Flows on surfaces are one of the fundamental examples of dynamical systems, studied since Poincaré; area preserving flows arise from many physical and mathematical examples, such as the Novikov model of electrons in a metal, unfolding of billiards in polygons, pseudo-periodic topology. In this course we will focus on smooth area-preserving -or locally Hamiltonian- flows and their ergodic properties. The course will be self-contained, so we will define basic ergodic theory notions as needed and no prior background in the area will be assumed. The course aim is to explain some of the many developments happened in the last decade. These include the full classification of generic mixing properties (mixing, weak mixing, absence of mixing) motivated by a conjecture by Arnold, up to very recent rigidity and disjointness results, which are based on a breakthrough adaptation of ideas originated from Marina Ratner’s work on unipotent flows to the context of flows with singularities. We will in particular highlight the role played by shearing as a key geometric mechanism which explains many of the chaotic properties in this setup. A key tool is provided by Diophantine conditions, which, in the context of higher genus surfaces, are imposed through a multi-dimensional continued fraction algorithm (Rauzy-Veech induction):  we will explain how and why they appear and how they allow to prove quantitative shearing estimates needed to investigate chaotic properties.

Flows on surfaces are one of the fundamental examples of dynamical systems, studied since Poincaré; area preserving flows arise from many physical and mathematical examples, such as the Novikov model of electrons in a metal, unfolding of billiards in polygons, pseudo-periodic topology. In this course we will focus on smooth area-preserving -or locally Hamiltonian- flows and their ergodic properties. The course will be self-contained, so we will define basic ergodic theory notions as needed and no prior background in the area will be assumed. The course aim is to explain some of the many developments happened in the last decade. These include the full classification of generic mixing properties (mixing, weak mixing, absence of mixing) motivated by a conjecture by Arnold, up to very recent rigidity and disjointness results, which are based on a breakthrough adaptation of ideas originated from Marina Ratner’s work on unipotent flows to the context of flows with singularities. We will in particular highlight the role played by shearing as a key geometric mechanism which explains many of the chaotic properties in this setup. A key tool is provided by Diophantine conditions, which, in the context of higher genus surfaces, are imposed through a multi-dimensional continued fraction algorithm (Rauzy-Veech induction):  we will explain how and why they appear and how they allow to prove quantitative shearing estimates needed to investigate chaotic properties.

One of the main open problems in the mathematical analysis of fluid flows is the understanding of the inviscid limit in the presence of boundaries. In the case of a fixed bounded domain, it is an open problem to know whether solutions to the Navier-Stokes system with no slip boundary condition (zero Dirichlet boundary condition) do converge to a solution to the Euler system when the viscosity goes to zero. The main problem here comes from the fact that we cannot impose a no slip boundary condition for the Euler system. To recover a zero Dirichlet condition, Prandtl proposed to introduce a boundary layer (a small neighborhood of the boundary) in which viscous effects are still present. It turns out that the system that governs the flow in this small neighborhood, namely the Prandtl system has many mathematical difficulties. The goal of this course is to discuss some of the recent development in the inviscid limit as well as the study of the Prandtl system. We will also discuss the singularity formation for both the stationary and non stationary Prandtl system.

One of the main open problems in the mathematical analysis of fluid flows is the understanding of the inviscid limit in the presence of boundaries. In the case of a fixed bounded domain, it is an open problem to know whether solutions to the Navier-Stokes system with no slip boundary condition (zero Dirichlet boundary condition) do converge to a solution to the Euler system when the viscosity goes to zero. The main problem here comes from the fact that we cannot impose a no slip boundary condition for the Euler system. To recover a zero Dirichlet condition, Prandtl proposed to introduce a boundary layer (a small neighborhood of the boundary) in which viscous effects are still present. It turns out that the system that governs the flow in this small neighborhood, namely the Prandtl system has many mathematical difficulties. The goal of this course is to discuss some of the recent development in the inviscid limit as well as the study of the Prandtl system. We will also discuss the singularity formation for both the stationary and non stationary Prandtl system.

One of the main open problems in the mathematical analysis of fluid flows is the understanding of the inviscid limit in the presence of boundaries. In the case of a fixed bounded domain, it is an open problem to know whether solutions to the Navier-Stokes system with no slip boundary condition (zero Dirichlet boundary condition) do converge to a solution to the Euler system when the viscosity goes to zero. The main problem here comes from the fact that we cannot impose a no slip boundary condition for the Euler system. To recover a zero Dirichlet condition, Prandtl proposed to introduce a boundary layer (a small neighborhood of the boundary) in which viscous effects are still present. It turns out that the system that governs the flow in this small neighborhood, namely the Prandtl system has many mathematical difficulties. The goal of this course is to discuss some of the recent development in the inviscid limit as well as the study of the Prandtl system. We will also discuss the singularity formation for both the stationary and non stationary Prandtl system.

One of the main open problems in the mathematical analysis of fluid flows is the understanding of the inviscid limit in the presence of boundaries. In the case of a fixed bounded domain, it is an open problem to know whether solutions to the Navier-Stokes system with no slip boundary condition (zero Dirichlet boundary condition) do converge to a solution to the Euler system when the viscosity goes to zero. The main problem here comes from the fact that we cannot impose a no slip boundary condition for the Euler system. To recover a zero Dirichlet condition, Prandtl proposed to introduce a boundary layer (a small neighborhood of the boundary) in which viscous effects are still present. It turns out that the system that governs the flow in this small neighborhood, namely the Prandtl system has many mathematical difficulties. The goal of this course is to discuss some of the recent development in the inviscid limit as well as the study of the Prandtl system. We will also discuss the singularity formation for both the stationary and non stationary Prandtl system.

In these lectures we will present the mathematical proofs of conformal invariance of a number of models coming from planar statistical mechanics, including the Ising and dimer models. In particular, we will explain how discrete notions of holomorphicity can be used to solve discrete versions of classical Boundary Value Problems, and how this analysis is related to conformal invariance of certain observables in planar statistical mechanics.

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Paraphrasing Alexander Polyakov, « Conformal Field Theory is a way to learn about elementary particles by studying boiling water ».
There is a technical statement behind this joke: Euclidean Conformal Field Theory, under certain conditions, can be rotated to the Lorentzian signature, and vice versa. This means that even statistical physicists studying finite-temperature phase transitions on a lattice should learn about the Minkowski space! The goal of this course will be to explain various classical and recent results pertaining to this somewhat surprising conclusion.

Plan of the course:
– elementary introduction to Euclidean CFT in d>2 dimensions
– the Osterwalder-Schrader theorem about the Wick rotation of general reflection-positive Euclidean Quantum Field Theories, and its limitations
– the Luescher-Mack theorem about continuation of CFT correlation functions to the Lorentzian cylinder, and its limitations
– recent results about the analytic structure of Lorentzian CFT correlators

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