de la Vallée Poussin Chair 2025

Programme

Workshop on Integrable Combinatorics

Inaugural lecture, Tuesday 18 novembre 2025, 16h15-17h15 :  "Integrable combinatorics"

Physics has always provided insights and inspiration into new mathematics. We will concentrate here on Combinatorics, namely the art of counting objects

in classes, and follow guidance from Statistical physics that attaches probability weights to those objects and tries to tackle fundamental questions such as correlations or thermodynamic behavior. Symmetries of the systems studied can sometimes drastically simplify them, and in the best cases lead to exact solutions (i.e. exact counting, or exact asymptotics of such counts). Discrete or continuous integrable systems have enough symmetries to guarantee the existence of exact solutions. In this lecture, we will explore and solve a number of combinatorial integrable problems, in relation to various areas of mathematics: random geometry, random surfaces, cluster algebras, etc.

Lecture 1, Wednesday 19 novembre 2025, 9h00-10h00 : "Tilings, Vertex Models and other remarkable combinatorial objects"
 

The study of Random tilings was initiated and first developed in physics, from the Dimer models to quasi-crystals, before it became a pure mathematics subject. On the other hand, Vertex models of statistical physics describe the long range effect of local interactions in crystals. In this lecture, we start from a remarkable coincidence between the number of configurations of the so-called six vertex model, describing an ideal square lattice crystal of ice in two dimensions and a rhombus tiling model of a suitable hexagon. This is a small part of an extraordinary sequence of combinatorial coincidences, none of which is yet understood in terms of natural bijections. We show how the integrability of the six-vertex model allows to count its weighted configurations via a determinant. The computation of this determinant and some of its limits allows to identify the counting with that of so-called Descending Plane Partitions, in bijection with some specific rhombus tiling problem. We then generalize the problem to triangular lattice ice, in the form of the twenty vertex model, and identify a suitable non-bijective domino tilings counting problem. Again, integrability allows to compute the numbers of configurations exactly, and to prove new coincidences, unsupported by natural bijections. 

Lecon 2, Wednesday 19 novembre 2025, 14h00-15h00 "Limit shapes"

A natural question in physics regards the thermodynamic limit, of large size and small mesh, where one expects to capture only the essential properties of a model, independently of its details. When it comes to tilings of specific domains a whole theory of limit shapes was developed in mathematics by Kenyon, Okounkov, Sheffield and others. These are generalizations of the famous arctic circle theorem of Cohn, Kenyon, Propp for the domino tilings of the so-called Aztec diamond. From the physics point of view, tiling problems are free fermion models, namely they can be expressed in terms of ``fermionic" degrees of freedom, generally in the form non-intersecting paths. However Vertex models such as the 6 vertex model are models of interacting fermions, where paths are allowed to touch thus violating the fermionic condition. In this lecture, we explore limit shapes of 6 and 20 vertex models, establishing their arctic phenomena by using their integrability. We find some remarkable relations between limit shapes of equinumerous yet non-bijective problems explored in lecture 1.

Lecon 3, Thursday 20 novembre 2025, 9h00-10h00 : "Meanders"

This lecture is devoted entirely to the ``Problème des Timbres poste" or stamp-folding problem first posed by Emile Lemoine in 1891, revisited by Poincaré in 1912 in purely geometric terms and later rebaptized ``meander problem" by Arnold in 1991. In the latter language, we wish to count inequivalent configurations of a non-self-intersecting road circuit crossing a straight infinite river through 2N bridges.

We show how the question relates to deep and fundamental problems in algebra and geometry, and how the physics theory of two-dimensional quantum gravity has allowed us to make remarkable predictions on the asymptotic count of meander configurations.