Université Paris Cité

Title: Qualitative Theory of ODEs and Its Applications to Sub-Riemannian Geometry

Abstract: This mini-course explores the connections between the qualitative theory of planar ordinary differential equations (ODEs) and sub-Riemannian geometry, with a particular focus on the Sard Conjecture. The course is divided into two parts. The first part revisits fundamental concepts in qualitative theory, emphasizing foliations, elementary singularities, focus points, polar coordinates, and divergence. The second part introduces key notions of sub-Riemannian geometry and develops a geometric framework to investigate the Sard Conjecture through a qualitative theory approach.

University of Pisa

Title: Causality Theory in General Relativity and Beyond

Abstract: The study of global differential geometric and topological aspects of general relativity relies on specialized tools tailored to the Lorentzian signature of spacetime. Unlike Riemannian geometry, the Lorentzian structure induces a distribution of light cones over the manifold, giving rise to a unique causal dynamics. This framework, known as "causality theory", underpins the celebrated singularity theorems of Hawking and Penrose and has recently seen renewed interest, particularly in low-regularity settings. Such developments are increasingly viewed as foundational for understanding the interplay between quantum mechanics and the structure of spacetime.  

In this series of three lectures, we provide an introduction to causality theory and explore its recent advancements, structured as follows:  

1. Classical Causality Theory: An overview of causality theory in the smooth Lorentzian setting, emphasizing its role in global results such as singularity theorems.  

2. Causality Theory for Cone Structures: Extensions to more general cone distributions on manifolds, including cases of upper semi-continuous, $C^0$, and Lipschitz regularity, as well as the Lorentz-Finsler generalization.  

3. Causality Beyond Manifolds: Exploration of causal structures in settings where manifolds are absent, drawing on techniques from topologically ordered spaces and metric geometry.  

These lectures aim to offer insights into both the mathematical structure and physical implications of causality theory.

Università di Roma, la Sapienza

Title: Overdetermined PDE's, geometric rigidity and isoparametric foliations

Abstract: In these talks we discuss overdetermined PDE's on compact Riemannian manifolds. We start from the classical problems on Rn: Pompeiu, Serrin etc. and discuss their extension to Riemannian manifolds. Actually, we  remark that a manifold as familiar as a round sphere already exhibits some "exotic" behavior for both problems. We then introduce a class of Riemannian domains, called isoparametric tubes, which are foliated by a parallel family of equidistant hypersurfaces having constant mean curvature, and show that this geometric structure in fact hosts a solution to many overdetermined PDE's. 

For two particular overdetermined heat equations we discuss geometric rigidity, that is, we actually classify all domains admitting a solution. The main theme is that the overdetermination forces geometric conditions on the mean curvature of the boundary of the domain and its equidistants and, often, minimality of the boundary.