Schedule

Abstract:

Chapoton initiated the study of $q$-Ehrhart polynomials: given a lattice polytope $P$ (i.e., $P$ is the convex hull of finitely many integer points in ${\bf R}^d$), we fix an integral linear form $\lambda$ and sum $q^{\lambda(m)}$ over all integer lattice points $m$ in the dilate $nP$, as a function ehr$(q,n)$. For $q=1$, this recovers the usual Ehrhart polynomial of $P$, counting integer lattice points in $nP$. From the viewpoint of the multigraded Hilbert series of the homogenization of $P$, the generating function of ehr$(q,n)$ is a simple specialization. However, ehr$(q,n)$ still carries an (a priori surprising) polynomial structure: Chapoton proved that that there exists a polynomial chap$(x)$, whose coefficients are rational functions of $q$, such that ehr$(q,n)$ equals the evaluation of chap$(x)$ at the $q$-integer $[n]_q$.

Our goal is twofold. First, we will show how Chapoton's results follow somewhat organically from Brion's Theorem, which decomposes the integer-point structure of $P$ into that of its vertex cones. This ansatz also yields several immediate extensions of Chapoton's work. Second, we will outline how similar $q$-polynomials might be useful in other settings, such as generalizing the chromatic polynomial of a graph, with connections to chromatic symmetric functions and the arithmetic of order cones.

Abstract:

The use of weighted contraction—particularly for vertex and edge weighted graphs—has emerged as a powerful and unifying technique in algebraic graph theory, with applications spanning statistical mechanics, symmetric functions, and coloring problems. In this overview, we trace the development of this idea, beginning with early examples such as the $U$- and $W$-polynomials. We then explore the role of edge weights, setting the stage for the introduction of the $V$-polynomial, which gives a universal framework for graph polynomials defined via state sums and deletion–contraction relations involving both vertex and edge weights. After reviewing the key structural features of the $V$-polynomial—including its recursion, activities expansion, and evaluations—we present a catalog of examples from the literature that illustrate how weighted contraction appears in diverse settings. These include chromatic symmetric functions, list and DP-coloring, Laplacians, and the Potts model with external fields. We show how these polynomials can be realized as specializations of the $V$-polynomial, and how this perspective yields new or simplified proofs of known results. This synthesis highlights the versatility of the $V$-polynomial as a unifying tool for graph-theoretic invariants arising from weighted combinatorial processes.

Abstract:

At the third edition of FPSAC, which took place in 1991 in Bordeaux, Doron Zeilberger gave an invited talk with the title "Identities in Search of Identity". Shortly before he had published his seminal paper on the holonomic systems approach to special function identities and, together with Herbert Wilf, had developed a framework for proving hypergeometric summation identities, nowadays known as WZ theory. During the following 35 years, the theory has been considerably extended and refined, and evolved into its own research area within symbolic computation. The corresponding implementations in computer algebra systems can help to solve various summation and integration problems, and likewise they support many operations on formal power series. Therefore it is not surprising that this algorithmic framework has found numerous applications, for instance in combinatorics. In our lecture, we aim at recapitulating the evolution of this research area, highlighting its main achievements, and discussing some recent trends. A special emphasis is put on the treatment of determinants and Pfaffians that became also amenable to symbolic methods, via the holonomic ansatz: the sought identity may be transformed into a set of summation identities, which themselves can be proven algorithmically. This procedure is elucidated with prominent examples, such as the q-enumeration of totally-symmetric plane partitions, the counting of configurations in the twenty-vertex model, and the evaluation of binomial determinants emerging from rhombus tilings.

Abstract:

In this talk, we will talk about extremal weight modules of type $A_{+\infty}$ or type $A$ of infinite rank, and their combinatorics. We consider a category generated by extremal weight modules, which is non-semisimple but still has a nice combinatorial structure due the theory of crystal base. We explain how it is related to or realizes well-known objects in the theory of symmetric functions including Fomin's Schur functions in non-commutative variables, generalized Cauchy identity, Koike's universal characters and so on. This talk is based on a joint work with Soo-Hong Lee.

Abstract:

Computing with any sort of object requires a way of encoding it on a computer, which poses a problem in enumerative combinatorics where the objects of interest are (infinite) combinatorial sequences. Thankfully, the generating function of a combinatorial sequence often satisfies natural algebraic/differential/functional equations, which can then be viewed as data structures for the sequence. In this talk we survey methods to take a sequence encoded by such data structures and automatically determine asymptotic behaviour using techniques from the field of analytic combinatorics. We also discuss methods to automatically characterize the asymptotic behaviour of multivariate sequences using analytic combinatorics in several variables (ACSV). The focus of each topic will be rigorous algorithms that have already been implemented in computer algebra systems and can be easily used by anyone. This talk is also complemented by a software presentation of Andrew Luo later in the day.

Abstract:

Volume polynomials have their origins in convex geometry, but their true potential emerges through connections with enumerative algebraic geometry. They are prototypical examples of Lorentzian polynomials and have played a central role in applications of the Hodge Index Theorem, notably in proving the log-concavity of the (absolute values of) coefficients of chromatic polynomials of graphs. Which properties, beyond Lorentzianity, do volume polynomials possess? Addressing this question requires a twofold approach: on one hand, we seek new inequalities for the coefficients of volume polynomials; on the other, we aim to construct families of examples realizing all possibilities. We present partial answers to both directions: the inequality aspect is based on the Calabi–Yau theorem and interactions between algebraic geometry, Kähler geometry, and mixed discriminants, while the constructive aspect draws from polytopes, toric geometry, and abelian varieties. We will also pose several interesting conjectures. The talk is based on joint works with Daoji Huang, June Huh, and Botong Wang.

Abstract:

The Schur function, the most important polynomial in the theory of symmetric functions, has been the subject of various profound theories in representation theory and combinatorics, including the study of the properties of the Macdonald function known as a generalization of the Schur function. In this talk, I will introduce some of the new theoretical developments obtained by applying this theory to zeta functions, which are the subject of research in analytic number theory. Specifically, I will introduce a generalization of the Riemann zeta function as a multivariate zeta function with a structure similar to that of the Schur function, which we call the Schur multiple zeta function, and discuss its determinant formulae, product formula, duality formula, and other properties and applications.

Abstract:

Many combinatorial objects with strikingly good enumerative formulae also have remarkable dynamical behavior and underlying algebraic structure. In this talk, we consider the promotion action on certain tableaux and explain its small, predictable order by reinterpreting promotion as a rotation in disguise. We show how the search for such a visual explanation of combinatorial dynamics led to solutions of algebraic problems involving web bases and a generalization of the six-vertex model of statistical physics. We find that this framework includes several other combinatorial objects of interest, including noncrossing partitions, alternating sign matrices, and plane partitions. This talk is based on joint works on webs with subsets of { Ashleigh Adams, Chris Fraser, Christian Gaetz, Rebecca Patrias, Stephan Pfannerer, Oliver Pechenik, Joshua P. Swanson }

Abstract:

I will overview how certain combinatorial graph invariants have symmetries agreeing with those of Feynman periods from quantum field theory. In this way these invariants behave somewhat as discrete analogues of the Feynman integrals themselves. We are starting to better understand how they relate to each other as well as to the underlying number theory and geometry and they are also just nice combinatorially.