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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-orde...

We review the theory of exponential sums due to Weyl and van der Corput and consider several applications. If time permits, we also look at the theory of p-adic exponent pairs, as developed by Milićević.

Our Cosmological Standard Model, LambdaCDM, is a remarkable success story. It describes our Universe’s evolution from the Big Bang until today in terms of only a small handful of parameters. Despite its many successes, LambdaCDM is not a fundamental theory. In particular, the microscopic origin of dark matter and dark energy remain among the greatest puzzles in modern physics. Of the two, dark energy poses a particularly vexing challenge, as we lack an understanding of the smallness of its value. At the same time, over the last decade, observations have revealed further cracks in the LambdaCDM model, manifesting as discrepancies between early and late universe determinations of its parameters. 
 

In this lecture, I will first review the LambdaCDM model and establish why it is considered our best model of the Universe. In the second part, I will discuss the intriguing possibility that the cosmic tensions, referring to the observational and theoretical challenges mentioned above, are...

A colored vertex model is a solution of the Yang--Baxter equation based on a higher-rank Lie algebra. These models generalize the famous six-vertex model, which may be viewed in terms of osculating lattice paths, to ensembles of colored paths. By studying certain partition functions within these models, one may define families of multivariate rational functions (or polynomials) with remarkable algebraic features. In these lectures, we will examine a number of these properties:
(a) Exchange relations under the Hecke algebra;
(b) Infinite summation identities of Cauchy-type;
(c) Orthogonality with respect to torus scalar products;
(d) Multiplication rules (combinatorial formulae for structure constants).

Our aim will be to show that all such properties arise very naturally within the algebraic framework provided by the vertex models. If time permits, applications to probability theory will be surveyed.

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 ...

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).

Our Cosmological Standard Model, LambdaCDM, is a remarkable success story. It describes our Universe’s evolution from the Big Bang until today in terms of only a small handful of parameters. Despite its many successes, LambdaCDM is not a fundamental theory. In particular, the microscopic origin of dark matter and dark energy remain among the greatest puzzles in modern physics. Of the two, dark energy poses a particularly vexing challenge, as we lack an understanding of the smallness of its value. At the same time, over the last decade, observations have revealed further cracks in the LambdaCDM model, manifesting as discrepancies between early and late universe determinations of its parameters. 
 

In this lecture, I will first review the LambdaCDM model and establish why it is considered our best model of the Universe. In the second part, I will discuss the intriguing possibility that the cosmic tensions, referring to the observational and theoretical challenges mentioned above, are...

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.

A colored vertex model is a solution of the Yang--Baxter equation based on a higher-rank Lie algebra. These models generalize the famous six-vertex model, which may be viewed in terms of osculating lattice paths, to ensembles of colored paths. By studying certain partition functions within these models, one may define families of multivariate rational functions (or polynomials) with remarkable algebraic features. In these lectures, we will examine a number of these properties:
(a) Exchange relations under the Hecke algebra;
(b) Infinite summation identities of Cauchy-type;
(c) Orthogonality with respect to torus scalar products;
(d) Multiplication rules (combinatorial formulae for structure constants).

Our aim will be to show that all such properties arise very naturally within the algebraic framework provided by the vertex models. If time permits, applications to probability theory will be surveyed.

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.
 

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.

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 Ge...