Professional activities

This project naturally extends my doctoral work, focusing on \(q\)-contact structures: geometric objects that generalise classical contact structures to higher codimensions. These structures induce natural \(\mathbb{R}^q\)-actions on their ambient manifolds, whose orbit foliation generalises the dynamics of Reeb vector field flows.

A central goal is to determine which properties of classical contact structures persist in this generalised setting. Current investigations include identifying useful invariants, studying rigidity phenomena, and exploring applications to the classification of Anosov actions for higher-rank groups.

This ongoing collaboration with Dr. Martin Leguil (École Polytechnique) continues work initiated during my postdoctoral year at CMLS. We study how much geometric information can be reconstructed from periodic data in hyperbolic billiards. Specifically, we investigate spectral rigidity for Sinai billiards: if two billiard tables share the same marked length spectrum, must they be isometric?

Our approach analyses the coarse geometry of Sinai billiard flows and examines whether classical tools (e.g., Otal and Croke's rigidity results for negatively/nonpositively curved surfaces) extend to CAT(0) settings.

Collaborative project with Dr. Christian Rodrigues and Dr. André Gomes (IMECC-UNICAMP/Applied Analysis Group), as part of the Geometry and Probability in Dynamical Systems research group of the Max Planck Institute for Mathematics in the Sciences. Our work explores Wasserstein spaces of probability measures, particularly their coarse geometric properties and connections to dynamical systems.

One of my main interests involves analysing the relationship between a dynamical system \(f: X \to X\) and the naturally induced dynamics on the space \(\mathcal{P}(X)\) given by the pushforward map \(\mu \mapsto f_\ast\mu\). When \(X\) is metrisable, the Wasserstein metric \(w\) enables more refined analyses of \(f_\ast\) – in terms of weak derivatives and properties related to expansiveness of the map, for instance – which are unavailable in the usual setting of topological dynamics where such maps are typically studied. Our aim is to systematically relate the dynamics of \(f\) on \(X\) to those of \(f_\ast\) on \(\mathcal{P}(X)\).

Additionally, there are other research directions, including metric rigidity questions for \((\mathcal{P}(X), w)\) and applications to continuity problems for Lyapunov exponents. The latter is an ongoing collaboration with Dr. Luiz A. B. San Martin (IMECC–UNICAMP).

This is a joint work with Dr. Elizeu França, investigating intrinsically harmonic differential forms and their topological implications. A primary objective is to establish a conjectured dual to Tischler’s theorem: whether closed orientable \(n\)-manifolds admitting closed nowhere-vanishing \((n-1)\)-forms necessarily fibre over the circle.