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. 

Title: Combinatorial Geometry meets Complexity Theory: Sylvester-Gallai Configurations and Polynomial Identity Testing

Abstract: In this talk we will survey the intrinsic connection between a classical family of problems in combinatorial geometry (the Sylvester-Gallai problem and its generalizations) and a fundamental question in algebraic complexity theory (the Polynomial Identity Testing problem - PIT). We will begin by establishing this connection in the linear algebraic setting, and we will see how complexity theory motivated algebraic-geometric generalizations of Sylvester-Gallai problems over algebraic varieties, and how their solutions have provided efficient algorithms for special and yet important cases of the PIT problem.

No background in complexity theory or advanced algebraic geometry will be needed in order to understand this talk and its connections.

This talk is based on joint works with Abhibhav Garg, Akash Kumar Sengupta, Shir Peleg, Nir Shalmon and Amir Shpilka

Title: Linear Spaces of Matrices of Bounded Rank 

Abstract: A linear subspace of the space of bxc matrix is of bounded rank r if no matrix in the space has rank greater than r. Classifications of them are an interesting and important problem with potential application to matrix multiplication complexity. The only case known is when r \leq 3 with two proofs given by Atkinson via the study of Atkinson normal form and Eisenbud-Harris via the study of the first Chern class of some sheaf associated with the space. I will discuss a connection between these two approaches as well as the recent progress in the classification of the basic space of matrices of bounded rank 4. This is joint work with J.M. Landsberg. 

Title: Weighted Ehrhart Theory

Abstract: In 1962 Ehrhart proved that the number of lattice points in integer dilates of a lattice polytope is given by a polynomial — the Ehrhart polynomial of the polytope. Since then Ehrhart theory has developed into a very active area of research at the intersection of combinatorics, geometry and algebra. The Ehrhart polynomial encodes important information about the polytope such as its volume and the dimension. An important tool to study Ehrhart polynomials is the h*-polynomial, a linear transform of the Ehrhart polynomial which is given by the numerator of the generating series. By a famous theorem of Stanley the coefficients of the h*-polynomial are always nonnegative integers. In this talk, we discuss generalizations of this result to weighted lattice point enumeration in rational polytopes where the weight function is given by a polynomial. In particular, we show that Stanley’s Nonnegativity Theorem continues to hold if the weight is a sum of products of linear forms that a nonnegative over the polytope. This is joint work with Esme Bajo, Robert Davis, Jesús De Loera, Alexey Garber, Sofía Garzón Mora and Josephine Yu.

Title: Generalized decompositions of a polynomial

Abstract: There are various notions of polynomial decomposition that are interconnected: the Generalized Additive Decomposition, the Generalized Affine Decomposition, and the natural apolar schemes. In this seminar, I will present these relationships with a particular focus on the regularity problem of the schemes that reveal the cactus rank of a given polynomial.

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