IMPA - O Instituto de Matemática Pura e Aplicada

Let $f$ be a partially hyperbolic diffeomorphism preserving a splitting $E^s\oplus E^c\oplus E^u$, where the unstable bundle $E^u$ is one-dimensional. When $f$ is homogeneous, like the time-one map of the geodesic flow on a surface of constant negative curvature or a linear automorphism of the torus, the unstable foliation admits a natural parametrization that makes invariant measures with absolutely continuous conditionals (the so-called uGibbs measures) correspond exactly to invariant measures of this flow. How can this picture be generalized when $f$ is not homogeneous? In this talk, I will explain a construction of a suitable suspension flow of $f$ for which the unstable foliation becomes the orbit foliation of a new flow, called the unstable flow. Ergodic uGibbs measures are in one-to-one correspondence with measures simultaneously invariant under both flows and ergodic for the suspension. As an application, I will deduce the following measure rigidity statement: if $E^c$ is expanding and the bundle $E^s \oplus E^u$ is integrable, then any uGibbs measure whose entropy exceeds its unstable entropy is necessarily an SRB measure. This is joint work with Sébastien Alvarez, Sylvain Crovisier, Martin Leguil, and Davi Obata.

A classe de Weil-Petersson é uma classe notável de homeomorfismos do círculo que se encontra na fronteira de vários campos da matemática contemporânea, incluindo a teoria de Teichmueller, a teoria de Schramm-Loewner, a geometria hiperbólica e assim por diante. Neste trabalho, apresentamos uma nova perspectiva, estudando essa classe do ponto de vista da geometria anti-de Sitter. Mostramos que um homeomorfismo do círculo é Weil-Petersson se e somente se seu gráfico limitar uma superfície máxima completa, semelhante ao espaço, em AdS^{2,1} de área finita renormalizada.

Este é um trabalho conjunto com F. Diaf, A. Moriani, R. Smai e E. Trebeschi.

In this talk, a general abstract duality result is proposed for equations that are governed by the sum of two operators (possibly multivalued). It allows to unify a large number of variational duality principles, including the Clarke-Ekeland least dual action principle and the Singer-Toland duality. Moreover, it offers a new duality approach to some central questions in the theory of variational inequalities and maximal monotone operators.

In this talk, I will discuss how to align audio, visual, spatial, and
semantic representations across multiple levels, from low-level
perceptual correspondence to object/event-level structure and
scene-level generation. The talk connects audio-visual learning with
scene understanding, generative modeling, and multimodal AI.

live @ https://www.youtube.com/live/MVUK-Bcw4fY

Mostramos que determinadas superfícies semi-aritméticas de Riemann têm um crescimento exponencial das multiplicidades médias em seu espectro de comprimento. Anteriormente, o crescimento exponencial das multiplicidades médias era conhecido apenas para superfícies aritméticas, embora os resultados experimentais que indicavam a possibilidade de exemplos não aritméticos já estivessem disponíveis. Explicarei alguns detalhes interessantes de nossa prova e indicarei algumas questões em aberto relacionadas. A palestra é baseada em um trabalho conjunto recente com Gregory Cosac, Cayo Dória e Gisele Teixeira Paula.

Grothendieck-Teichmüller theory originated in arithmetic geometry through the study of Galois actions on geometric étale fundamental groups. In this talk, we argue that arithmetic topology provides a natural setting for applications of Grothendieck–Teichmüller theory. More specifically, we explain how our theory of automorphisms of profinite mapping class groups may lead to new results on profinite rigidity questions for mapping class groups.