Monday, 21 October 2024

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.

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.

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 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 $p=q \geqslant 0.10883$

In our study, we model the heart's biological system using percolation on a 2D lattice with three types of cells: Active, Waiting, and Inactive. The system incorporates inhibitory and refractory effects, influenced by two parameters: p_{act}, the probability of a Waiting cell becoming Active, and p_{switch}, the probability of an Inactive cell becoming Waiting. Inactive cells undergo a refractory period before reverting to Waiting.

Our findings show that inhibition raises the percolation threshold, slowing signal propagation. Conversely, reducing the refractory time lowers the threshold and speeds up signal transmission, but it can also trap the signal within the system. We analyzed the distribution of Inactive cells and critical exponents, observing that the Rushbrooke inequality is satisfied.

Tuesday, 22 October 2024

TBA

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.

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.

TBA

Wednesday, 23 October 2024

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.

No background beyond linear algebra will be assumed.

TBA

We show that the partial sums of the long Plucker relations for pairs of weakly separated Plucker coordinates oscillate around 0 on the totally nonnegative part of the Grassmannian. Our result generalizes the classical oscillating inequalities by Gantmacher–Krein (1941) and recent results on totally nonnegative matrix inequalities by Fallat–Vishwakarma (2023). In fact we obtain a characterization of weak separability, by showing that no other pair of Plucker coordinates satisfies this property. Weakly separated sets were initially introduced by Leclerc and Zelevinsky and are closely connected with the cluster algebra of the Grassmannian. Moreover, our work connects several fundamental objects such as weak separability, Temperley–Lieb immanants, and Plucker relations, and provides a very general and natural class of additive determinantal inequalities on the totally nonnegative part of the Grassmannian. This is joint work with Soskin.

For the unharmonic oscillator equation I will show how discriminants, i.e. locus of degenerate eigenvalues, can be plotted inside stability space, literally a space of framed quadratic differentials. They respect chamber structure.

Thursday, 24 October 2024

TBA

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.

Airy_1 and Airy_2 processes are stationary stochastic processes on the real line that arise in various contexts in integrable probability. In particular, they are obtained as scaling limits of passage time profiles in planar exponential last passage percolation (LPP) models with different initial conditions. In this talk, we shall present law of fractional logarithms with optimal constants for maxima and minima of Airy processes over growing intervals, extending and complementing the work of Pu. We draw upon the recently established sharp tail estimates for various passage times in exponential LPP by Baslingker et al., as well as geometric properties of exponential LPP landscape. The talk is based on a recent work with Riddhipratim Basu

(https://doi.org/10.48550/arXiv.2406.11826).

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.

A notion of generalized cluster algebras was introduced by L.Chekhov and M.Shapiro, where the exchange relations are given by polynomials of arbitrary order. A natural question arises; can these generalized cluster algebras be embedded into some standard cluster algebra? In a joint work in progress with R.Ramos, we provide a construction and show that the answer is yes. In this talk, I will briefly describe what a generalized cluster algebra is, and provide a few examples of this embedding.

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.

In this talk I will try to describe how the interplay between the system environment coupling and external driving frequency shapes the dynamical properties and steady state behavior in a periodically driven transverse field Ising chain subject to measurement. I will describe fate of the steady state entanglement scaling properties as a result of measurement induced phase transition. I will briefly explain how such steady state entanglement scaling can be exactly computed using asymptotic analysis of the determinant of associated correlation matrix which turned out to be of block Toeplitz form. I will try to point out the differences from the Hermitian systems in understanding entanglement scaling behaviour in regimes where the asymptotic analysis can be performed using Fisher-Hartwig conjecture. I will end the talk with some open questions in this direction.

The magnetohydrodynamics (1 + 1) dimension equation, with a force and force-free term, is analysed with respect to its point symmetries. Interestingly, it reduces to an Abel’s Equation of the second kind and, under certain conditions, to equations specified in Gambier’s family. The symmetry analysis for the force-free term leads to Euler’s Equation and to a system of reduced second-order odes for which singularity analysis is performed to determine their integrability.

Friday, 25 October 2024

TBA

Monday, 28 October 2024

TBA

TBA

We introduce two properties that characterise integrable discrete systems: singularity confinement and low growth. The latter is quantified through the dynamical degree, a quantity that is equal to 1 for integrable systems and larger than 1 for non-integrable ones. We show how the structure of singularities conditions the growth properties of a given system. We introduce the full deautonomisation discrete integrability criterion and illustrate its application through concrete examples. Starting from the results of R. Halburd we show how one can obtain the dynamical degree of a given mapping based on its singularity structure. The notion of Diophantine approximation is introduced as a practical way to obtain the dynamical degree. We show how one can obtain the degree growth of a given birational mapping in an algorithmic way using only the information on its singularities. Several examples of second-order mappings are presented and we show how our approach can be extended to higher-order ones. This is joint work with A. Ramani, R. Willox and T. Mase.

We study the structure of singularities in the discrete Korteweg–deVries equation and its modified sibling. Four different types of singularities are identified. The first type corresponds to localised, ‘confined’, singularities. Two other types of singularities are of infinite extent and consist of oblique lines. The fourth type of singularity corresponds to horizontal strips where the product of the values on vertically adjacent points is equal to 1. Due to its orientation this singularity can, in fact, interact with the other types. This type of singularity was dubbed ‘taishi’. The taishi can interact with singularities of the other two families, giving rise to very rich and quite intricate singularity structures. Nonetheless, these interactions can be described in a compact way through the formulation of a symbolic representation of the dynamics. We give an interpretation of this symbolic representation in terms of a box & ball system related to the ultradiscrete KdV equation.

Tuesday, 29 October 2024

I will discuss the relation between non-compact spin chains and the zero-range processes introduced by Sasamoto-Wadati, Povolotsky and Barraquand-Corwin. The main difference compared to the prime examples of integrable particle processes, namely the SSEP and the ASEP, is that for the models discussed in this talk several particles can occupy one and the same site. Guided by the desire to maintain the integrable structure, I will introduce boundary conditions for these models that are obtained from the boundary Yang-Baxter equation. This allows to define analogues of the open SSEP and ASEP with boundary reservoirs. Some recent exact results concerning these types of integrable non-equilibrium zero-range processes will be discussed.

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 this talk we shall present different approaches to the direct and inverse spectral problems for a family of solutions of KP2 , highlight the interplay of tropical geometry and combinatorics in solving such problems, and present unexpected connections to other problems in theoretical physics and statistical mechanics.

I will discuss quantum Ruijsenaars and Toda integrable systems from the quantum cluster varieties point of view.

Wednesday, 30 October 2024

In [1] we introduced the skew RSK dynamics, which is a time evolution for a pair of skew Young tableaux (P,Q). This gives a connection between the q-Whittaker measure and the periodic Schur measure, which immediately implies a Fredholm determinant formula for various KPZ models[2]. The dynamics exhibits interesting solitonic behaviors similar to box ball systems (BBS) and is related to the theory of crystal.

In this talk we explain basics of the skew RSK dynamics. We also plan to introduce a column version of the skew RSK dynamics and discuss its properties.

The talk is based on a collaboration with T. Imamura, M. Mucciconi and T. Scrimshaw.

[1] T. Imamura, M. Mucciconi, T. Sasamoto, Skew RSK dynamics: Greene invariants, affine crystals and applications to $q$-Whittaker polynomials, Forum of Mathematics, Pi (2023), e27 1–101 (arXiv: 2106.11922).

[2] T. Imamura, M. Mucciconi, T. Sasamoto, Solvable models in the KPZ class: approach through periodic and free boundary Schur measures

In this talk, I will introduce a connection between the Kardar-Parisi-Zhang solvable models and free fermions at finite temperature. This connection results from an identity between restricted Cauchy sums associated to the q-Whittaker/modified Hall-Littlewood polynomials and the skew Schur polynomials. The former describes the KPZ models while the later does the free fermions. We obtain this identity by introducing a new combinatorial approach called the skew RSK dynamics. This is a joint work with Matteo Mucciconi, Tomohiro Sasamoto and Travis Scrimshaw.

I will first present a generic argument to derive large deviations of a stochastic process when large deviations of certain functionals of that process are available. I will then apply such a general argument to the analysis of the lower tail of the height functions of the stochastic six vertex model starting with step initial conditions. One of the main novelties will be a proof of weak logarithmic concavity of the cumulative distribution function of the height function. This is a joint work with Sayan Das and Yuchen Liao.

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.

We propose a four-dimensional analogue of a multiplicative QRT mapping and construct its space of initial conditions as a rational variety of which the mapping becomes a pseudo-automorphism. We embed this variety into a family which admits symmetries forming an extended affine Weyl group and use this to construct a non-autonomous version of the mapping which can be regarded as a fourth order $q$ Painlevé equation. We also construct another $q$-Painlevé type mapping from a different translation element of the symmetry group.

Based on joint work with Adrian Stefan Carstea and Tomoyuki Takenawa.

Friday, 01 November 2024

In this talk, the problem of constructing the stationary states of the multispecies asymmetric simple exclusion process on a one-dimensional periodic lattice is revisited. A central role is played by a quantum oscillator-weighted five vertex model, which features an unusual weight conservation distinct from the conventional one. This approach clarifies the interrelations among several known results and refines their derivations, including the multiline queue construction and matrix product formulas. (Joint work with Masato Okado and Travis Scrimshaw)

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 $P_{\lambda}$.

The quantum K-theory of the flag variety is a ring defined by introducing a quantum product to the K-theory of the flag variety. Under appropriate localization, it is known that the following three rings (i), (ii), and (iii) are isomorphic, and this property allows for a detailed investigation of each ring: (i)the coordinate ring of the phase space of the relativistic Toda lattice, (ii) the quantum equivariant K-theory of the flag variety, and (iii) the K-equivariant homology ring of the affine Grassmannian.

The isomorphism between (i) and (ii) is derived from the Lax formalism of the relativistic Toda lattice [Ikeda-Iwao-Maeno]. The isomorphism between (ii) and (iii) is referred to as the K-Peterson isomorphism [Lam-Li-Mihalcea-Shimozono, Kato, Chow-Leung, Ikeda-Iwao-Maeno]. In this talk, I will outline how techniques from classical integrable systems, such as the construction of algebraic solutions and Bäcklund transformations, are applied to the study of geometry. This talk is based on a joint work with Takeshi Ikeda (Waseda University), Satoshi Naito (Tokyo Institute of Technology / Institute of Science Tokyo), and Kohei Yamaguchi (Nagoya University).