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I will present results on the dynamics of horocyclic flows on the unit tangent bundle of hyperbolic surfaces, density and equidistribution properties in particular. I will focus on infinite volume hyperbolic surfaces. My aim is to show how these properties are related to dynamical properties of geodesic flows, as product structure, ergodicity, mixing, ...

37D40

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*Outreach;Mathematics Education and Popularization of Mathematics*

Alexander Bufetov got his Diploma in Mathematics at the Independent University of Moscow in 1999 and his PhD at Princeton University in 2005. After one year as a Postdoctoral student at the University of Chicago, he was employed as an Assistant Professor at Rice University where he also held the 'Edgar Odell Lovett Junior Chair'. In 2009, Alexander Bufetov joined the Steklov Mathematical Institute where he passed his habilitation thesis in order to supervise PhD students. In 2012, he became a CNRS Senior Researcher for the LATP (Laboratoire d’Analyse, Topologie, Probabilités) department at Aix-Marseille University.
Alexander Bufetov has received several prizes: a Prize by Moscow Mathematical Society in 2005, a grant by the Sloan Foundation and a grant from the President of the Russian Federation in 2010 and also a grant from the Simons Foundation at the Independent University of Moscow in 2011.
His research area is the Ergodic theory of dynamical systems.
Alexander Bufetov got his Diploma in Mathematics at the Independent University of Moscow in 1999 and his PhD at Princeton University in 2005. After one year as a Postdoctoral student at the University of Chicago, he was employed as an Assistant Professor at Rice University where he also held the 'Edgar Odell Lovett Junior Chair'. In 2009, Alexander Bufetov joined the Steklov Mathematical Institute where he passed his habilitation thesis in order ...

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I will present results of three studies, performed in collaboration with M.Benli, L.Bowen, A.Dudko, R.Kravchenko and T.Nagnibeda, concerning the invariant and characteristic random subgroups in some groups of geometric origin, including hyperbolic groups, mapping class groups, groups of intermediate growth and branch groups. The role of totally non free actions will be emphasized. This will be used to explain why branch groups have infinitely many factor representations of type $II_1$.
I will present results of three studies, performed in collaboration with M.Benli, L.Bowen, A.Dudko, R.Kravchenko and T.Nagnibeda, concerning the invariant and characteristic random subgroups in some groups of geometric origin, including hyperbolic groups, mapping class groups, groups of intermediate growth and branch groups. The role of totally non free actions will be emphasized. This will be used to explain why branch groups have infinitely ...

20E08 ; 20F65 ; 37B05

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Two important examples of the determinantal point processes associated with the Hilbert spaces of holomorphic functions are the Ginibre point process and the set of zeros of the Gaussian Analytic Functions on the unit disk. In this talk, I will talk such class of determinantal point processes in greater generality. The main topics concerned are the equivalence of the reduced Palm measures and the quasi-invariance of these point processes under certain natural group action of the group of compactly supported diffeomorphisms of the phase space. This talk is based partly on the joint works with Alexander I. Bufetov and partly on a more recent joint work with Alexander I. Bufetov and Shilei Fan.
Two important examples of the determinantal point processes associated with the Hilbert spaces of holomorphic functions are the Ginibre point process and the set of zeros of the Gaussian Analytic Functions on the unit disk. In this talk, I will talk such class of determinantal point processes in greater generality. The main topics concerned are the equivalence of the reduced Palm measures and the quasi-invariance of these point processes under ...

60G55 ; 46E20 ; 30H20

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Our first purpose is to show how aspects of the representation theory of (non-amenable) algebraic groups can be utilized to derive effective ergodic theorems for their actions. Our second purpose is to demonstrate some the many interesting applications that ergodic theorems with a rate of convergence have in a variety of problems. We will start by a discussion of property $T$ and show how to extend the spectral estimates it provides considerably beyond their usual formulations. We will also show how to derive best possible spectral estimates via representation theory in some cases. In turn, such spectral estimates will be used to derive effective ergodic theorems. Finally we will show how the rate of convergence in the ergodic theorem implies effective solutions in a host of natural problems, including the non-Euclidean lattice point counting problem, fast equidistribution of lattice orbits on homogenous spaces, and best possible exponents of Diophantine approximation on homogeneous algebraic varieties.
Our first purpose is to show how aspects of the representation theory of (non-amenable) algebraic groups can be utilized to derive effective ergodic theorems for their actions. Our second purpose is to demonstrate some the many interesting applications that ergodic theorems with a rate of convergence have in a variety of problems. We will start by a discussion of property $T$ and show how to extend the spectral estimates it provides considerably ...

37A30 ; 37A15 ; 37P55 ; 11F70

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Our first purpose is to show how aspects of the representation theory of (non-amenable) algebraic groups can be utilized to derive effective ergodic theorems for their actions. Our second purpose is to demonstrate some the many interesting applications that ergodic theorems with a rate of convergence have in a variety of problems. We will start by a discussion of property $T$ and show how to extend the spectral estimates it provides considerably beyond their usual formulations. We will also show how to derive best possible spectral estimates via representation theory in some cases. In turn, such spectral estimates will be used to derive effective ergodic theorems. Finally we will show how the rate of convergence in the ergodic theorem implies effective solutions in a host of natural problems, including the non-Euclidean lattice point counting problem, fast equidistribution of lattice orbits on homogenous spaces, and best possible exponents of Diophantine approximation on homogeneous algebraic varieties.
Our first purpose is to show how aspects of the representation theory of (non-amenable) algebraic groups can be utilized to derive effective ergodic theorems for their actions. Our second purpose is to demonstrate some the many interesting applications that ergodic theorems with a rate of convergence have in a variety of problems. We will start by a discussion of property $T$ and show how to extend the spectral estimates it provides considerably ...

37A30 ; 37A15 ; 37P55 ; 11F70

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Our first purpose is to show how aspects of the representation theory of (non-amenable) algebraic groups can be utilized to derive effective ergodic theorems for their actions. Our second purpose is to demonstrate some the many interesting applications that ergodic theorems with a rate of convergence have in a variety of problems. We will start by a discussion of property $T$ and show how to extend the spectral estimates it provides considerably beyond their usual formulations. We will also show how to derive best possible spectral estimates via representation theory in some cases. In turn, such spectral estimates will be used to derive effective ergodic theorems. Finally we will show how the rate of convergence in the ergodic theorem implies effective solutions in a host of natural problems, including the non-Euclidean lattice point counting problem, fast equidistribution of lattice orbits on homogenous spaces, and best possible exponents of Diophantine approximation on homogeneous algebraic varieties.
Our first purpose is to show how aspects of the representation theory of (non-amenable) algebraic groups can be utilized to derive effective ergodic theorems for their actions. Our second purpose is to demonstrate some the many interesting applications that ergodic theorems with a rate of convergence have in a variety of problems. We will start by a discussion of property $T$ and show how to extend the spectral estimates it provides considerably ...

37A30 ; 37A15 ; 37P55 ; 11F70

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Our first purpose is to show how aspects of the representation theory of (non-amenable) algebraic groups can be utilized to derive effective ergodic theorems for their actions. Our second purpose is to demonstrate some the many interesting applications that ergodic theorems with a rate of convergence have in a variety of problems. We will start by a discussion of property $T$ and show how to extend the spectral estimates it provides considerably beyond their usual formulations. We will also show how to derive best possible spectral estimates via representation theory in some cases. In turn, such spectral estimates will be used to derive effective ergodic theorems. Finally we will show how the rate of convergence in the ergodic theorem implies effective solutions in a host of natural problems, including the non-Euclidean lattice point counting problem, fast equidistribution of lattice orbits on homogenous spaces, and best possible exponents of Diophantine approximation on homogeneous algebraic varieties.
Our first purpose is to show how aspects of the representation theory of (non-amenable) algebraic groups can be utilized to derive effective ergodic theorems for their actions. Our second purpose is to demonstrate some the many interesting applications that ergodic theorems with a rate of convergence have in a variety of problems. We will start by a discussion of property $T$ and show how to extend the spectral estimates it provides considerably ...

37A30 ; 37A15 ; 37P55 ; 11F70

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*Mathematical Physics;Probability and Statistics*

The two-periodic Aztec diamond is a dimer or random tiling model with three phases, solid, liquid and gas. The dimers form a determinantal point process with a somewhat complicated but explicit correlation kernel. I will discuss in some detail how the Airy point process can be found at the liquid-gas boundary by looking at suitable averages of height function differences. The argument is a rather complicated analysis using the cumulant approach and subtle cancellations. Joint work with Vincent Beffara and Sunil Chhita.
The two-periodic Aztec diamond is a dimer or random tiling model with three phases, solid, liquid and gas. The dimers form a determinantal point process with a somewhat complicated but explicit correlation kernel. I will discuss in some detail how the Airy point process can be found at the liquid-gas boundary by looking at suitable averages of height function differences. The argument is a rather complicated analysis using the cumulant approach ...

60K35 ; 60G55 ; 60C05 ; 82B20 ; 05B45

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A determinantal point process governed by a Hermitian contraction kernel $K$ on a measure space $E$ remains determinantal when conditioned on its configuration on a subset $B \subset E$. Moreover, the conditional kernel can be chosen canonically in a way that is "local" in a non-commutative sense, i.e. invariant under "restriction" to closed subspaces $L^2(B) \subset P \subset L^2(E)$. Using the properties of the canonical conditional kernel we establish a conjecture of Lyons and Peres: if $K$ is a projection then almost surely all functions in its image can be recovered by sampling at the points of the process.

Joint work with Alexander Bufetov and Yanqi Qiu.
A determinantal point process governed by a Hermitian contraction kernel $K$ on a measure space $E$ remains determinantal when conditioned on its configuration on a subset $B \subset E$. Moreover, the conditional kernel can be chosen canonically in a way that is "local" in a non-commutative sense, i.e. invariant under "restriction" to closed subspaces $L^2(B) \subset P \subset L^2(E)$. Using the properties of the canonical conditional kernel ...

60G55 ; 60C05

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It is well-known that a large class of determinantal processes including the sine-process satisfies the Central Limit Theorem. For many dynamical systems satisfying the CLT the Donsker Invariance Principle also takes place. The latter states that, in some appropriate sense, trajectories of the system can be approximated by trajectories of the Brownian motion. I will present results of my joint work with A. Bufetov, where we prove a functional limit theorem for the sine-process, which turns out to be very different from the Donsker Invariance Principle. We show that the anti-derivative of our process can be approximated by the sum of a linear Gaussian process and small independent Gaussian fluctuations whose covariance matrix we compute explicitly.
It is well-known that a large class of determinantal processes including the sine-process satisfies the Central Limit Theorem. For many dynamical systems satisfying the CLT the Donsker Invariance Principle also takes place. The latter states that, in some appropriate sense, trajectories of the system can be approximated by trajectories of the Brownian motion. I will present results of my joint work with A. Bufetov, where we prove a functional ...

60G55 ; 60F05 ; 60G60

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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. ...

47B35

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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. ...

47B35

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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. ...

47B35

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Determinantal point processes arise in a wide range of problems in asymptotic combinatorics, representation theory and mathematical physics, especially the theory of random matrices. While our understanding of determinantal point processes has greatly advanced in the last 20 years, many open problems remain. The course will give an elementary introduction to determinantal point processes, starting from the basics and leading on to open problems.

PROGRAMME.

1. Examples.

2. Limit theorems.

3. Palm-Khintchine theory. Quasi-symmetries.

4. Determinantal point processes and extrapolation.
Determinantal point processes arise in a wide range of problems in asymptotic combinatorics, representation theory and mathematical physics, especially the theory of random matrices. While our understanding of determinantal point processes has greatly advanced in the last 20 years, many open problems remain. The course will give an elementary introduction to determinantal point processes, starting from the basics and leading on to open ...

60G55

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Determinantal point processes arise in a wide range of problems in asymptotic combinatorics, representation theory and mathematical physics, especially the theory of random matrices. While our understanding of determinantal point processes has greatly advanced in the last 20 years, many open problems remain. The course will give an elementary introduction to determinantal point processes, starting from the basics and leading on to open problems.

PROGRAMME.

1. Examples.

2. Limit theorems.

3. Palm-Khintchine theory. Quasi-symmetries.

4. Determinantal point processes and extrapolation.
Determinantal point processes arise in a wide range of problems in asymptotic combinatorics, representation theory and mathematical physics, especially the theory of random matrices. While our understanding of determinantal point processes has greatly advanced in the last 20 years, many open problems remain. The course will give an elementary introduction to determinantal point processes, starting from the basics and leading on to open ...

60G55

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Determinantal point processes arise in a wide range of problems in asymptotic combinatorics, representation theory and mathematical physics, especially the theory of random matrices. While our understanding of determinantal point processes has greatly advanced in the last 20 years, many open problems remain. The course will give an elementary introduction to determinantal point processes, starting from the basics and leading on to open problems.

PROGRAMME.

1. Examples.

2. Limit theorems.

3. Palm-Khintchine theory. Quasi-symmetries.

4. Determinantal point processes and extrapolation.
Determinantal point processes arise in a wide range of problems in asymptotic combinatorics, representation theory and mathematical physics, especially the theory of random matrices. While our understanding of determinantal point processes has greatly advanced in the last 20 years, many open problems remain. The course will give an elementary introduction to determinantal point processes, starting from the basics and leading on to open ...

60G55

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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 ...

60B20 ; 60K35 ; 37K10

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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 ...

60B20 ; 60K35 ; 37K10

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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 ...

47B35 ; 35Q15

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