The theory of De Giorgi (1958) and Nash (1959) solved Hilbert’s 19th problem and was a major contribution to 20th century PDE analysis. It is concerned with the Hölder regularity of solutions to elliptic and parabolic PDEs with rough (merely measurable) coefficients; it was developed by Moser (1960-1964) to include the Harnack inequality. These lectures  are an introduction to a recent active research area (Pascucci-Polidoro, Wang-Zhang, Golse-Imbert-M-Vasseur, Imbert-Silvestre, Imbert-Guerand, Guerand-M, Anceschi-Rebucci, Loher, Niebel-Zacher…): the extension of this theory to the hypoelliptic PDEs, local and nonlocal, that appear naturally in kinetic theory. The simpler prototypical case is the Kolmogorov equation (aka kinetic Fokker-Planck equation) with a rough matrix of coefficients in the kinetic diffusion. The course will in particular emphasize the recent quantitative robust methods based on the construction of trajectories and their connexions to control theory and hypocoercivity (works with Dieter, Hérau, Hutridurga, Niebel, Zacher).

The theory of De Giorgi (1958) and Nash (1959) solved Hilbert’s 19th problem and was a major contribution to 20th century PDE analysis. It is concerned with the Hölder regularity of solutions to elliptic and parabolic PDEs with rough (merely measurable) coefficients; it was developed by Moser (1960-1964) to include the Harnack inequality. These lectures  are an introduction to a recent active research area (Pascucci-Polidoro, Wang-Zhang, Golse-Imbert-M-Vasseur, Imbert-Silvestre, Imbert-Guerand, Guerand-M, Anceschi-Rebucci, Loher, Niebel-Zacher…): the extension of this theory to the hypoelliptic PDEs, local and nonlocal, that appear naturally in kinetic theory. The simpler prototypical case is the Kolmogorov equation (aka kinetic Fokker-Planck equation) with a rough matrix of coefficients in the kinetic diffusion. The course will in particular emphasize the recent quantitative robust methods based on the construction of trajectories and their connexions to control theory and hypocoercivity (works with Dieter, Hérau, Hutridurga, Niebel, Zacher).

One of the main goals of statistical physics is to study how spins displayed along the lattice $Z^d$ interact together and fluctuate as the temperature changes. When the spins belong to a discrete set (which is the case for example in the celebrated Ising model, where spins $sigma_x$ take values in ${-1,+1}$) the nature of the phase transitions which arise as one varies the temperature is now rather well understood. When the spins belong instead to a continuous space (for example the unit circle $S^1$ for the so-called XY model, the unit sphere $S^2$ for the classical Heisenberg model etc.), the nature of the phase transitions differs drastically from the discrete symmetry setting. The case where the (continuous) symmetry is non-Abelian is even more mysterious (especially when $d = 2$) than the Abelian case. In the latter case, Berezinskii, Kosterlitz and Thouless have predicted in the 70’s that these spins systems undergo a new type of phase transition in $d = 2$ — now called the BKT phase transition — which is caused by a change of behaviour of certain monodromies called « vortices ». 
In this course, I will give an introduction to this intriguing BKT phase transition. 
Lecture 1. Introduction to the Berezinskii-Kosterlitz-Thouless (BKT) phase transition.Main examples which undergo a BKT phase transition (XY and Villain models on $Z^2$, Coulomb gas, clock models, integer-valued height functions).Physics explanations of the BKT transition and difference between $S^1$ and $S^2$.
Lecture 2. Mathematical approach to BKT. 
Lecture 3. Domain of attraction of the BKT phase.
Lecture 4. Non-linear sigma models and curvature.
 

One of the main goals of statistical physics is to study how spins displayed along the lattice $Z^d$ interact together and fluctuate as the temperature changes. When the spins belong to a discrete set (which is the case for example in the celebrated Ising model, where spins $sigma_x$ take values in ${-1,+1}$) the nature of the phase transitions which arise as one varies the temperature is now rather well understood. When the spins belong instead to a continuous space (for example the unit circle $S^1$ for the so-called XY model, the unit sphere $S^2$ for the classical Heisenberg model etc.), the nature of the phase transitions differs drastically from the discrete symmetry setting. The case where the (continuous) symmetry is non-Abelian is even more mysterious (especially when $d = 2$) than the Abelian case. In the latter case, Berezinskii, Kosterlitz and Thouless have predicted in the 70’s that these spins systems undergo a new type of phase transition in $d = 2$ — now called the BKT phase transition — which is caused by a change of behaviour of certain monodromies called « vortices ». 
In this course, I will give an introduction to this intriguing BKT phase transition. 
Lecture 1. Introduction to the Berezinskii-Kosterlitz-Thouless (BKT) phase transition.Main examples which undergo a BKT phase transition (XY and Villain models on $Z^2$, Coulomb gas, clock models, integer-valued height functions).Physics explanations of the BKT transition and difference between $S^1$ and $S^2$.
Lecture 2. Mathematical approach to BKT. 
Lecture 3. Domain of attraction of the BKT phase.
Lecture 4. Non-linear sigma models and curvature.
 

One of the main goals of statistical physics is to study how spins displayed along the lattice $Z^d$ interact together and fluctuate as the temperature changes. When the spins belong to a discrete set (which is the case for example in the celebrated Ising model, where spins $sigma_x$ take values in ${-1,+1}$) the nature of the phase transitions which arise as one varies the temperature is now rather well understood. When the spins belong instead to a continuous space (for example the unit circle $S^1$ for the so-called XY model, the unit sphere $S^2$ for the classical Heisenberg model etc.), the nature of the phase transitions differs drastically from the discrete symmetry setting. The case where the (continuous) symmetry is non-Abelian is even more mysterious (especially when $d = 2$) than the Abelian case. In the latter case, Berezinskii, Kosterlitz and Thouless have predicted in the 70’s that these spins systems undergo a new type of phase transition in $d = 2$ — now called the BKT phase transition — which is caused by a change of behaviour of certain monodromies called « vortices ». 
In this course, I will give an introduction to this intriguing BKT phase transition. 
Lecture 1. Introduction to the Berezinskii-Kosterlitz-Thouless (BKT) phase transition.Main examples which undergo a BKT phase transition (XY and Villain models on $Z^2$, Coulomb gas, clock models, integer-valued height functions).Physics explanations of the BKT transition and difference between $S^1$ and $S^2$.
Lecture 2. Mathematical approach to BKT. 
Lecture 3. Domain of attraction of the BKT phase.
Lecture 4. Non-linear sigma models and curvature.
 

One of the main goals of statistical physics is to study how spins displayed along the lattice $Z^d$ interact together and fluctuate as the temperature changes. When the spins belong to a discrete set (which is the case for example in the celebrated Ising model, where spins $sigma_x$ take values in ${-1,+1}$) the nature of the phase transitions which arise as one varies the temperature is now rather well understood. When the spins belong instead to a continuous space (for example the unit circle $S^1$ for the so-called XY model, the unit sphere $S^2$ for the classical Heisenberg model etc.), the nature of the phase transitions differs drastically from the discrete symmetry setting. The case where the (continuous) symmetry is non-Abelian is even more mysterious (especially when $d = 2$) than the Abelian case. In the latter case, Berezinskii, Kosterlitz and Thouless have predicted in the 70’s that these spins systems undergo a new type of phase transition in $d = 2$ — now called the BKT phase transition — which is caused by a change of behaviour of certain monodromies called « vortices ». 
In this course, I will give an introduction to this intriguing BKT phase transition. 
Lecture 1. Introduction to the Berezinskii-Kosterlitz-Thouless (BKT) phase transition.Main examples which undergo a BKT phase transition (XY and Villain models on $Z^2$, Coulomb gas, clock models, integer-valued height functions).Physics explanations of the BKT transition and difference between $S^1$ and $S^2$.
Lecture 2. Mathematical approach to BKT. 
Lecture 3. Domain of attraction of the BKT phase.
Lecture 4. Non-linear sigma models and curvature.
 

The key property of linear dispersive flows is that waves with different frequencies travel with different group velocities, which leads to the phenomena of dispersive decay. Nonlinear dispersive flows also allow for interactions of linear waves, and their long time behavior is determined by the balance of linear dispersion on one hand, and nonlinear effects on the other hand. The first goal of these lectures  will be to present and motivate a new set of conjectures which aim to describe the global well-posedness and the dispersive properties of solutions in the most difficult case when the nonlinear effects are dominant, assuming only small initial data. This covers many interesting physical models, yet, as recently as a few years ago, there was no clue even as to what one might reasonably expect. The second objective of the lectures will be to describe some very recent results in this direction, in joint work with my collaborator Mihaela Ifrim from University of Wisconsin, Madison.

The key property of linear dispersive flows is that waves with different frequencies travel with different group velocities, which leads to the phenomena of dispersive decay. Nonlinear dispersive flows also allow for interactions of linear waves, and their long time behavior is determined by the balance of linear dispersion on one hand, and nonlinear effects on the other hand. The first goal of these lectures  will be to present and motivate a new set of conjectures which aim to describe the global well-posedness and the dispersive properties of solutions in the most difficult case when the nonlinear effects are dominant, assuming only small initial data. This covers many interesting physical models, yet, as recently as a few years ago, there was no clue even as to what one might reasonably expect. The second objective of the lectures will be to describe some very recent results in this direction, in joint work with my collaborator Mihaela Ifrim from University of Wisconsin, Madison.

The key property of linear dispersive flows is that waves with different frequencies travel with different group velocities, which leads to the phenomena of dispersive decay. Nonlinear dispersive flows also allow for interactions of linear waves, and their long time behavior is determined by the balance of linear dispersion on one hand, and nonlinear effects on the other hand. The first goal of these lectures  will be to present and motivate a new set of conjectures which aim to describe the global well-posedness and the dispersive properties of solutions in the most difficult case when the nonlinear effects are dominant, assuming only small initial data. This covers many interesting physical models, yet, as recently as a few years ago, there was no clue even as to what one might reasonably expect. The second objective of the lectures will be to describe some very recent results in this direction, in joint work with my collaborator Mihaela Ifrim from University of Wisconsin, Madison.

The key property of linear dispersive flows is that waves with different frequencies travel with different group velocities, which leads to the phenomena of dispersive decay. Nonlinear dispersive flows also allow for interactions of linear waves, and their long time behavior is determined by the balance of linear dispersion on one hand, and nonlinear effects on the other hand. The first goal of these lectures  will be to present and motivate a new set of conjectures which aim to describe the global well-posedness and the dispersive properties of solutions in the most difficult case when the nonlinear effects are dominant, assuming only small initial data. This covers many interesting physical models, yet, as recently as a few years ago, there was no clue even as to what one might reasonably expect. The second objective of the lectures will be to describe some very recent results in this direction, in joint work with my collaborator Mihaela Ifrim from University of Wisconsin, Madison.

In this talk, we first review the local monodromy at infinity of the Bessel F-isocrystals following Dwork, Sperber. Then we explain a generalization of this story for theta connections. Theta connections are certain rigid connections over P1 minus two points, related to epipelagic representations under the geometric Langlands correspondence. As an application, we verify a conjecture of Reeder-Yu on the epipelagic Langlands parameters under some technical conditions. The talk is based on my joint work with Xinwen Zhu and a work in progress with Lingfei Yi. 
 
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Motivated by the study of spontaneously broken non-invertible symmetries, I will examine the connection between N=1* gauge theories and elliptic Calogero-Moser (CM) integrable systems, with a focus on the well-studied type A case. N=1* gauge theories are mass-deformations of N=4 super Yang-Mills theories, while elliptic Calogero-Moser systems describe integrable systems of particles on a torus associated with compact or complex simple Lie algebras. A puzzling feature of this correspondence is the fact that there seems to be a one-to-one mapping between the isolated extrema of CM systems and the massive vacua of N=1* theories on R4. These extrema, however, should rather be in one-to-one correspondence with the massive vacua of N=1∗ theories compactified on a circle, which are typically more numerous. I will explain how this apparent discrepancy is resolved by distinguishing global variants of N=1* theories, leading to an association of CM systems with global variants of Lie groups rather than Lie algebras.