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.

TBA

TBA

TBA

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

TBA

Bridgeland stability structure/condition on a triangulated category is a vast generalization of the notion of an ample line bunlde (or polarization) in algebraic geometry. The origin of the notion lies in string theory, and is applicable to derived categories of coherent sheaves, quiver representations and Fukaya categories. In a category with Bridgeland stability every objects carries a canonical filtration with semi-stable pieces, an analog of Harder-Narasimhan filtration.

It is expected that for categories over complex numbers Bridgeland stability structures often admit analytic enhancements, similar to the relation between ample bundles and usual Kaehler metrics. In a sense, this should be a generalization Donaldson-Uhlenbeck-Yau theorem which syas that a holomorphic vector bundle over compact Kaehler manifold is polystable if and only if it admits a metrization satisfying hermitean Yang-Mills equation.

In my course I will talk about a non-archimedean analog of analytic Bridgeland stability. I will show several examples, results and conjectures. In particular, I’ll introduce non-archimedean moment map equations, generalized honeycomb diagrams, and hypothetical stability on derived categories of coherent sheaves on maximally degenerating varieties over non-archimedean fields.

Bridgeland stability structure/condition on a triangulated category is a vast generalization of the notion of an ample line bunlde (or polarization) in algebraic geometry. The origin of the notion lies in string theory, and is applicable to derived categories of coherent sheaves, quiver representations and Fukaya categories. In a category with Bridgeland stability every objects carries a canonical filtration with semi-stable pieces, an analog of Harder-Narasimhan filtration.

It is expected that for categories over complex numbers Bridgeland stability structures often admit analytic enhancements, similar to the relation between ample bundles and usual Kaehler metrics. In a sense, this should be a generalization Donaldson-Uhlenbeck-Yau theorem which syas that a holomorphic vector bundle over compact Kaehler manifold is polystable if and only if it admits a metrization satisfying hermitean Yang-Mills equation.

In my course I will talk about a non-archimedean analog of analytic Bridgeland stability. I will show several examples, results and conjectures. In particular, I’ll introduce non-archimedean moment map equations, generalized honeycomb diagrams, and hypothetical stability on derived categories of coherent sheaves on maximally degenerating varieties over non-archimedean fields.

Bridgeland stability structure/condition on a triangulated category is a vast generalization of the notion of an ample line bunlde (or polarization) in algebraic geometry. The origin of the notion lies in string theory, and is applicable to derived categories of coherent sheaves, quiver representations and Fukaya categories. In a category with Bridgeland stability every objects carries a canonical filtration with semi-stable pieces, an analog of Harder-Narasimhan filtration.

It is expected that for categories over complex numbers Bridgeland stability structures often admit analytic enhancements, similar to the relation between ample bundles and usual Kaehler metrics. In a sense, this should be a generalization Donaldson-Uhlenbeck-Yau theorem which syas that a holomorphic vector bundle over compact Kaehler manifold is polystable if and only if it admits a metrization satisfying hermitean Yang-Mills equation.

In my course I will talk about a non-archimedean analog of analytic Bridgeland stability. I will show several examples, results and conjectures. In particular, I’ll introduce non-archimedean moment map equations, generalized honeycomb diagrams, and hypothetical stability on derived categories of coherent sheaves on maximally degenerating varieties over non-archimedean fields.