Jesús M. Ruiz

Facultad de Matemáticas
Despacho 308-C
(+34) 913944511
jesusr@ucm.es

Fields: Differential and Algebraic Topology; Real Algebraic, Nash and Analytic Geometry; Semidefinite functions and sums of squares; Real Algebra.

Teaching concerns

Several advanced undergraduate courses on Analytic Geometry, 5th year in Mathematics. A sample of the topics covered is this, and the book that came from part of them is [2009b].
Standard undergraduate courses on Linear Algebra and Geometry, 1st year in Mathematics, jointly with J.M. Gamboa. A typical contents table is this, and a pdf file with a collection of 287 problems and exercises can be downloaded here. This subject becomes Linear Algebra in the 1st year of the new 2009/10 Maths Degree. Up to 250 problems and exercises are here; they are compiled jointly with José F. Fernando, Concepción Fuertes, José Manuel Gamboa and Celia Martínez. Some of us wrote several texts on the topic, with final version [2019a,b].
Undergraduate courses on Affine and Projective Geometry, 2nd year in Mathematics on these topics, jointly with J.M. Rodríguez-Sanjurjo, based on prior courses by J.M. Sánchez Abril. Then we wrote a book on Projective Geometry: the last edition is [2020a].
Since 1999: An exhibition on Projective Geometry: Origins and Foundations, organized jointly with R. Mallavibarrena; follow this link. Posters are available on demand.
Several courses on Riemannian Geometry, 4th year in Mathematics, with these contents. Here there is a pdf with a collection of 200 problems on the topics treated. The first half of this became Calculus on manifolds, following [2020c].
Undergraduate courses on Computational Geometry, 4th year in Computer Sciences, with these topics covered. We have this draft on splines.
Undergraduate courses on Elements of Topology (see the contents), 3rd year in the Maths Degree. A sample of problems and exercises is here. A more advanced course on General Topology, 4th year in the Maths Degree, with R. Díaz, on these contents by R. Díaz. Here there is a collection of exercises.
Undergraduate courses on Differential Calculus, 2nd year in the Maths Degree with these contents, following prior courses by J.M. Martínez Ansemil and Socorro Ponte. Also here there are problems sets, more than 160 exercises collected jointly with Pilar Cembranos, Ana Gallardo, J.M. Martínez Ansemil and Socorro Ponte; from some very easy to others not so, to check how far you can go. A few exam models are these.
Postgraduate courses on Real Algebraic Geometry, following this famous book, the bible for the topic, universally known as BCR. Then, this evolved into Semialgebraic Geometry (contents), jointly with Carlos Andradas and Antonio Díaz-Cano, more focused on this monograph by Michel Coste.
Postgraduate courses on Differential Topology, jointly with E. Outerelo, devoted mainly to transversality as described here, a part of it included in a book, [2020b] is a highly refined edition written with E. Outerelo and J.A. Rojo. This was the germ of a postgraduate course on Mapping Degree Theory at Pisa University, with this contents; all of it in this [2009b] book with E. Outerelo.

2024/25: Courses on Smooth Manifolds, Elementary Topology, Differential Topology.

Advisor: PhD thesis and degree/master final projects.

The anniversary of Professor Enrique Outerelo.

Professor Enrique Outerelo is a member of our mathematical community since 1961. All these years, he has been teaching mathematics, and many other important things, to several generations of us all. The many who have been lucky to be first his students, then his colleagues, and sometimes his collaborators, joined to celebrate his 65th anniversary. On that occasion a volume including mathematical contributions by many of us was published and dedicated to him in appreciation of his generous devotion to Mathematics. Those contributions can be found in this link, as well as the witty address Professor Outerelo offered us on 5 October, 2004.