ICTS

It is well known that the box-ball system discovered by Takahashi and Satsuma can be obtained by the ultra-discrete analogue of the discrete integrable system, including both the ultra-discrete analogue of the KdV lattice and the ultra-discrete analogue of the Toda lattice. This mini-course will demonstrate that it is possible to derive extended models of the box-ball systems related to the relativistic Toda lattice and the fundamental Toda orbits, which are obtained from the theory of orthogonal polynomials and their extensions. We will first introduce an elementary procedure for deriving box-ball systems from discrete KP equations. Then, we will discuss the relationship between discrete Toda lattices and their extensions based on orthogonal polynomial theory, and outline the exact solutions and ultra-discretization procedures for these systems. Additionally, we will introduce the box-ball system on R, which is obtained by clarifying its relationship with the Pitman transformation in probability theory. Furthermore, we will also discuss how extended box-ball systems can compute the Smith normal form of bidiagonal integer matrices.

We consider \emph{bond percolation games} on the $2$-dimensional square lattice in which each edge (that is either between the sites $(x,y)$ and $(x+1,y)$ or between the sites $(x,y)$ and $(x,y+1)$, for all $(x,y) \in \mathbb{Z}^{2}$) has been assigned, \emph{independently}, a label that reads \emph{trap} with probability $p$, \emph{target} with probability $q$, and \emph{open} with probability $1-p-q$. Once a realization of this labeling is generated, it is revealed in its entirety to the players before the game starts. The game involves a single token, initially placed at the origin, and two players who take turns to make \emph{moves}. A \emph{move} involves relocating the token from where it is currently located, say the site $(x,y)$, to any one of $(x+1,y)$ and$(x,y+1)$. A player wins if she is able to move the token along an edge labeled a target, or if she is able to force her opponent to move the token along an edge labeled a trap. The game is said to result in a draw if it continues indefinitely (i.e.\ with the token always being moved along open edges). We ask the question: for what values of $p$ and $q$ is the probability of draw equal to $0$? By establishing a close connection between the event of draw and the \emph{ergodicity} of a suitably defined \emph{probabilistic cellular automaton}, we are able to show that the probability of draw is $0$ when $p > 0.157175$ and $q=0$, and when $p=q \geqslant 0.10883$

Totally positive (TP) matrices are matrices in which each minor is positive. First introduced in 1930's by I. Schoenberg and F. Gantmakher and M. Krein, these matrices proved to be important  in many areas of pure and applied mathematics. The notion of total positivity was generalized by G. Lusztig in the context of reductive Lie groups and inspired the discovery  of cluster algebras by S. Fomin and A. Zelevinsky.

In this mini-course, I will first review some basic features of TP matrices, including their spectral properties and discuss some of their classical applications. Then I will focus on weighted networks parametrization of TP matrices due to A. Berenstein, S. Fomin and A. Zelevinsky. I will show how elementary transformations of planar networks lead to criteria of total positivity and important examples of mutations in the theory of cluster algebras. Finally, I will explain how particular sequences of mutations can be used to construct exactly solvable nonlinear dynamical systems including the Toda lattice and the pentagram map.

No background beyond linear algebra will be assumed.

Totally positive (TP) matrices are matrices in which each minor is positive. First introduced in 1930's by I. Schoenberg and F. Gantmakher and M. Krein, these matrices proved to be important  in many areas of pure and applied mathematics. The notion of total positivity was generalized by G. Lusztig in the context of reductive Lie groups and inspired the discovery  of cluster algebras by S. Fomin and A. Zelevinsky.

In this mini-course, I will first review some basic features of TP matrices, including their spectral properties and discuss some of their classical applications. Then I will focus on weighted networks parametrization of TP matrices due to A. Berenstein, S. Fomin and A. Zelevinsky. I will show how elementary transformations of planar networks lead to criteria of total positivity and important examples of mutations in the theory of cluster algebras. Finally, I will explain how particular sequences of mutations can be used to construct exactly solvable nonlinear dynamical systems including the Toda lattice and the pentagram map.

No background beyond linear algebra will be assumed.

We will present a method for computing the stationary measures of integrable probabilistic systems on finite domains. Focusing on the example of a well-studied model called last passage percolation, we will describe the stationary measure in various ways, and emphasize the key role played by Schur symmetric functions. The method works as well for other models and their associated families of symmetric functions, suchas Whittaker functions or Hall-Littlewood polynomials. We will also discuss how this is related to the traditional approach for computing stationary measures of interacting particle systems between boundary reservoirs: the matrix product ansatz.

Alternating sign matrices (ASM) and descending plane partitions (DPP) both have the concept of rank, and it has been known that the same number of them exist for each rank (conjectured in 1983 by W. H. Mills, David P. Robbins and Howard Rumsey, Jr., and proved in 1996 by Doron Zeilberger and by Greg Kuperberg independently). However, no explicit bijections between them have been found so far. This problem is known as the ASM-DPP bijection problem.

In 2020, Fischer and Konvalinka constructed a bijection between ASM(n)xDPP(n-1) and ASM(n-1)xDPP(n), where ASM(n) denotes the set of ASMs with rank n, and it is similar for DPP(n). This bijection was developed using the concept of signed bijections. I introduce the notion of compatibility of signed bijections to measure the naturalness of signed bijections and to simplify the construction. In this talk, I present the definition of compatibility and some of the results obtained from it. For example, these include the refined structure of Gelfand-Tsetlin patterns and some simplified signed bijections related to ASMs.

In this talk, we consider the skein algebra (or cluster algebra) of an oriented marked surface with boundary. Each triangulation of the surface determines an algebraic torus in the corresponding variety, and in some cases these tori collectively cover the variety. In other cases, there are points not contained in any algebraic torus, which we call deep points. In joint work with Greg Muller, we have classified the deep points of skein algebras of unpunctured marked surfaces. We will discuss this classification, beginning with convex polygons and then extending those same ideas to other unpunctured surfaces.

Two-dimensional integrable lattice models that can be described in terms of (non-intersecting, possibly osculating) paths with suitable boundary conditions display the arctic phenomenon: the emergence of a sharp phase boundary between ordered cristalline phases (typically near the boundaries) and disordered liquid phases (away from them). We show how the so-called tangent method can be applied to models such as the 6 Vertex model or its triangular lattice variation the 20 Vertex model, to predict exact arctic curves. A number of companion combinatorial results are obtained, relating these problems to tiling problems of associated domains of the plane.

I will revisit some integrable difference equations arising in the study of the distance statistics of random planar maps (discrete surfaces built from polygons). In a paper from 2003 written jointly with P. Di Francesco and E. Guitter, we conjectured a general formula for the so-called ``two-point function'' characterizing these statistics. The first proof of this formula was given much later in a paper from 2012 joint with E. Guitter, where we used bijective arguments and the combinatorial theory of continued fractions. I will present a new elementary and purely analytic proof of the result, obtained by considering orthogonal polynomials with respect to a polynomial deformation of the Wigner semicircle distribution. This talk is based on a work in progress with Sofia Tarricone.
 

In our previous study (TI-Mucciconi-Sasamoto, Forum of Mathematics, Pi 11(e27) 1-101,2023) we introduced a deterministic time evolution of a pair of skew semistandard Young tableaux called the skew RSK dynamics.

In this talk we introduce a variant based on the column bumping in the RSK correspondence, which we call the column skew RSK dynamics. Utilizing (bi) crystal structure in the pair of skew tableaux, we show that the column skew RSK dynamics can be mapped to the single species box and ball system (BBS).  Using the mapping we obtain a relation between restricted Cauchy sums about the modified Hall-Littlewood polynomials and the skew Schur polynomials. This talk is based on the joint work with Matteo Mucciconi, Tomohiro Sasamoto and Travis Scrimshaw.

The theory of degree growth and algebraic entropy plays a crucial role in the field of discrete integrable systems. However, a general method for calculating degree growth for lattice equations (partial difference equations) is not yet known. In this talk, I will propose a new method to rigorously compute the exact degree of each iterate for lattice equations. The strategy is to extend Halburd's method, which is a novel approach to computing the exact degree of each iterate for mappings (ordinary difference equations) from the singularity structure, to lattice equations.

First, I will illustrate, without rigorous discussion, how to calculate degrees for lattice equations using the lattice version of Halburd's method and discuss what problems we need to solve to make the method rigorous. Then, I will provide a framework to ensure that all calculations are accurate and rigorous. If time permits, I would also like to discuss how to detect the singularity structure of a lattice equation.

Reference: T. Mase, Exact calculation of degrees for lattice equations: a singularity approach, arXiv:2402.16206.

In this talk, I'll describe some recently discovered connections between one-dimensional interacting particle models (the ASEP and the TAZRP) and Macdonald polynomials and show the combinatorial objects that make these connections explicit. Recently, a new tableau formula was found for the modified Macdonald polynomial $\widetilde{H}_{\lambda}$ in terms of a queue inversion statistic that is naturally related to the dynamics of the TAZRP. We give a new compact tableau formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ using the same queue inversion statistic on certain sorted non-attacking tableaux. The nonsymmetric components of our formula are the ASEP polynomials, which specialize to the probabilities of the asymmetric simple exclusion process (ASEP) on a circle, and the queue inversion statistic encodes to the dynamics of the ASEP.  Our tableaux are in bijection with Martin's multiline queues, from which we obtain an alternative multiline queue formula for $P_{\lambda}$.