Algebra, Geometry, and Combinatorics Seminar

Title: Families of degenerations from mutations of polytopes

Abstract: Theory of Newton-Okounkov bodies has led to the extension of the geometry-combinatorics dictionary from toric varieties to varieties which admit a toric degeneration. In a paper with Harada, we gave a piecewise-linear bijection between Newton-Okounkov bodies of a single variety. This involves a collection of lattices connected by piecewise-linear bijections. Inspired by these ideas in joint work in progress with Harada and Manon we propose a generalized notion of polytopes in $\Lambda=(\{M_i\}_{i\in I},\{\mu_{ij}\}_{i,j\in I})$, where the $M_i$ are lattices and the $\mu_{ij}:M_i\to M_j$ are piecewise-linear bijections. Roughly, these are $\{P_i\mid P_i\subseteq M_i\otimes \mathbb{R}\}_{I\in I}$ such that $\mu_{ij}(P_i)=P_j$ for all $i,j$. In analogy with toric varieties a generalized polytope can encode a compactification of an affine variety as well as toric degenerations for the compactification.

Title: Singular curves, compactified Jacobians and knots

Abstract: Given a plane curve singularity C, its compactified Jacobian is a certain moduli space of sheaves on C. In the talk, I will define compactified Jacobians and review their properties, results and conjectures about them. In particular, the topological invariants of the knot corresponding to C will make a surprising appearance. All notions will be defined in the talk.

Title: Bi-colored bosonic solvable lattice models

Abstract: The study of solvable lattice models originated in statistical mechanics, and has since formed rich connections with areas of math including combinatorics, probability, and representation theory. Lattice models are called solvable when they can be studied using the Yang-Baxter equation. The partition function of a system, which captures global information about the lattice model, is at the heart of many of these connections with other areas. To compute the partition function, one method is to identify boundary conditions that give systems with a unique state, from which other systems can be computed by Demazure recursion relations coming from the Yang-Baxter equation. Bosonic and colored variants of solvable lattice models have been studied in recent years by Aggarwal, Borodin, Brubaker, Buciumas, Bump, Gustafsson, Naprienko, Wheeler, and others. We will define a class of these models which are bosonic and include two types of colors, generalizing the now widely-studied colored models. These bicolored bosonic models satisfy the Yang-Baxter equation, which gives a four-term recurrence relation on the partition function. We will give conditions on the number of states of the model based on boundary conditions in terms of the Bruhat order, and discuss connections with Gelfand-Tsetlin patterns.

Title: Counting Homogeneous Einstein Metrics

Abstract: The problem of finding Einstein metrics on a compact homogeneous space reduces to solving a system of Laurent polynomial equations. We prove that the number of isolated solutions of this system is bounded above by the central Delannoy numbers and we describe the discriminant locus where the number of isolated solutions drops in terms of the principal A-determinant. 

Federico Ardila Mantilla has served the SF State Community for 20 years. During his time at SF State, Federico has been internationally recognized for his excellence in research, teaching, and building inclusive mathematical communities.

Please join us to celebrate Federico as he transitions to a new phase of his career. Light refreshments will be served from 3:00-4:00, followed by a Colloquium by Federico.

Title: Inequalities for Trees and Matroids

Abstract: In their 1971 study of telephone switching circuitry, Graham and Pollak designed a novel addressing scheme that was better suited for the faster communication required by computers. They introduced the distance matrix of a graph, and used its eigenvalues to bound the size of the addresses in their scheme. We continue their investigation, obtaining more precise spectral information about tree distance matrices. These results, combined with the theory of Lorentzian polynomials, allow us to prove some conjectural inequalities about graphs and matroids that are very easy to state but have taken decades to prove. Along the way we uncover a surprising appearance of Lorentzian polynomials in optimization and economics.

This is joint work with Sergio Cristancho, Graham Denham, Chris Eur, June Huh, and Botong Wang. The talk will assume no previous knowledge of these topics; it will be accessible to anyone with some knowledge of linear algebra.

Title: Statistics of character tables of symmetric groups

Abstract: In 2017, Miller computed the character tables of S_n for all n up to 38 and looked at various statistical properties of the entries. Characters of symmetric groups take only integer values, and, based on his computations, Miller conjectured that almost all entries of the character table of S_n are divisible by any fixed prime power as n tends to infinity. This was proven by Soundararajan and I, building on earlier work of ours in the case of primes. I will describe the ideas going into our proof, and discuss some related open problems. 

Title: Moduli theory through examples

Abstract: A natural endeavor in mathematics is to classify objects according to their properties. For example, we can readily identify straight lines in the plane, or recognize different kinds of triangles depending on their symmetries. Less intuitive, however, is that given a class of mathematical objects, it is often possible to construct a geometric space parametrizing those objects. Known as "moduli spaces," the study of these spaces has been a major driving force in modern geometry. In the first half of this talk, we will explore some of the main ideas behind moduli theory through examples, ranging from the moduli space of lines in the plane to that of points on the sphere. In the second half, we will discuss ongoing work with P. Gallardo and J.L. Gonzalez on two novel moduli spaces of labeled points in flags of affine spaces, which exhibit surprising connections with the theory of polymatroids.

Title: Blocks of General Linear Groups over Finite 

Abstract: The representation theory of finite groups has a long and rich history, dating back to the nineteenth century. In this talk, we will focus on a specific aspect of that theory: the study of blocks and their applications to particular classes of groups. The goal is to provide an introduction for non-specialists interested in understanding the key ideas and techniques that arise in block theory—ranging from ring-theoretic to module-theoretic approaches. With this background in place, we will apply these methods to study the representation theory of the groups GL_n(Z/p^2Z) and GL_n(F_p[t]/t^2) where p is a prime. 

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