MenuCombinatorial Algebraic Geometry
Spring 2025
| Date: | January 14, 2025 |
| Time: | 12:45pm |
| Location: | Bloc 302 |
| Speaker: | Ruzho Sagayaraj, Texas A&M University |
| Title: | Chebyshev Varieties |
| Abstract: | Chebyshev polynomials are useful in function approximation as the root finding problem is better-conditioned in the basis of Chebyshev polynomials than in the familiar monomial basis. In this talk, I will introduce multivariate generalizations of Chebyshev polynomials and use them to define Chebyshev varieties parametrized by Chebyshev polynomials analogous to toric varieties parametrized by monomials. I will also discuss the geometry of Chebyshev varieties and list some applications. This talk is based on the paper Chebyshev varieties by Z. Bel-Afia, C. Meroni and S. Telen ( arXiv:2401.12140 ). Following this presentation, there will be a more general discussion of arithmetic toric varieties. |
| Date: | January 28, 2025 |
| Time: | 12:45pm |
| Location: | Bloc 302 |
| Speaker: | Frank Sottile, Texas A&M University |
| Title: | Computing Schubert Problems I |
| Abstract: | This is the first in a series of informal discussions about Grassmannians and flag varieties, with the goal of describing both how to represent Schubert problems on a computer and some goals of this computational study. An outline of the discussions as they are given is at https://franksottile.github.io/talks/25/SchubertProblems.txt |
| Date: | February 4, 2025 |
| Time: | 12:45pm |
| Location: | Blocker 302 |
| Speaker: | Frank Sottile, Texas A&M University |
| Title: | Computing Schubert Probems II |
| Abstract: | This is the second in a series of informal discussions about Grassmannians and flag varieties, with the goal of describing both how to represent Schubert problems on a computer and some goals of this computational study. This will begin with the decomposition of the Grassmannian into Schubert cells and describe Schubert varieties. |
| Date: | February 11, 2025 |
| Time: | 12:45pm |
| Location: | Bloc 302 |
| Speaker: | Frank Sottile, Texas A&M University |
| Title: | More on Schubert Problems and Numerical Algebraic Geometry |
| Abstract: | This continues my informal discussion about Schubert problems in the Grassmannian. After describing Schubert problems, I will pivot to discuss some basics of numerical homotopy continuation, in preparation for the visit of Thomas Yahl in two weeks. |
| Date: | February 18, 2025 |
| Time: | 12:50pm |
| Location: | Bloc 302 |
| Speaker: | Frank Sottile |
| Title: | Introduction to Numerical Algebraic Geometry |
| Abstract: | I will continue to provide general background to the topic of numerical algebraic geometry. This is in part to prepare for the visit of Thomas Yahl and projects to study Galois groups in enumerative geometry. |
| Date: | March 18, 2025 |
| Time: | 12:50pm |
| Location: | Bloc 302 |
| Speaker: | Frank Sottile, Texas A&M University |
| Title: | A bestiary of Bloch varieties |
| Abstract: | (This is a reprise of the talk in the MPHA seminar on Friday, March 7.) To a Zd-periodic weighted graph with vertices V, we may associated a periodic graph operator that acts on ℓ2(V). After Floquet transform, we obtain its Bloch variety, which is an algebraic hypersurface in Td×R whose projection to R is the spectrum of the operator. Features on Bloch varieties such as Dirac (double) points, critical points, and their level sets (Fermi varieties) reflect spectral properties of the operator. With students Faust and Robinson, and using the Brazos cluster, we investigated over 2.1 million small periodic graphs, recording invariants and features of their Bloch and Fermi varieties. In this talk, I will briefly discuss the background and present some examples of interesting behavior of Bloch varieties that we uncovered. |
| Date: | March 25, 2025 |
| Time: | 12:50pm |
| Location: | Bloc 302 |
| Speaker: | Tianyi Yu, Université du Québec à Montréal |
| Title: | TBA |
