talk - Josephine Yu
1:45pm-2:45pm Josephine Yu, Tropical Positivity and Symmetric Low Rank Matrices
Abstract: In 2003, Speyer and Williams introduced two definitions for positive tropicalizations of algebraic varieties over complex Puiseux series and proved their equivalence for Grassmannians. The two corresponding notions of positive tropical generators were studied by Brandenburg, Loho, and Sinn in 2022. In this talk, I will provide a brief introduction to tropical geometry and then present a combinatorial description of the tropicalization of the space of symmetric rank two matrices, along with their real and positive parts. We will compare the two notions of tropical positivity for rank 2 or corank 1 symmetric matrices. The results are based on joint works with Abeer Al Ahmadieh, May Cai, and Kisun Lee.
2:45pm-3:15pm Coffee break
3:15-4:15pm Ritvik Ramkumar, Hilbert scheme of points on threefolds
Abstract: The Hilbert scheme of d points on a smooth variety X, denoted by Hilb^d(X), is an important moduli space in algebraic geometry. In this talk, I will focus on the case where X is a threefold, a setting in which several questions about its singularities remain open. I will discuss some of these questions along with recent results. Although not apparent at first glance, the theory is often guided by subtle underlying combinatorial structures, which I will highlight.
4:30-5:30pm Poster session
9-9:30am Coffee / snacks available
9:30am-10:30am Ada Stelzer, Gröbner degeneration and equivariant homological invariants
Abstract: When a reductive group G acts on an embedded variety X, the coordinate ring C[X] is a G-representation. The data of this representation may be recorded directly as the G-equivariant Hilbert series of C[X], or more compactly as its K-polynomial or twisted K-polynomial (which are connected to the minimal free resolution and multidegree of C[X] respectively). Non-cancellative combinatorial rules for the coefficients in all three polynomials are therefore desirable. We focus on determinantal varieties, where the combinatorics of pipe dreams and the RSK correspondence naturally arise. We present joint work with Abigail Price and Alexander Yong computing the G-equivariant Hilbert series of generalized determinantal varieties, along with open problems and directions for future research.
11am-11:30am: Pre-Talk - Sam Payne
11:45-12:45 Sam Payne, Tropical linear series and matroids
Abstract: I will present a theory of linear series on tropical curves that balances two notions of rank: the Baker-Norine rank from the tropical Riemann-Roch theorem and another idea of rank based on tropical independence. The resulting theory is surprisingly closely related to matroids. Every tropical linear series contains an open dense subset of nondegenerate divisors in a neighborhood of which the tropical linear series of dimension r is locally isomorphic to the Bergman fan of a matroid of rank r + 1. Moreover, every matroid appears as such a local matroid of a tropical linear series at a nondegenerate divisor. For instance, cographic matroids appear as local matroids of canonical linear series at the canonical divisor. Based on joint work with Chih-Wei Chang, Matthew Dupraz, Hernán Iriarte, David Jensen, Dagan Karp, and Jayden Wang. https://arxiv.org/abs/2508.20062