Seminars and short courses

In this talk, I will discuss recent results regarding the intermediate nonlinear Schrödinger equation (INLS). Analytically, INLS is a one-dimensional completely integrable nonlinear Schrödinger equation with a cubic derivative nonlinearity and is $L^2$-critical. A limiting form of INLS is the continuum Calogero-Moser equation (CCM), which is also completely integrable. Interestingly, CCM keeps the Hardy space $L^2_+$ invariant, and, under this assumption, tools from complete integrability have recently resolved the well-posedness problem for CCM in $L^2_+$. I will discuss progress on the well-posedness for INLS and CCM (not relying on complete integrability), outside of the Hardy space and in low-regularity. Our approach combines a gauge transformation, bilinear Strichartz estimates and a refined decomposition for smooth solutions. This is based on joint work with A. Chapouto (CNRS, Monash) and T. Laurens (UW-Madison).

In this talk, we consider the low regularity well-posedness problem for the Korteweg-de Vries equation (KdV) on the real line. Aiming to bridge the regularity gap between the scaling critical space and the known optimal well-posedness in $L^2$-based Sobolev spaces, we consider rough data in Fourier-Lebesgue spaces. Via infinite normal form reductions and exploiting algebraic cancellations, we introduce a new gauged KdV equation, equivalent to the original one at high regularity, but better behaved for rough solutions below the $H^{-1}$-scale. Surprisingly, our method does not rely on the completely integrable structure of KdV and is easily adapted to other equations with quadratic derivative nonlinearities, such as the dispersion-generalized Benjamin-Ono equations.

This talk is based on joint work with Simão Correia (IST, U. Lisboa) and João Pedro Ramos (IMPA).

We show that, under a division condition, the tangential Cauchy-Riemann cohomology of a compact Lie group with a left-invariant CR structure can be computed on a suitable maximal torus. As a consequence, we conclude that the tangential Cauchy-Riemann cohomology is finite-dimensional. We also show that, for a class of CR structures, this division condition is necessary for the total cohomology to be finite-dimensional. The proof combines Fourier analysis on compact Lie groups, highest-weight representations and Lie algebra cohomology. This not only generalizes but provides completely new proofs for the analogous result due to Pittie and for its extensions to Levi-flat CR structures, obtained by Jacobowitz and Jahnke.

We study the family of Cartan--Schouten connections on Lie groups, parameterized by $\lambda\in[0,1]$, whose geodesics through the identity are one-parameter subgroups. We compute their curvature, torsion, parallel transport, and geodesics, and develop Euler--Poincar\'e and Lie--Poisson reduction for mechanical systems via these connections, unifying the "minus'" and `"plus" cases. These inspire us to introduce a connection-dependent variational principle where the Lagrangian is expressed in terms of the parallel-transported velocity, leading to an integro-differential Euler--Lagrange equation that explicitly involves torsion and curvature memory terms. The general framework is illustrated on two concrete examples: the Heisenberg group, where the equations simplify to an ODE system, and the rotation group SO(3), where the integro-differential system is solved numerically via a Magnus expansion.