Classical course on multi-variate calculus following Marsden and Tromba book.
Homotopical Topology, Master/Phd Course, CIMAT, Spring 2021.
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.
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.
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.
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.
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.
