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