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[-]

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).[-]

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.[-]

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.[-]

I will present for each integer d > 3 a K-trivial log canonical variety over the complex numbers of dimension d that does not admit a Beauville-Bogomolov decomposition. That is, for the universal cover X of the variety, there is no decomposition of X as a product of an affine space and of three types of projective varieties: strict Calabi-Yau, symplectic and rationally connected varieties. Note: the counterexample is sharp in the sense that for Kawamata log terminal varieties the decomposition does hold.[-]

We'll discuss our joint work with Ron Donagi and Tony Pantev on the construction of the Higgs bundles associated to Hecke eigensheaves for the geometric Langlands program in the case of rank 2 local systems on a curve of genus 2 . Recall that the moduli space of bundles in this case has two connected components: $\mathbb{P}^3$ and the intersection of two quadrics in $\mathbb{P}^5$. We look for Higgs bundles on these spaces with parabolic structure and logarithmic poles along the wobbly locus. This leads to the study of the geometry of the wobbly locus and its singularities, and the use of our Dolbeault higher direct image construction for the calculation of Hecke operators.[-]

I will first explain the joint work with Walter van Suijlekom on a new result about th zeros of the Fourier transform of extremal eigenvectors for quadratic forms associated to distributions on a bounded interval and its relation with the spectral action. Then I will explain how these results allow to advance in the joint work which I am doing with Consani and Moscovici on the zeta spectral triple. Finally, if time permits, I will discuss several ideas in connection with physics and non-commutative geometry.[-]