Research interests (AMS subject classification)
  • Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills) (53C07)
  • Fundamental groups (14H30)
  • Vector bundles on curves and their moduli (14H60)
I work in the area at the intersection between the theory of Vector bundles on curves and Real algebraic geometry. Themes which are of special interest to me are:
  • Topology of moduli spaces in Real algebraic geometry (connected components, Betti numbers).
  • Narasimhan-Seshadri, Hitchin-Kobayashi-Simpson and Donaldson-Corlette correspondences over real algebraic curves.
Preprints
  1. With Victoria Hoskins. Rational points of quiver moduli spaces. 36 pages. Submitted. arXiv:1704.08624 (pdf).
  2. With Victoria Hoskins. Group actions on quiver varieties and applications. 33 pages. Submitted. arXiv:1612.06593 (pdf).
Refereed full papers
  1. Finite group actions on moduli spaces of vector bundles. To appear in Sémin. Théor. Spectr. Géom. (Grenoble). arXiv:1608.03977 (pdf).
  2. On the Narasimhan-Seshadri correspondence for Real and Quaternionic vector bundles. J. Differential Geom. (2017) 105 (1), 119-162. doi:10.4310/jdg/1483655861. arXiv:1509.02052 (pdf).
  3. With Indranil Biswas. Vector bundles over a real elliptic curve. Pacific J. Math. 283 (2016), no. 1, 43–62. doi:10.2140/pjm.2016.283.43. arXiv:1410.6845 (pdf).
  4. With Chiu-Chu Melissa Liu. The Yang-Mills equations over Klein surfaces. J. Topol. 6 (2013), no. 3, 569-643. doi:10.1112/jtopol/jtt001. arXiv:1109.5164 (pdf).
  5. Real points of coarse moduli schemes of vector bundles on a real algebraic curve. J. Symplectic Geom. 10 (2012), no.4, 503-534. doi:10.4310/JSG.2012.v10.n4.a2. arXiv:1003.5285 (pdf).
  6. Moduli spaces of vector bundles over Klein surfaces. Geom. Dedicata 151 (2011), no. 1, 187-206. doi:10.1007/s10711-010-9526-3. arXiv:0912.0659 (pdf).
  7. Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups. Math. Annalen 342 (2008), no. 2, 405-447. doi:10.1007/s00208-008-0241-4. arXiv:0703869 (pdf).
  8. A note on quasi-Hamiltonian geometry and representation spaces of surface groups. In The COE Seminar on Mathematical Sciences 2005 (2007), vol. 36 of Sem. Math. Sci., 35-48 (pdf).
  9. Anti-symplectic involutions on quasi-Hamiltonian quotients. In Proceedings of the School on Poisson Geometry and Related Topics. Trav. Math. 17 (2007), 57-64 (pdf).
  10. Un théorème de convexité réel pour les applications moment à valeurs dans un groupe de Lie. C.R. Math. Acad. Sci. Paris 345 (2007), no. 1, 25-30. doi:10.1016/j.crma.2007.05.023 (pdf).
  11. Representations of the fundamental group of an L-punctured sphere generated by products of Lagrangian involutions. Canad. J. Math. 59 (2007), no. 4, 845-879. doi:10.4153/CJM-2007-036-9 (pdf).
  12. With Elisha Falbel and Jean-Pierre Marco. Classifying triples of Lagrangians in a Hermitian vector space. Topology Appl. 144 (2004), no. 1-3, 1-27. doi:10.1016/j.topol.2003.08.027 (pdf).
Refereed conference publications, book chapters and research announcements
  1. Lectures on Klein surfaces and their fundamental group. In Geometry and Quantization of Moduli Spaces. Advanced Courses in Mathematics - CRM Barcelona, Springer (2016), 67-108. ISBN 978-3-319-33577-3. arXiv:1509.01733 (pdf). doi:10.1007/978-3-319-33578-0.
  2. Differential geometry of holomorphic vector bundles on a curve. In Geometric and Topological Methods for Quantum Field Theory, Proceedings of the 2009 Villa de Leyva Summer School, Cambridge University Press (2013), 39-80. ISBN: 9781107026834 (pdf).
  3. Quasi-Hamiltonian quotients as disjoint unions of symplectic manifolds. Noncommutative geometry and physics 2005 (2007), 31-54, World Sci. Publ., Hackensack, NJ, 2007. ISBN: 978-981-4476-20-1. arXiv:0701905 (pdf). doi:10.1142/9789812779649_0002.
Theses
  1. Représentations décomposables et sous-variétés lagrangiennes des espaces de modules associés aux groupes de surfaces (Doctoral dissertation). Thèse en ligne : HAL tel-00264370.
Books
  1. With Jean-Pierre Marco, Laurent Lazzarini, Hakim Boumaza, Benjamin Collas, Stéphane Collion, Marie Dellinger and Zoé Faget. Analyse L3 (2009). Pearson Education France. 6 chapters. ISBN 9782744073502.
Lecture notes
  1. Differential geometry of holomorphic vector bundles on a curve (expanded version of the 2009 Villa de Leyva conference publication above). arxiv:1509.01734 (pdf).
  2. Una introducción a la geometría hiperbólica (en preparación).
Popularization and other short texts
  1. El plano hiperbólico, de la geometría a la teoría de números. Aceptado, Hipótesis 22 (2018) (pdf).
  2. MCA 2017: A report on the Gender and Mathematics Panel for the Colombian Society of Mathematics.
  3. Congreso Internacional de Matemáticos 2014. Hipótesis 17 (2014), 17-18 (pdf).
Other