I am a postdoctoral researcher at CIMAT in Guanajuato, Mexico. Previously, I was a postdoctoral researcher for one year at UNAM (Unidad Cuernavaca). I received my PhD from Universidad Complutense de Madrid in June 2018. I was a student of Javier Fernández de Bobadilla and María Pe Pereira.

My general area of research is singularity theory. I am mainly interested in the interactions of singularity theory with other areas such as low dimensional topology. For example, attacking problems regarding mapping class groups using techniques from singularity theory and viceversa. I also work on the understanding of the geometric monodromy of higher dimensional Milnor fibrations and so sympletic topology and Lipschitz geometry are of my interest as well.

Here you can download my CV (updated december 2020).

Currently, I am an organizer of the Iberosing online seminar

5. On a quadratic form associated with the nilpotent part of the monodromy of a curve, 2020, (Preprint) Joint work with Lilia Alanís-López, Enrique Artal Bartolo, Christian Bonatti, Xavier Gómez-Mont and Manuel González Villa.
   Abstract arXiv
   We study the nilpotent part of certain pseudoperiodic automorphisms of surfaces appearing in singularity theory. We associate a quadratic form \tilde{Q} defined on the first (relative to the boundary) homology group of the Milnor fiber F of any germ analytic curve on a normal surface. Using the twist formula and techniques from mapping class group theory, we prove that the form \tilde{Q} obtained after killing \ker N is definite positive, and that its restriction to the absolute homology group of F is even whenever the Nielsen-Thurston graph of the monodromy automorphism is a tree. The form \tilde{Q} is computable in terms of the Nielsen-Thurston or the dual graph of the semistable reduction, as illustrated with several examples. Numerical invariants associated to \tilde{Q} are able to distinguish plane curve singularities with different topological types but same spectral pairs or Seifert form. Finally, we discuss a generic linear germ defined on a superisolated surface with not smooth ambient space..
6. Vanishing cycles, plane curve singularities, and framed mapping class groups, 2020, Geom. Topol. (Accepted) Joint work with Nick Salter.
   Abstract arXiv
   Let f be an isolated plane curve singularity with Milnor fiber of genus at least 5. For all such f, we give (a) an intrinsic description of the geometric monodromy group that does not invoke the notion of the versal deformation space, and (b) an easy criterion to decide if a given simple closed curve in the Milnor fiber is a vanishing cycle or not. With the lone exception of singularities of type A_n and D_n, we find that both are determined completely by a canonical framing of the Milnor fiber induced by the Hamiltonian vector field associated to f. As a corollary we answer a question of Sullivan concerning the injectivity of monodromy groups for all singularities having Milnor fiber of genus at least 7.
7. Positive factorizations of pseudoperiodic homeomorphisms, 2020, Math. Ann. (Accepted).
   Abstract arXiv Journal
   We generalize a classical result concerning smooth germs of surfaces, by proving that monodromies on links of isolated complex surface singularities associated with reduced holomorphic map germs admit a positive factorization. As a consequence of this and a topological characterization of these monodromies by Anne Pichon, we conclude that a pseudoperiodic homeomorphism on an oriented surface with boundary with positive fractional Dehn twist coefficients and screw numbers, admits a positive factorization. We use the main theorem to give a sufficiency criterion for certain pseudoperiodic homeomorphisms with negative screw numbers to admit a positive factorization.
8. Mixed tête-à-tête twists as monodromies associated with holomorphic function germs, 2019, Geom. Dedicata (Accepted). Joint work with Baldur Sigurðsson.
   Abstract arXiv Journal
   Tête-à-tête graphs were introduced by N. A'Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed tête-à-tête graphs provide a generalization which define mixed tête-à-tête twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed tête-à-tête twists as those pseudo-periodic automorphisms that have a power which is a product of right-handed Dehn twists around disjoint simple closed curves, including all boundary components. It follows that the class of tête-à-tête twists coincides with that of monodromies associated with reduced function germs on isolated complex surface singularities.
9. Tête-à-tête twists, monodromies and representation of elements of Mapping Class Group, 2018, Ann. Inst. Fourier (Accepted). Joint work with Norbert A'Campo, Javier Fernández de Bobadilla and María Pe Pereira.
   Abstract arXiv
   We study monodromies of plane curve singularities and pseudo-periodic homeomorphisms of oriented surfaces with boundary, following an original idea of the first author: tête-à-tête graphs and twists. We completely characterize mapping classes that can be represented by tête-à-tête twists, and generalize the notion to be able to represent any class of the mapping class group relative to the boundary which is boundary-free periodic. This improves previous work on the subject by C. Graf. Furthermore, we introduce the class of mixed tête-à-tête graphs and twists, and prove that mixed tête-à-tête twists contain monodromies of irreducible plane curve singularities. In a sequel paper, the fourth author and B. Sigurdsson have extended this to the reducible case.
10. (PhD Thesis) Monodromies as tête-à-tête graphs , 2018, Under the supervision of Javier Fernández de Bobadilla and María Pe Pereira.
    PDF

Upcoming

Selected invited talks

• Characterizing the geometric monodromy group of an isolated plane curve singularity, Singularity Theory from Modern Perspectives, Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi (Vietnam), 16th September, 2021
• Characterizing the geometric monodromy group of a plane curve singularity, Singularity Theory from Modern Perspectives, CIRM (France), 4th-8th September, 2021
• Framed mapping class groups and plane curve singularities, Singularity Theory Seminar, UFC, (Brazil), June 8th, 2021
• Caracterizando el grupo de monodromía geométrico de una singularidad de curva plana , Algebraic Topology Seminar, CIMAT, Mérida (Mexico), March 16th, 2021
• Positive factorizations of monodromies on links of isolated complex surface singularities, Graduate/Postdoc Topology and Singularities Seminar, University of Wisconsin–Madison, (USA), March 1st, 2021
• Positive factorizations of monodromies, Seminar, Aix-Marseille, Marseille (France), January 2020.
• (Mini-Course) Geometric monodromy via tête-à-tête graphs, Workshop on Topological and Analytical Methods in Singularity Theory, CIMAT, Guanajuato (Mexico), November 2019.
• Monodromies as tête-à-tête graphs, BCM-SINGULARITIES, 4th meeting on singularities Brazil-Chile-Mexico , Universidad de Talpa, Talpa (Chile), March 2019.
• Monodromies as tête-à-tête graphs, Coloquio investigación, CIMAT, Guanajuato (Mexico), February 2019.
• Combinatorial representation of pseudo-periodic automorphisms, Seminar, Université de Lille, Lille (France), January 2019.
• Factorizations of monodromies on links of complex surface singularities, Singularity Theory Seminar, IMATE, Cuernavaca (Mexico), July 2019.
• (Mini-Course) Topological aspects of singularity theory, IMATE, Cuernavaca (Mexico), October 2018.
• Tête-à-tête graphs and Lagrangian skeleta Arborealization of singularities of Lagrangian skeleta, American Institute of Mathematics, San Jose, California (USA), March 2018.
• Tête-à-tête graphs and pseudo-periodic homeomorphisms, Congreso de jóvenes investigadores (organized by Real Sociedad Matemática Española), Universidad de Valencia, Valencia (Spain), September 2017.
• Tête-à-tête graphs and Seifert manifolds, Invited Talk, Worldwide Center of Mathematics, Boston (USA), October 2016.
• Tête-à-tête graphs and monodromies in singularity theory, Northeastern University, Boston (USA), October 2016.
• Tête-à-tête graphs and fibered knots, IUMA day on knots, Universidad de Zaragoza, Zaragoza (Spain), May 2016.
• Combinatorial monodromy, Congreso de jóvenes investigadores (organized by Real Sociedad Matemática Española), Universidad de Murcia, Murcia (Spain), September 2015.
• The Hodge decomposition, Hodge structures seminar, Columbia University, New York (USA), April 2015.
• Expository talk on work by Dusa McDuff, Symplectic Geometry seminar, Columbia University, New York (USA), March 2015.
• Circle fiberings and characteristic classes, PhD students seminar, ICMAT, Madrid (Spain), October 2014.
• The Whitney's trick, Workshop of young researchers in Mathematics, Universidad Complutense de Madrid (Spain), September 2013.

• Third Latin American school on Algebraic Geometry and its applications, CIMAT, Guanajuato, (Mexico), Augusts 2017.
• Introduction to Geometric Analysis: The Atiyah-Singer Index Theorem (3-week course), BCAM, Bilbao, (Spain), June 2017.
• CIMPA school on singularity theory in topology, geometry and foliations, IMATE, Cuernavaca, Mexico, November-December 2014.
• 13th international workshop on real & complex singularities, Sao Carlos (Brazil), July 2014.

• Homotopical Topology, Master/Phd Course, CIMAT, Spring 2021.
  Syllabus
  This was an advanced course on algebraic topology covering some selected topics from the book ``Homotopical topology'' by A. Fomenko and D. Fuchs and including a first review of classical topics including: homotopy theory (fundamental group and higher homotopy groups, covering spaces, Seifert-Van Kampen Theorem and long exact sequence associated to a fibration), homology (different homology theories, Mayer-Vietoris, CW-complexes and a brief introduction to obstruction theory) and an introduction to spectral sequences including some applications..
• Singularities of Plane Curves, Master/Phd Course, CIMAT, Fall 2020.
  Syllabus
  Introductory course covering some selected topics from ``Singular points of plane curves'' C.T.C. Wall and ``Singular points of complex hypersurfaces'' by J. Milnor. The course initiated with a complete proof of Milnor's fibration theorem following his book and including necessary and related results such as the Curve Selection Lemma and the Conical Structure Theorem. The second part of the course was devoted to the study of plane curve singularities following Wall's book. In this second part chapters 1 to 9 were covered including the following topics: Puiseux theorem, embedded resolution of plane curve singularities, contact and intersection multiplicity of two branches, topological classification of plane curve singularities and the construction of the Milnor fiber and its geometric monodromy from the decorated resolution graph..
• Mapping Class Groups, Master/Phd Course, CIMAT, Spring 2020.
  Syllabus
  Introductory course covering some selected topics from ``A primer on mapping class groups'' by B. Farb and D. Margalit. The selected topics treated include: Hyperbolic Geometry, Dehn twists and their relations, generation theorems (Dehn-Lickorish theorems and related versions), Teichmüller spaces, Nielsen realization theorem for periodic homeomorphisms and Nielsen-Thurston decomposition theorem..
• Algebraic Structures/Estructuras algebraicas, 2nd year undergraduate course, Universidad Complutense de Madrid (assistant teacher), 2015-2016.

• Organizer of the Iberoamerican Webminar of Young Researchers in Singularity Theory and related topics, website .
• Tête-à-tête graphs library , mathematical software SAGE contribution, (ongoing).
• Ribbon Graphs library, mathematical software SAGE contribution, 2016
• Guide at IMAGINARY exhibition, Madrid (Spain), 2012.