Fall 2024

Title: Genocchi numbers and hyperplane arrangements

Abstract:  In joint work with Alex Lazar, we refine a result of Gabor Hetyei relating the number of regions of a homogenized version of the Linial hyperplane arrangement to the median Genocchi numbers. We do so by obtaining combinatorial interpretations of the coefficients of the characteristic polynomial of the arrangement and by deriving generating functions for the characteristic polynomials, which reduce to known generating functions for the Genocchi and median Genocchi numbers. Our work involves the Ferrers graphs of Ehrenborg and van Willigenburg, a class of permutations related to Dumont permutations, the surjective staircase tableaux of Dumont, and a result of Chung and Graham on chromatic polynomials of incomparibility graphs.  Our techniques also yield type B analogs, and Dowling arrangement generalizations.

Title: Constructing vertex expanding graphs

Abstract:  In a vertex expanding graph, every small subset of vertices neighbors many different vertices. Random graphs are near-optimal vertex expanders; however, it has proven difficult to create families of deterministic near-optimal vertex expanders, as the connection between vertex and spectral expansion is limited. We discuss successful attempts to create unique neighbor expanders (a weak version of vertex expansion), as well as limitations in using common combinatorial methods to create stronger expanders. This is based on joint work with Jun-Ting Hsieh, Sidhanth Mohanty, and Pedro Paredes.

Title:  Klyachko's Formula

Abstract:  The Grassmannian Gr(k,n) has an action of the algebraic n-torus. This talk concerns the (closures of) orbits of this action, which are subvarieties of Gr(k,n). Klyachko's formula expresses the cohomology class of the generic orbit in terms of dimensions of representations of GL_n. We will see how to leverage this formula and some matroid theory to compute cohomology classes of the other orbits. 

Title: Identifiability: Using math and trees to solve problems from biology

Abstract: Recovering parameter values from mathematical models is a primary interest of those that use them to model the physical and biological world.  This recovery, or identification, of parameters within models is also an interesting mathematical problem that we call Identifiability.  In this talk, we will explore the identifiability of a specific type of model called Linear Compartmental Models, which are often used to understand biological phenomena and have an underlying graphical structure.  Starting with an introduction to graph theory, we will explore the relationship that this graphical structure has to Linear Compartmental Models and their defining differential equations.  At the end of the talk, we classify identifiability criteria for an interesting subclass of Linear Compartmental Models called tree models. 

Title: A Spectral Approach to Polytope Diameter

Abstract: A classic question in discrete geometry is: what is the maximum possible diameter of the skeleton of a polytope P = {x in R^d : Ax ≤ b} in R^d defined by m constraints, as a function of m and d? We describe a new approach to this problem which uses classical inequalities from convex geometry to bound the eigenvalues of a certain weighted adjacency matrix of the skeleton, thereby yielding an upper bound in terms of m, d and some quantities depending on the polar of P. When A is an integer matrix this yields improved bounds on the diameter in terms of the subdeterminants of A.

 Joint work with H. Narayanan and R. Shah.

Title: Hypercube decompositions and combinatorial invariance for Kazhdan-Lusztig polynomials

Abstract: Kazhdan-Lusztig polynomials are of foundational importance in geometric representation theory. Yet the Combinatorial Invariance Conjecture, due to Lusztig and to Dyer, suggests that they only depend on the combinatorics of Bruhat order on permutations. I'll explain what these things are and describe joint work with Grant Barkley in which we adapt the hypercube decompositions introduced by Blundell-Buesing-Davies-Veličković-Williamson to prove this conjecture for Kazhdan-Lusztig R-polynomials in the case of elementary intervals. This significantly generalizes the main previously known case of the conjecture, that of lower intervals.

Title: On valuation polytopes of height one posets

Abstract: Geissinger defined the valuation polytope as the set of all [0,1]-valuations on a finite distributive lattice. Dobbertin showed the valuation polytope is equivalently defined as the convex hull of vertices characterized by all the chains of a given poset.  In this project, we study the valuation polytope, VAL(P), arising from a poset P of height one on n elements. We consider height one posets, generally, and the zig-zag poset and complete bipartite poset, specifically. We will present results on their normalized volume, the existence of a unimodular triangulation, and their f-vector. This is joint work with Federico Ardila, Jessica De Silva, Jose Luis Herrera Bravo, and Andrés R. Vindas-Meléndez.

Title: Legendrian links and Newton polytopes

Abstract: Legendrian curves appear in our daily life as wavefronts of light and in ice-skating patterns. They are curves in 3-space whose tangent vectors satisfy a constraint imposed by a hyperplane field known as a contact structure. Legendrians are a central object of study in contact and symplectic geometry, two types of geometries that arose from the study of Hamiltonian dynamics. Both share the property that they can have no local invariants. However, by considering Legendrian submanifolds one can distinguish contact manifolds. I will talk about results on Lagrangian surfaces in the symplectic 4-ball that rely on Legendrian invariants and Newton polytopes.