
Catedrático de Universidad Departamento de Álgebra, Geometría y Topología Facultad de Ciencias Matemáticas Universidad Complutense de Madrid Despacho 308C (Facultad de Matemáticas) Tf.: (+34) 913944511 Fax: (+34) 913944564 email: jesusr@mat.ucm.es 
Research interests
— Member of the Spanish UCM group Real Algebra and Geometry since 1984.
 Real algebraic sets and semialgebraic sets.
 Real analytic sets (and germs) and semianalytic sets (and germs).
 Semidefinite functions and sums of squares.
 Nash sets and functions.
 Real spectra and spaces of signs.
— Macintosh user since 1986.
— Member of the European Network Real Algebraic and Analytic Geometry since 1994.
— Member of the Royal Spanish Mathematical Society (RSME) since 1998.
— Member of the Interdisciplinary Mathematics Institute (IMI UCM) since 2006.Preprints, papers, books, PhD's, DFP's.
A somehow sentimental statement.
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 this.
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 (contents) in the 1st year of the new 2009/10 Maths Degree. Up to 250 problems and exercises can be downloaded here; they arecompiled jointly with José F. Fernando, Concepción Fuertes, José Manuel Gamboa and Celia Martínez. Some more by Raquel Mallavibarrena are in RM. Some of us wrote four fascicles (first, second, third and fourth) on the topic, now fully revised in two volumes (vol 1, vol 2).
Undergraduate courses on Affine and Projective Geometry, 2nd year inMathematics, jointly with J.M. RodríguezSanjurjo, on these topics. This book was the first version of that cooperation, and this second one is the final elaborated presentation. Also, here there is a pdf with 92 problems on Projective Geometry, mainly due to J.M. Sánchez Abril.
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, with this program, following this text, from which this is a selection of exercises.
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. More advanced problem classes on General Topology, 4th year in the Maths Degree, corresponding to lectures 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 these Lecture Notes by J.M. Martínez Ansemil and Socorro Ponte, who keep them updated. Also here there are problem 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. Also, try these very simple questions, this first test for the asking plus this 20 minute companion, and then this second test.
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 DíazCano, 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 this book. This was the germ of a postgraduate course on Mapping Degree Theory at Pisa University, with this contents; all of it in this book with E. Outerelo. Now there is a new quite refined version of our old AddisonWesley book, written with E. Outerelo and José Ángel Rojo. Here there are three outstanding Lectures by J. Milnor.
Participation in a project to narrow the bridge between high school and university, which includes a book and some additional lessons on elementarygeometry. The book is this, and the lessons can be downloaded here.
A course on Smooth Manifolds, last year in the Maths Degree and Double Degrees (with Computer Sciences and Physics) (VarDif), with main reference this textbook written jointly with J.M. Gamboa. The sections for the course are I.14, II.13, III.14, IV.14, V.14. This is a basic collection of exercises and these are some exam samples. An extra bonus: the text of a lecture introducing de Rham cohomology, notes by P. CalvoSotelo, G. Gallego and P. Manchón.
A course on Elements of Topology (TOEL, see detailed contents), 3rd year in the Maths Degree and Double Degrees (with Computer Sciences and Physics). These notes are a very basic introduction to the fundamental group. Samples of problems and exercises are these 234 and for exam models look here. And from classroom life, this first quiz on the spot. Also, this is a quite general theorem, illustrating the beauty of General Set Topology, kindly written by Manuel Morán. Here there is a bonus on the fundamental group of a surface. In the end, enjoy these two 3blue1brown's videos: Unsolved and Fair Division, brought to my atention by David Pérez.
A course on Differential Topology, first year in the Maths Advanced Master (TopDif), with main reference this textbook written jointly with E. Outerelo and J.A. Rojo. Here there is a list with a few presentation proposals.
Advisor for degree final projects on Topology and Geometry, in the Maths Degree and Double Degrees. These are some projects so far completed.
And everybody will find some essential additional references here.
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 October the 5th, 2004.
Other activities
 Since 1999: Exhibition on Projective Geometry: Origins and Foundations.
 2001: Workshop on Real Geometry, coorganized with Carlos Andradas.
 2004: Conference on Real Geometry, coorganized with Carlos Andradas and Antonio DíazCano.
Erdös number ≤ 3
And here our much admired Wallace and Gromit
who most likely are to recommend some further readings, like these or these other...
Quotations
Jacob Bronowski, Kenneth Clark, John Ruskin, Donald Knuth, Tony Judt, Josep Pla, Rudyard Kipling, Eric Hobsbawm, Albert Camus, Robert Hughes, Walter Pater, Giuseppe Verdi, Marc Fumaroli, Johan Huizinga, Caritas italiana, Gilbert K. Chesterton, Nuccio Ordine, Paul Simon, Piers, Jubal E. Harshaw, Julian Barnes/Shostakóvich, Andrei Toom, Peter F. Drucker, John Ruskin, Naftalí Hertz Mussman, Aldous Huxley, June Huh.