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).
- Wrinkling of submanifolds of jet spaces (with L.E. Toussaint)
- Classification of generic distributions through convex integration (with F.J. Martínez Aguinaga)
- A control theoretic version of convex integration (with F.J. Martínez Aguinaga)
- Microflexibility and local integrability of regular integral curves in bracket-generating distributions (with T. Shin)
- Non-holonomic Morse theory (with L. Accornero, F. Gironella, and L.E. Toussaint)
- The Engel-Lutz twist and overtwisted Engel structures. To appear in G&T. arXiv:1712.09286 (with T. Vogel)
- Loose Engel structures. To appear in Compos. Math. 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)
- 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.