Andreï Kolmogorov est un mathématicien russe (1903-1987) qui a apporté des contributions frappantes en théorie des probabilités, théorie ergodique, turbulence, mécanique classique, logique mathématique, topologie, théorie algorithmique de l'information et en analyse de la complexité des algorithmes. Alexander Bufetov, Directeur de recherche CNRS (I2M - Aix-Marseille Université, CNRS, Centrale Marseille) et porteur local de la Chaire Jean-Morlet (Chaire Tamara Grava 2019 - semestre 1) donnera une conférence sur les contributions exceptionnelles et la vie dramatique d'un grand génie du XXe siècle.[-]

We consider the domino tilings of a large class of Aztec rectangles. For an appropriate scaling limit, we show that, the disordered region consists of roughly two arctic circles connected with a finite number of paths. The statistics of these paths is governed by a kernel, also found in other models (universality). The kernel thus obtained is believed to be a master kernel, from which the kernels, associated with critical points, can all be derived.[-]

These lectures will focus on understanding properties of classical operators and their connections to other important areas of mathematics. Perhaps the simplest example is the asymptotics of determinants of finite Toepltiz matrices, which have constants along the diagonals. The determinants of these $n$ by $n$ size matrices, have (in appropriate cases) an asymptotic expression that is of the form $G^n \times E$ where both G and E are constants. This expansion is useful in describing many statistical quantities variables for certain random matrix models.

In other instances, where the above expression must be modified, the asymptotics correspond to critical temperature cases in the Ising Model, or to cases where the random variables are in some sense singular.

Generalizations of the above result to other settings, for example, convolution operators on the line, are also important. For example, for Wiener-Hopf operators, the analogue of the determinants of finite matrices is a Fredholm determinant. These determinants are especially prominent in random matrix theory where they describe many quantities including the distribution of the largest eigenvalue in the classic Gaussian Unitary Ensemble, and in turn connections to Painleve equations.

The lectures will use operator theory methods to first describe the simplest cases of the asymptotics of determinants for the convolution (both discrete and continuous) operators, then proceed to the more singular cases. Operator theory techniques will also be used to illustrate the links to the Painlevé equations.[-]

These lectures will focus on understanding properties of classical operators and their connections to other important areas of mathematics. Perhaps the simplest example is the asymptotics of determinants of finite Toepltiz matrices, which have constants along the diagonals. The determinants of these $n$ by $n$ size matrices, have (in appropriate cases) an asymptotic expression that is of the form $G^n \times E$ where both G and E are constants. This expansion is useful in describing many statistical quantities variables for certain random matrix models.

In other instances, where the above expression must be modified, the asymptotics correspond to critical temperature cases in the Ising Model, or to cases where the random variables are in some sense singular.

Generalizations of the above result to other settings, for example, convolution operators on the line, are also important. For example, for Wiener-Hopf operators, the analogue of the determinants of finite matrices is a Fredholm determinant. These determinants are especially prominent in random matrix theory where they describe many quantities including the distribution of the largest eigenvalue in the classic Gaussian Unitary Ensemble, and in turn connections to Painleve equations.

The lectures will use operator theory methods to first describe the simplest cases of the asymptotics of determinants for the convolution (both discrete and continuous) operators, then proceed to the more singular cases. Operator theory techniques will also be used to illustrate the links to the Painlevé equations.[-]

These lectures will focus on understanding properties of classical operators and their connections to other important areas of mathematics. Perhaps the simplest example is the asymptotics of determinants of finite Toepltiz matrices, which have constants along the diagonals. The determinants of these $n$ by $n$ size matrices, have (in appropriate cases) an asymptotic expression that is of the form $G^n \times E$ where both G and E are constants. This expansion is useful in describing many statistical quantities variables for certain random matrix models.

In other instances, where the above expression must be modified, the asymptotics correspond to critical temperature cases in the Ising Model, or to cases where the random variables are in some sense singular.

Generalizations of the above result to other settings, for example, convolution operators on the line, are also important. For example, for Wiener-Hopf operators, the analogue of the determinants of finite matrices is a Fredholm determinant. These determinants are especially prominent in random matrix theory where they describe many quantities including the distribution of the largest eigenvalue in the classic Gaussian Unitary Ensemble, and in turn connections to Painleve equations.

The lectures will use operator theory methods to first describe the simplest cases of the asymptotics of determinants for the convolution (both discrete and continuous) operators, then proceed to the more singular cases. Operator theory techniques will also be used to illustrate the links to the Painlevé equations.[-]

We will investigate the form of spatio-temporal correlation functions for integrable models of systems of particles on the line. There are few analytical results for nonlinear systems, and so we start developing intuition from harmonic chains, where steepest descent analysis yields detailed asymptotic behaviour of the correlation functions in a variety of scaling limits. We will introduce integrable nonlinear lattices, explain the integrable solution procedure, as well as computational simulations to see dynamics of correlation functions in action.[-]

We will investigate the form of spatio-temporal correlation functions for integrable models of systems of particles on the line. There are few analytical results for nonlinear systems, and so we start developing intuition from harmonic chains, where steepest descent analysis yields detailed asymptotic behaviour of the correlation functions in a variety of scaling limits. We will introduce integrable nonlinear lattices, explain the integrable solution procedure, as well as computational simulations to see dynamics of correlation functions in action.[-]

Starting with Onsager's celebrated solution of the two-dimensional Ising model in the 1940's, Toeplitz determinants have been one of the principal analytic tools in modern mathematical physics; specifically, in the theory of exactly solvable statistical mechanics and quantum field models. Simultaneously, the theory of Toeplitz determinants is a very beautiful area of analysis representing an unusual combinations of profound general operator concepts with the highly nontrivial concrete formulae. The area has been thriving since the classical works of Szegö Fisher and Hartwig and Widom, and it very much continues to do so.

In the 90s, it has been realized that the theory of Toeplitz and Hankel determinants can be also embedded in the Riemann-Hilbert formalism of integrable systems. The new Riemann-Hilbert techniques proved very efficient in solving some of the long-standing problems in the area. Among them are the Basor-Tracy conjecture concerning the asymptotics of Toeplitz determinants with the most general Fisher-Hartwig type symbols and the double scaling asymptotics describing the transition behavior of Toeplitz determinants whose symbols change from smooth, Szegö to singular Fisher-Hartwig types. An important feature of these transition asymptotics is that they are described in terms of the classical Painlevè transcendents. The later are playing an increasingly important role in modern mathematics. Indeed, very often, the Painlevé functions are called now ``special functions of 21st century''.

In this mini course, the essence of the Riemann-Hilbert method in the theory of Topelitz determinants will be presented. The focus will be on the use of the method to obtain the Painlevé type description of the transition asymptotics of Toeplitz determinants. The Riemann-Hilbert view on the Painlevé function will be also explained.[-]

Starting with Onsager's celebrated solution of the two-dimensional Ising model in the 1940's, Toeplitz determinants have been one of the principal analytic tools in modern mathematical physics; specifically, in the theory of exactly solvable statistical mechanics and quantum field models. Simultaneously, the theory of Toeplitz determinants is a very beautiful area of analysis representing an unusual combinations of profound general operator concepts with the highly nontrivial concrete formulae. The area has been thriving since the classical works of Szegö Fisher and Hartwig and Widom, and it very much continues to do so.

In the 90s, it has been realized that the theory of Toeplitz and Hankel determinants can be also embedded in the Riemann-Hilbert formalism of integrable systems. The new Riemann-Hilbert techniques proved very efficient in solving some of the long-standing problems in the area. Among them are the Basor-Tracy conjecture concerning the asymptotics of Toeplitz determinants with the most general Fisher-Hartwig type symbols and the double scaling asymptotics describing the transition behavior of Toeplitz determinants whose symbols change from smooth, Szegö to singular Fisher-Hartwig types. An important feature of these transition asymptotics is that they are described in terms of the classical Painlevè transcendents. The later are playing an increasingly important role in modern mathematics. Indeed, very often, the Painlevé functions are called now ``special functions of 21st century''.

In this mini course, the essence of the Riemann-Hilbert method in the theory of Topelitz determinants will be presented. The focus will be on the use of the method to obtain the Painlevé type description of the transition asymptotics of Toeplitz determinants. The Riemann-Hilbert view on the Painlevé function will be also explained.[-]