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 II*(with L.E. Toussaint)*Classification of corank-2 distributions through convex integration*(with M. Jovanovic, F.J. Martínez Aguinaga and I. Zelenko)*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)*Regularisation of Lie algebroid contact structures*(with A. Witte)

**Preprints**

*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**

*Introducing sub-Riemannian and sub-Finsler Billiards*. Discr. Cont. Dyn. Sys. (2022). arXiv:2011.12136 (with L. Dahinden)*The Engel-Lutz twist and overtwisted Engel structures.*Geometry & Topology 24(5) (2020), 2471–2546. arXiv:1712.09286 (with T. Vogel)*Loose Engel structures*. Compos. Math. 156(2) (2020), 412-434. arXiv:1712.09283 (with R. Casals, F. Presas).*Classification of Engel knots.*Math. Ann. 371(1-2) (2018), 391-404. arXiv:1710.11034 (with R. Casals)*Tight contact foliations that can be approximated by overtwisted ones.*Archiv der Mathematik, 110(4), 413-419. arXiv:1709.03773*On the classification of prolongations up to Engel homotopy.*Proc. Amer. Math. Soc. 146 (2018) 891-907. arXiv:1708.00295*Flexibility for tangent and transverse immersions in Engel manifolds.*Rev. Mat. Comp 32(1) (2019), 215-238. arXiv:1609.09306 (with F. Presas)*The foliated Weinstein conjecture*. Int. Math. Res. Not. 16 (2018), 5148-5177. arXiv:1509.05268 (with F. Presas)*Existence h-Principle for Engel structures*. Invent. Math. 210 (2017), 417-451. arXiv:1507.05342 (with R. Casals, J.L. Pérez, F. Presas)*Foliated vector fields without periodic orbits.*Isr. J. Math. 214 (2016), 443-462.arXiv:1412.0123 (with D. Peralta-Salas, F. Presas)*The foliated Lefschetz hyperplane theorem*. Nag. Math. J. 231 (2018), 115-127. arXiv:1410.3043 (with D. Martínez Torres, F. Presas)*h-Principle for Contact Foliations*. Int. Math. Res. Not. 20 (2015), 10176-10207. arXiv:1406.7238 (with R. Casals, F. Presas)

**Expository writing**

- Topological aspects in the study of tangent distributions. Textos de Matemática. Série B [Texts in Mathematics. Series B], 48. Universidade de Coimbra, Departamento de Matemática, Coimbra, 2019.

**My thesis**

- Engel structures and symplectic foliations. PhD thesis at Instituto de Ciencias Matemáticas (ICMAT) and Universidad Autónoma de Madrid (UAM). Supervisor: Fran Presas. 2017