Invitation to Ehrhart theory: integer points, volume, and more! (Max Hlavacek)
Polytopes are discrete geometric objects that appear in many fields of mathematics, including applied fields such as optimization and statistics. Often, we are interested in computing the volume of these objects. In this talk, we explore discrete methods of computing volumes, leading towards Ehrhart theory, the study of counting integer points in polytopes. In particular, we will introduce the Ehrhart and h^* polynomials of lattice polytopes and discuss some open questions surrounding these polynomials. This will lead us to a discussion of properties that combinatorial polynomials can have, such as positivity, unimodality, and real-rootedness. We will end with a survey of classification results and conjectures surrounding properties of polynomials in Ehrhart theory.
Zero Forcing as a Tool for Maximum Nullity of a Graph (Ryan Moruzzi, Jr.)
Inverse problems have long played a central role in mathematics, focusing on identifying models of a system that realize particular solutions of interest. Within graph theory, the inverse eigenvalue problem of a graph (IEP-G) asks which sets of real numbers can be realized as the spectra of matrices described by a graph. The difficulty often arises when eigenvalues have high multiplicities, tying the IEP-G to minimum rank and maximum nullity.
In 2008, zero forcing on a graph was shown to provide a combinatorial upper bound on the maximum nullity of a graph, opening graph theoretic avenues for progress on the IEP-G. Zero forcing is an iterative graph coloring process in which, starting with an initial set of blue vertices, we try to force other non-blue vertices blue according to a color change rule. In this talk, we will use zero forcing to study a family of graphs that we call Hopi Rectangle graphs, highlighting how it can help determine their maximum nullity. If time permits, motivated by monitoring an electrical network, we will also introduce a robust version of zero forcing. Although we lose the connection to the maximum nullity of a graph, this version of graph coloring creates new avenues of research within zero forcing.
On the classification of modular categories (Agustina Czenky)
Modular tensor categories sit at the crossroads of quantum algebra and condensed-matter physics, providing an algebraic model for anyon systems arising from topological phases of matter, which are currently viewed as potential hardware for fault-tolerant quantum computing. Although these categories play a central role in both mathematics and physics, their overall landscape remains only partially understood, making their classification a rich and challenging problem. In this talk, I will give an overview of the classification program for fusion and modular categories, focusing on key invariants—such as rank, quantum dimensions, and modular data—and on the construction techniques that allow new examples to be built from known ones.
Coloring Graphs: From Chromatic Polynomials to Polyhedral Geometry, (Andrés R. Vindas Meléndez)
How many ways can you color the vertices of a graph so that no two connected vertices share the same color? This classic question is answered by the chromatic polynomial, a fundamental tool in combinatorics that counts "proper colorings". A key theme of the talk is the connection between graph coloring and polyhedral geometry. I will explain how proper colorings can be interpreted as lattice points inside geometric objects called polytopes, and how tools from polyhedral geometry (such as Ehrhart theory) help us understand these polynomials. This perspective leads to different formulas, structural insights, and a concrete bridge between combinatorics and geometry.
Time permitting, I will introduce a new "weighted" version of this concept: the q-chromatic polynomial. Instead of simply counting colorings, this invariant assigns weights to them, producing a richer algebraic/combinatorial object that recovers the classical chromatic polynomial when q=1.
No background necessary… just a willingness to stare at graphs and polytopes.