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, TBA
8:30-9:30am 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, TBA