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?” There is a branch of Mathematics, called the h-principle, dedicated to answering things like this.

My main line of research has to do with a particular class of structures called tangent distributions (i.e. subbundles of the tangent bundle). This is a very broad subject so, more concretely: 

  • My most recent work has to do with foundational aspects of h-principle (particularly, two techniques known as wrinkling and convex integration). Both play an important role in the construction of submanifolds tangent/transverse to distributions.
  • I often think about the classification of Engel structures, which are distributions particular to 4-manifolds.
  • In the first half of my PhD I studied foliations possessing a leafwise contact or symplectic structure.

I am very interested in Contact and Symplectic Topology, an area in which the h-principle has been extremely successful (while other techniques, like pseudoholomorphic curves or gauge theory, can be used to test its limits). I have also been trying to get more into the geometric aspects in the study of distributions (Control Theory, Geometry of PDEs, and Subriemannian Geometry).

In preparation

  • Wrinkling h-principles for integral submanifolds of jet spaces (with L.E. Toussaint)
  • Classification of generic distributions through convex integration (with F.J. Martínez Aguinaga)
  • h-Principle for knots tangent to bracket-generating distributions (with F.J. Martínez Aguinaga)
  • Horizontal Morse Theory (with L. Accornero, F. Gironella, and L.E. Toussaint)

Preprints

  • Introducing sub-Riemannian and sub-Finsler Billiards. Submitted. arXiv:2011.12136 (with L. Dahinden)
  • Microflexiblity and local integrability of horizontal curves. Submitted. arXiv:2009.14518 (with T. Shin)

Accepted/published articles

Expository writing