Pablo Portilla Cuadrado

About me

defense_picture

I hold a "Ramón y Cajal" tenure track position at Universidad Politécnica de Madrid. Before this, I was a postdoctoral researcher at Laboratoire Paul Painlevé at Université de Lille in Lille, France. Previously, I was a postdoctoral researcher at CIMAT in Guanajuato, Mexico for two years and 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 September 2023).

Currently, I am one of the organizers of the Iberosing seminars and international workshops.

Research

(Preprint) The total spine of the Milnor fibration of a plane curve singularity, 2023, Joint work with Baldur Sigurðsson.
Abstract arXiv

We study the separatrices at the origin of the vector field -\nabla \log |f| where f is any plane curve singularity. Under some genericity conditions on the metric we produce a natural partition of the set of separatrices S into segments and disks. As a byproduct of this theory we construct a smooth fibration equivalent to the Milnor fibration that lives on a quotient of the Milnor fibration at radius 0, and, furthermore we see how the strict transform of S in this space induces a spine for each Milnor fiber of this fibration..
On a quadratic form associated with a surface automorphism and its applications to Singularity Theory, 2022, Indag. Math. Joint work with Lilia Alanís-López, Enrique Artal Bartolo, Christian Bonatti, Xavier Gómez-Mont and Manuel González Villa.
Abstract arXiv Journal

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. Joint work with Norbert A'Campo, Javier Fernández de Bobadilla and María Pe Pereira.
Abstract arXiv Journal

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.
PDF

Teaching

Courses

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.

Other courses

Classical tools in differential topology, Phd Course, BCAM, October 2022.
Syllabus

20 hours. The course was a comprehensive introduction to some of the most used tools in differential topology with a particular emphasis on transversality and Morse theory. More info
Introduction to Singularities of Plane Curves, Mini-Course, Seoul National University, July 2022.
Syllabus

The study of plane curve singularities is one of the most classical parts of singularity theory going back to Newton in the XVII century. When one studies complex polynomials in two variables, singularities appear in a very natural way. Although many times this topic is treated from an algebraic point of view, one quickly sees that it has many ties with low dimensional topology topics such as knot theory and mapping class groups. In this mini-course we will make a gentle introduction to singularity theory through the world of plane curves. We will focus on the topological aspect of singularities and we will mainly learn techniques through rich examples. By the end of the course we will be able to compute many invariants of a plane curve singularity and we will understand the topology around a singular point of an algebraic plane curve. In particular we will learn how to find parametrizations of each irreducible component of a plane curve singularity. We will see how these parametrizations can result very useful in computing the embedded topology of each branch and how each branch interacts with the rest. We will learn to find smooth models (resolve) of plane curve singularities by repeatedly blowing up the ambient space and, from the final picture, we will understand the topology of the Milnor fibration and its geometric monodromy. We will end the course by introducing the versal unfolding of a plane curve singularity and posing some questions that naturally emanate from it.

Contact

You may contact me at: p (dot) portilla (at) upm (dot) es