The focusing nonlinear Schrödinger equation serves as a universal model for the amplitude of a wave packet in a general one-dimensional weakly-nonlinear and strongly-dispersive setting that includes water waves and nonlinear optics as special cases. Rogue waves of infinite order are a novel family of solutions of the focusing nonlinear Schr¨odinger equation that emerge universally in a particular asymptotic regime involving a large-amplitude and near-field limit of a broad class of solutions of the same equation. In this talk, we will present several recent results on the emergence of these special solutions along with their interesting asymptotic and exact properties. Notably, these solutions exhibit anomalously slow temporaldecay and are connected to the third Painlev´e equation. Finally, we will extend the emergence of rogue waves of infinite order to the first several flows of the AKNS hierarchy — allowing for arbitrarily many simultaneous flows — and report on recent work regarding their space-time asymptotic behavior under a general flow from the hierarchy.[-]

We prove that the solution to the Benjamin-Ono equation on the line, with initial data given by minus a soliton, exhibits scattering in infinite time. Our approach relies on an explicit formula for solutions with rational initial data in L2 having only simple poles. This formula is expressed as a ratio of determinants involving contour integrals. Additionally, we develop some spectral properties of the Lax operator associated with the Benjamin-Ono equation. This work is in collaboration with Elliot Blackstone, Patrick Gérard, and Peter D. Miller[-]

The focusing Continuum Calogero–Moser (CCM) equation is a completely integrable PDE that describes a continuum limit of a particle gas interacting pairwise through an inverse square potential. This system is well-posed in the scaling-critical space L2 below the mass of the soliton, but above this threshold there are solutions that blow up in finite time. In this talk, we will discuss some new and existing results about solutions below the soliton mass threshold. This is based on joint work with Rowan Killip and Monica Visan.[-]

In this talk, I discuss the energy-critical half-wave maps equation (HWM). It has been known for quite some time that (HWM) is completely integrable with a Lax pair structure. However, the question about global-in-time existence of solutions has been completely open so far — even for smooth and sufficiently small initial data. I will present very recent results that prove global well-posedness for rational initial data (with no size restriction) along with a general soliton resolution result in the large-time limit. The proofs strongly exploit the Lax structure of (HWM) in combination with an explicit flow formula. This is joint work with Patrick Gérard (Paris-Saclay).[-]

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.[-]

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 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 problems.

PROGRAMME.
1. Examples.
2. Limit theorems.
3. Palm-Khintchine theory. Quasi-symmetries.
4. Determinantal point processes and extrapolation.[-]