Research

I am a differential topologist that studies geometric structures. The questions I consider are often of the form: “What is the homotopy type of the space of all geometric structures of class X on a given manifold M?

This is a rather broad question, since it can be posed in any geometric setting you like. I mostly spend my time thinking about:

  • h-Principle (i.e. about developing techniques, with the goal of answering the question above, that are general and not geometry-specific). Some keywords: wrinkling, convex integration, removal of singularities.
  • Tangent distributions (i.e. subbundles of the tangent bundle), particularly under the assumption that they are bracket-generating (and often maximally non-integrable). Some keywords: horizontal submanifolds, Engel structures, (2,3,5)-distributions, (4,6)-distributions, Cartan distribution.
  • Poisson Geometry and Foliation Theory. Some keywords: symplectic foliation, contact foliation, Lie algebroid, transverse geometry.

Furthermore, I am very interested in Contact and Symplectic Topology (as the main inspiration behind the things I do), Sub-Riemannian Geometry (as the field that studies distributions from a geometric perspective), and Homotopy Theory (concretely, its relation to h-principle).

In preparation

  • Wrinkling h-principles for integral submanifolds of jet spaces II (with L.E. Toussaint)
  • Classification of corank-2 distributions through convex integration (with M. Jovanovic, F.J. Martínez Aguinaga and I. Zelenko)
  • Horizontal Morse Theory (with L. Accornero, F. Gironella, and L.E. Toussaint)
  • Some construction and classification statements for singular foliations (with A. Witte)

Preprints

  • Regularisation of Lie algebroid contact structures. arXiv:2211.14891 (with A. Witte)
  • Classification of tangent and transverse knots in bracket-generating distributions. arXiv:2210.00582 (with F.J. Martínez Aguinaga)
  • Wrinkling h-principles for integral submanifolds of jet spaces. arXiv:2112.14720 (with L.E. Toussaint)
  • Convex integration with avoidance and hyperbolic (4,6) distributions. arXiv:2112.14632 (with F.J. Martínez Aguinaga)
  • Microflexiblity and local integrability of horizontal curves. arXiv:2009.14518 (with T. Shin)

Accepted/published articles

Expository writing

My PhD thesis