Seminários e cursos curtos

Let us consider two notions of concentration for homogeneous polynomials in $d$ complex variables on the unit sphere: a local notion measuring the fraction of the $L^2$-norm supported on a measurable subset and a global notion given by the generalized Wehrl entropy. Lieb and Solovej proved that the extremizers in both cases are monomials up to a unitary rotation. Their result generalizes the one by Lieb in 1978 on the Wehrl entropy conjecture for coherent states in representations of the Heisenberg group to symmetric representations of the groups $\operatorname{SU}(d)$.

In this talk, we will focus on the stability of the previous inequalities. Namely, if the concentration is close to the optimal one, we will quantify how close the polynomial is to the extremizers. This is obtained in full generality in the case $d=2$, while in the case of higher dimensions restrictions on the size of the subset or on the degree of the polynomials arise. We will finally recover analogous stability results in the Bargmann–Fock space.

This is a joint work with Joaquim Ortega-Cerdà (UB-CRM).

A compact symplectic manifold $(M, \omega)$ is called positive monotone if its first Chern class is a positive multiple of $[\omega]$ in the second de Rham group $H^2(M)$. A Fano variety is a smooth complex variety that admits a holomorphic embedding into $\mathbb{C} P^N$ for some $N$. Such a variety can be endowed with a symplectic form such that it becomes a positive monotone symplectic manifold. For this reason, positive monotone symplectic manifolds are considered the symplectic counterparts of smooth Fano varieties.

In the field of symplectic geometry, a general outstanding issue is understanding in what context positive monotone symplectic manifolds differ from Fano varieties. In low dimensions, namely two and four, it has been proven by Gromov, Taubes, McDuff, and Ohat-Ono that any positive monotone symplectic manifold is symplectomorphic to a Fano variety. Starting from dimension twelve, work by Fine and Panov provides examples of positive monotone symplectic manifolds that are not even homotopy equivalent to a Fano variety.

In this talk, I will explain what is known about the differences between Fano varieties and positive monotone symplectic manifolds endowed with a Hamiltonian action of a compact torus $T$. In particular, I will present new results for the case where the complexity of the action is one, i.e., $\frac{1}{2}\dim(M)-\dim(T)=1$.

This talk is based on joint work with Liat Kessler, Silvia Sabatini, and Daniele Sepe.

There are many works on geometric representation theory of quiver varieties and their relation to quantum loop algebras and Yangians. Recently, I have been interested in their variants, where quiver varieties are replaced by σ-quiver varieties, the fixed point loci of involutions on quiver varieties. I will explain my recent work on geometric representation theory of σ-quiver varieties and twisted Yangian, focusing on the special case of cotangent bundles of l-step isotropic flag varieties.

The Zakharov-Kuznetsov equation (ZK) is a model for the propagation of waves in the context of plasma physics and can be viewed as a two-dimensional analogue of the celebrated Korteweg-de Vries equation (KdV). In this talk, we study the Cauchy problem associated with the $k$-generalized Zakharov-Kuznetsov equation (gZK) posed on $\mathbb{R} \times \mathbb{T}$, where $k \geq 2$ is an integer. We establish several new Strichartz-type estimates in the framework of Jean Bourgain's $X_{s,b}$ spaces, with the main contributions being an almost optimal linear $L^4$-estimate and a family of bilinear refinements of this bound. As a direct application, we prove multilinear $X_{s,b}$-estimates that lead to improved local well-posedness thresholds for gZK via a fixed-point iteration.