Research

I am a differential geometer who studies geometric structures by investigating their symmetries. The naive idea is that a lot of information about a geometric structure can be extracted from its set of symmetries, assuming this set is well behaved. This idea appears ubiquitously in geometry. In the past, I focused on the following two topics:

• the relationship between symmetries and conserved quantities in mathematical physics – the starting point being the Noether theorem;
• constructing cohomological invariants for geometric structures on manifolds with a well behaved set of symmetries – having in mind Haefliger’s construction of characteristic classes for foliated manifolds.

One of my main interests are higher structures in differential geometry. In particular, Lie groupoids and multiplicative structures over them, as well as the theory of Morita equivalences and differentiable stacks, are central in my research.

I am also very much interested in Poisson geometry (especially its intereaction with almost complex geometry) and Jacobi geometry. Furthermore, I am working on a version of Morse theory that captures the homotopy type of a hyperplane field on a manifold.

Publications

Below is the list of my publications and manuscripts in preparations.

In preparation

• Horizontal Morse Theory

  (with Fabio Gironella, Álvaro del Pino Gómez, and Lauran Toussaint)

  This paper develops a version of Morse theory compatible with a codimension one distribution on a manifold.

• Pseudogroups and geometric structures

  (with Francesco Cattafi, Marius Crainic, and María Amelia Salazar)

  This is a book describing a modern approach to the theory of Lie pseudogroups and geometric structure on manifolds that builds upon the theory of Lie groupoids, principal groupoid actions, and multiplicative structures. The main achievements include a clarification of the foundations of Élie Cartan’s work on the subject and a framework capable of dealing with non transitive pseudogroups.

Preprints

• Pseudogroups of symmetries and Morita equivalences

  (with Francesco Cattafi)

  arXiv:2211.01319

• A groupoid approach to transitive differential geometry

  (with Francesco Cattafi)

  arXiv:2211.16639

Peer reviewed

• Haefliger’s differentiable cohomology

  (with Marius Crainic)

  Journal of Noncommutative Geometry (2023), accepted

  arXiv:2012.07777

• Symmetry transformations of extremals and higher conserved quantities: invariant Yang–Mills connections

  (with Marcella Palese)

  J. Math. Phys. 62, 043504 (2021)

  arXiv:1910.11688

• The Jacobi morphism and the Hessian in higher order field theory; with applications to a Yang-Mills theory on a Minkowskian background

  (with Marcella Palese)

  Int. Jour. Geom. Meth. Mod. Phys. (2020) 2050114

  arXiv:1710.09100