Seminars

Foliated Plateau problems in Cartan-Hadamard manifolds

Abstract: We solve a foliated Plateau problem for surfaces of constant extrinsic curvature in 3-dimensional Cartan-Hadamard manifolds, and review its applications to the study of asymptotic Plateau problems à la Calegari-Marques-Neves. This is the content of arXiv:2212.13604 and is joint work with Ben Lowe (U. Chicago) and Sébastien Alvarez (Universidad de la República, Montevideo).

Positivstellensatz, Semidefinite Programming and Global Optimization with Polynomials

Abstract: We present Putinar and Schmüdgen theorems for certifying positivity of polynomials over closed basic semialgebraic sets. These certificates, based in sum of squares (SOS), can be approached from a hierarchy of semidefinite programming problems. This metodology is called Lasserre hierarchy relaxation, or moment/SOS hierarchy, and allow us try to find global solution for optimization problems with polynomials. Some examples of Optimal Power Flow problems illustrate the techniques.

If time isn't enough, we will do this seminar in two parts.

Rigidity of u-Gibbs measures for uniformly expanding partially hyperbolic endomorphisms

Abstract: We consider expanding endomorphisms of the two dimensional torus admitting a dominated splitting with a weak (center) expanding direction and a strong expanding direction, such as perturbations of the map A(x,y)=(3x mod 1, 2x mod 1). In this context we have a well defined center foliation. The strong unstable direction may not to be uniquely integrable (to a foliation) in this case, even though strong unstable curves do exist. The endomorphism is called special when the strong unstable bundle uniquely integrates into a foliation. Using the inverse limit space one can define u-Gibbs measures. The linear map A, for instance, presents uncountably many different u-Gibbs measures, all of them supported in unions of horizontal strong unstable compact curves. In this talk I will discuss some recent results concerning rigidity of u-Gibbs measures for partially hyperbolic systems. In the case of partially hyperbolic uniformly expanding endomorphisms we show that, for non-special maps, any fully supported and ergodic u-Gibbs measure must coincide with the unique absolutely continuous invariant measure of the map. We give examples of non-special maps having unstable curves with dense forward orbit, and for these examples maps we can classify all u-Gibbs measures. This is joint work with Marisa Cantarino (UFF).

Variations of Hodge structures and Hodge loci II

Abstract: This will be a continuation of the talk given last week. We will discuss period domains and period maps induced by polarized variations of Hodge structures. Then we will see how one can prove algebraicity of Hodge loci in one interesting special case, namely when the period domain is a Hermitian symmetric space. In the end, if I have time, I will sketch the modern approach to proving the algebraicity of Hodge loci that relies on the theory of o-minimal structures.

Multivariate Method of Moments: From Sample Complexity to Implicit Computations

Abstract: The focus of this talk is on theoretical and computational aspects of the multivariate method of moments for parameter estimation. First, from a theoretical standpoint, we show that in problems where the noise is high, the sample complexity (number of observations necessary to estimate parameters) is dictated by the moments of the distribution. Second, from a computational standpoint, we address the curse of dimensionality: the number of entries of higher-order moments of multivariate random variables scale exponentially with the order of the moments. For Gaussian Mixture Models (GMMs), we develop numerical methods for implicit computations with the empirical moment tensors. This reduces the computational and storage costs, and opens the door to the competitiveness of the method of moments as compared to expectation maximization methods. Time permitting, I will also mention related results on symmetric CP tensor decomposition and sketch a recent algorithm we proposed that is faster than the state-of-the-art and comes with guarantees.

Magic Functions

Abstract: We will present some challenging problems from different areas of mathematics that were solved via the construction of certain magic functions with constraints on physical and/or frequency space. In particular, we will discuss sphere packings and kissing numbers.

Geometria não linear de espaços de Banach

Abstract: Nesta palestra, farei um apanhado geral da teoria não linear de espaços de Banach. De forma mais detalhada: Como espaços de Banach são espaços lineares normados, a noção de isometria linear é o tipo correto de equivalência a ser considerado caso o objetivo seja obter um controle completo sobre todos os aspectos dos espaços envolvidos. Relaxando um pouco o controle desejado, isomorfismo linear é a próxima equivalência natural entre espaços de Banach que ainda exerce certo controle sobre suas estruturas lineares (historicamente, esta é a noção de equivalência mais considerada por pesquisadores na área). No entanto, espaços de Banach são, em particular, espaços métricos e noções de equivalência puramente métricas também podem ser estudadas, e.g., isometrias (não lineares) e equivalências de Lipschitz. O principal objetivo de pesquisadores nessa área é entender quando características puramente métricas são capazes de determinar características lineares de espaços de Banach. Um dos primeiros marcos nesta área é o teorema de Mazur-Ulam: isometrias (não lineares) entre espaços de Banach (reais) que preservam zero são automaticamente lineares. Já na década de 70, P. Enflo mostrou que a estrutura linear dos espaços de Hilbert é determinada por uma noção muito mais fraca: se um espaço de Banach for uniformemente equivalente a um espaço de Hilbert, então ele deve ser (linearmente) isomórfico a ele. Nesta palestra, detalharei essa área de pesquisa e descreverei alguns dos principais problemas da área. Na sua parte final, apresentarei noções novas de equivalências e mergulhos entre espaços de Banach e descreverei alguns resultados recentes na área.

Reticulados, álgebras e moonshine

Abstract: A matemática contempla inúmeras interações, inesperadas e produtivas, entre objetos de diversas naturezas: discretas, contínuas e algébricas. Nessa palestra, gostaria de examinar algumas instâncias deste tema, abrangendo reticulados, partições de inteiros, álgebras de Lie e formas modulares. Essa seleção é motivada pela minha área de pesquisa na teoria de representações, em que as conexões entre estes objetos são utilizadas para avançar o estudo de álgebras de dimensão infinita que surgem da física teórica, como as álgebras de vértices, e vice-versa.