Algebra, Geometry, and Combinatorics Seminar
San Francisco State University
Spring 2025
San Francisco State University
Spring 2025
Pontificia Universidad Católica de Chile
Title: The cd-index of a semi-Eulerian poset
Abstract: We generalize the definition of the cd-index of a Eulerian poset to the class of semi-Eulerian posets. For connected simplicial semi-Eulerian Buchsbaum posets, we show that all coefficients of the cd-index are nonnegative. This proves a conjecture of Novik for odd-dimensional manifolds and extends it to the even-dimensional case. Joint work with Martina Juhnke and Lorenzo Venturello.
Freie Universität Berlin
Title: Tropical Linear Systems
Abstract: In this talk I want to introduce abstract tropical curves and their systems of divisors. I will explain how these linear systems admit the structure of a tropical module, but also of an abstract polyhedral complex. We will see that the dimension of this polyhedral complex is closely related to the Baker-Norine rank, which adds to existing analogies between tropical curves and algebraic ones.
San Francisco State University
Title: A geometer’s guide to chromatic polynomials
Abstract: Chromatic polynomials count the number of ways to color a graph’s vertices so that no two adjacent vertices have the same color. One of the great combinatorial conjectures of the 20th century claimed that the chromatic polynomial of any graph is log-concave, meaning that the square of each of its interior coefficients is at least as big as the product of its neighbors. This conjecture remained unsolved for over 50 years until, in a major breakthrough, June Huh finally resolved it in 2012. Huh's work around this conjecture ultimately led to his receiving the Fields Medal in 2022. In this talk, we’ll explore chromatic polynomials, log-concavity, and a recently-discovered method by which we can give a new proof of Huh’s result by viewing it through the lens of classical ideas in convex geometry.
The research that this talk is based on is joint work with quite a few current and former SF State master's students: Jordan Guillory, Rebeca Hernandez Valdez, Anastasia Nathanson, Lauren Nowak, and Patrick O'Melveny.
University of North Carolina at Chapel Hill
Title: Lagrangian fibrations by Prym surfaces
Abstract: Holomorphic symplectic manifolds (aka hyperkahler manifolds) are complex analogues of real symplectic manifolds. They have a rich geometric structure, though few compact examples are known. In this talk I will describe attempts to construct and classify holomorphic symplectic manifolds that also admit a holomorphic fibration. In particular, we will consider examples in four dimensions that are fibred by abelian surfaces known as Prym varieties.
University of Massachusetts Amherst
Title: The generalized Pitman-Stanley polytope
Abstract: The eponymous Pitman-Stanley polytope introduced in 1999 is related to plane partitions of skew shape with entries 0 and 1. It has been well studied because of its connections to probability, parking functions, generalized permutahedra, and flow polytopes. We consider a generalization of this polytope related to plane partitions with entries 0, 1, ... , m. We show that this polytope can also be realized as a flow polytope of a grid graph. We give multiple characterizations of its vertices in terms of plane partitions of skew shape and integer flows. We also study formulas for the volume as well as for the number of lattice points and vertices of this polytope. This is joint work with William Dugan, Maura Hegarty and Alejandro Morales.
University of California Davis
Title: What is the Best Way to Slice a Polytope?
Abstract: For literally hundreds of years mathematicians have been fascinated with slicing high-dimensional mathematical objects as a way to get knowledge and intuition of higher dimensions. There are many classical results and conjectures (e.g., Bourgain’s conjecture, recently a theorem, on the relation of volume and area of slices).
This talk is yet a computational contribution to this subject. I will present:
We concentrate on polytopes. Given a d-dimensional convex polytope P, what is the "best’’ slice of P by a hyperplane? Here "best’’ can mean many possible things, analytics e.g., a slice with the largest volume? Or combinatorial, e.g. a slice with the largest number of vertices? etc. This touches on classical work by Lagrange, Bourgain, Ball, Koldobsky, Milman, and many other mathematicians. Not only we investigate the above optimization theorem but also, as we slice P with hyperplanes, we create many combinatorially different (d-1)-slices, which are also polytopes of course. E.g., for a 3-dimensional regular cube there are 4 combinatorial types of slices (triangles, quadrilaterals, pentagons, hexagons). We investigate: How many different slices are there for a polytope P? How can we count them all? Can we give lower/upper bounds on their number? What are extremal cases?
I will explain a powerful new geometric model (a moduli space of slices) and algorithmic framework that answers these problems (and others) in polynomial time when dim(P) is fixed. Moreover, we show the problems have hard complexity otherwise. This is joint work with Marie-Charlotte Brandenburg (MPI/KTH) and Chiara Meroni (Harvard/ETH).
If time allows I will also show open questions that my research group is attacking right now.
University of California, Berkeley
Title: Gram Matrices for Isotropic Vectors
Abstract: We investigate determinantal varieties for symmetric matrices that have zero blocks along the main diagonal. In theoretical physics, these arise as Gram matrices for kinematic variables in quantum field theories. We study the ideals of relations among functions in the matrix entries that serve as building blocks for conformal correlators. We introduce computational methods for finding the defining equations of these varieties as well as numerical methods to calculate their degree.
University of California, Berkeley
Title: From Combinatorics to Knot Theory (and Back Again)
Abstract: Catalan numbers are among the most ubiquitous objects in mathematics, arising naturally in combinatorics, representation theory, geometry, and many other areas. Although there are various polynomial generalizations of these numbers, particularly fruitful are the so-called (q,t)-Catalan polynomials. Among many other things, these polynomials provide a direct link between combinatorial objects, such as Dyck paths and parking functions, and the Khovanov-Rozansky homology (a particular homological link invariant) of so-called torus knots. In this talk, I will explain some of the fascinating connections—both known and conjectural—between various Catalan objects and knot theory. I will also present new families of Catalan polynomials, constructed by my collaborators and me, that generalize previous formulations and provide new insights into the Khovanov-Rozansky homology for the larger family of Coxeter knots.
UC Berkeley and Max Planck Institute for Mathematics in Sciences, Leipzig
Title: Kinematic Stratifications
Abstract: We study stratifications of regions in the space of symmetric matrices. Their points are Mandelstam matrices for momentum vectors in particle physics. Kinematic strata in these regions are indexed by signs and rank two matroids. Matroid strata of Lorentzian quadratic forms arise when all signs are non-negative. We characterize the posets of strata, for massless and massive particles, with and without momentum conservation. This is joint work with Veronica Calvo and Hadleigh Frost.
Naval Postgraduate School
Title: Tropical Fermat-Weber Points over Tree Spaces
Abstract: It is well-known that a tropical Fermat-Weber point with respect to the tropical symmetric metric might not be in a space of phylogenetic trees on m leaves even though all observations in the input are inside of the space. In this talk we discuss a set of tropical Fermat-Weber points with respect to the tropical symmetric metric over the tree space and its application to the species tree reconstruction under the multi-species coalescent model. This is joint work with S. Cox, R. Talbut, and J. Sabol.
San Francisco State University
Title: Decomposition of topological Azumaya algebras with and without involution
Abstract: Topological Azumaya algebras (TAAs) —with or without involution—serve as the topological analogues of their richer algebraic counterparts. Because tensor product endows them with a natural multiplication, any algebra A of degree mn (for positive integers m and n) prompts the question: can A be factored in accordance with the decomposition of its degree? In this talk, I will define TAAs (and their involutive variants) over a space X, and present conditions on m, n, and the topology of X under which an algebra of degree mn admits a decomposition into factors of degrees m and n.
University of Michigan
Title: Spherical friezes
Abstract: A frieze pattern is an array of numbers that obeys simple local algebraic rules. Come build your own frieze pattern, and see how frieze patterns connect to triangulations of polygons. We will explore how a new type of frieze pattern can solve a problem in distance geometry. The speaker proudly holds a Master's in Math from SFSU!