Pablo Portilla Cuadrado

About me

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Currently, I am a postdoctoral researcher at Laboratoire Paul Painlevé at Université de Lille in Lille, France. Before this, I was a postdoctoral researcher at CIMAT in Guanajuato, Mexico for two years and previously, I worked 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 October 2021). And here you can download an extended version of it with precise dates and some extra information.

Currently, I am an organizer of the Iberosing online seminar

Research

On a quadratic form associated with the nilpotent part of the monodromy of a curve, 2022, Indag. Math. (Accepted) 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..
Vanishing cycles, plane curve singularities, and framed mapping class groups, 2021, Geom. Topol. Joint work with Nick Salter.
Abstract arXiv Journal

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.
Positive factorizations of pseudoperiodic homeomorphisms, 2020, Math. Ann.
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.
Mixed tête-à-tête twists as monodromies associated with holomorphic function germs, 2019, Geom. Dedicata. 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.
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.
(PhD Thesis) Monodromies as tête-à-tête graphs, 2018, Under the supervision of Javier Fernández de Bobadilla and María Pe Pereira.
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Teaching

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.

Contact

You may contact me at: pablo dot portilla at univ-lille dot fr.