Fall 2026

Title: Linear optimization degrees of varieties

Abstract: An affine variety is the set of common solutions to a system of polynomial equations. Given a generic linear objective function, we can ask for its number of critical points restricted to a fixed variety. This integer measures the algebraic complexity of linear optimization over the variety. In practice, optimization problems often involve slicing a variety with linear spaces, leading to the study of sectional linear optimization (LO) degrees. We analyze the relationship between sectional LO degrees and classical polar degrees, demonstrating that the gap between them can be surprisingly large. We derive an exact formula for the sectional LO degrees of arbitrary Segre-Veronese varieties. This yields new bounds for the algebraic complexity of computing the Wasserstein distance to independence models in statistics.

Title: The Eigenvalue Method in coding theory

Abstract: One of the main goals in spectral graph theory is to deduce the principal properties and structure of a graph from its graph spectrum. In this talk we will show how spectral graph theory provides powerful methods for obtaining results concerning substructures of graphs, and also how these results can be useful in other mathematical fields such as coding theory. In particular, we will derive sharp eigenvalue bounds for the k-independence number of a graph (or equivalently, the independence number of the k-th graph power), which is known to be very hard to compute. We will see how to use polynomials and mixed integer linear programming in order to optimize and compute such spectral bounds. Finally, we will illustrate some recent applications of the obtained eigenvalue bounds to coding theory. The obtained results are encouraging and strongly suggest that spectral graph theory can uncover structural properties of ambient spaces that are relevant to coding theory, but that are often not captured by classical coding theory techniques. There is no question that eigenvalues play a central role in our fundamental understanding of graphs. The goal of this talk is to show that we can also use them for deepening our understanding of codes.

Title: Quantum representations and sightings of QFT 

Abstract: In this talk I will try to "approach" quantum field theory (QFT) from the perspective of quantum algebra. I'll begin with a casual discussion of classical, versus quantum phenomena in mathematics. Classical mathematics is, from the algebraic perspective, typified in the representation theory of finite groups and Lie groups. One can consider, for example, beautiful analyses of representations of SL_n(C) or Gl_n(C). Starting from this point, we'll travel into the world of QFT, approaching the subject through the lease of so-called quantum group representations.

Title: A flag refinement of the h*-formula for the hypersimplices

Abstract: The hypersimplex Δ_{k,n} is the convex hull of all 0/1-vectors of length n with coordinate sum k.

Early conjectured, and Kim proved, a combinatorial formula for the h*-polynomial of hypersimplices in terms of hypersimplicial decorated ordered set partitions. In this paper, we refine the Early--Kim formula to the flag h-numbers, introduced by Stanley, of the alcove triangulation complex of hypersimplices. We also give a representation-theoretic interpretation of its flag f-numbers in terms of Young permutation modules.

Title: Toric ideals of 0/1 polytopes through triangulations

Abstract: Over three decades ago, Sturmfels established a direct connection between the existence of quadratic Gröbner bases and regular flag unimodular triangulations. We solve a famous question posed by Herzog and Hibi about quadratic Gröbner bases for matroid toric ideals: we show that the base polytopes of the Fano plane and its dual do not have regular flag unimodular triangulations. Our proof relies on several novel tools: a lemma connecting the 1-skeleton of a lattice polytope to the lattice points in its dilations, an encoding with Boolean formulas and SAT solvers, and symmetry-breaking arguments. These tools can also decide whether a given lattice polytope has a unimodular or regular unimodular flag triangulation. Using them, we classify all full-dimensional (0,1)-polytopes of dimension at most five according to the existence of unimodular, flag unimodular, and regular flag unimodular (quadratic) triangulations. Among the consequences: there is an infinite family of polytopes with a flag unimodular triangulation whose toric ring is not Koszul, and the proportion of normal (0,1)-polytopes tends to zero as the dimension grows.

Title: A1 Gromov--Witten invariants

Abstract: There is a unique line through 2 points in the plane, and a unique conic through 5. These counts generalize to a count of degree d genus 0 curves in the plane passing through 3d-1 points. Surprisingly, the problem of determining these numbers is connected to mathematical physics, and it was not until the 1990's that it was completely solved. For example, Kontsevich determined them with a celebrated WDVV equation. Such formulas are valid when you allow your curves to be defined with complex coefficients.  Some of the solutions may be real, or integral, or defined over Q[i], but the fixed count records the number of solutions over the algebraic closure. A1-homotopy theory provides a framework to define analogous counts which are sensitive to the field of definition of the curves. A1 or quadratic Gromov--Witten define an arithmetically meaningful count of genus 0 plane curves. They were developed in joint work with Kass, Levine, and Solomon. This talk will completely determine the count of degree d rational curves passing through 3d-1 points over an arbitrary field of characteristic not 2 or 3. This is joint work with Erwan Brugallé and Johannes Rau.

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