I have taught in various undergraduate and graduate mathematics programs from February 1985 to November 2024. My main topics in these programs were related to algebra and geometry, in various forms. With the implantation of the ECTS, the University of Cantabria interpreted that 1 ECTS credit must contain 10 hours on a blackboard for the professor. The total number of hours I have devoted to teaching in differente formats of degrees in Mathematics exceeds, 3763 hours on a blackboard.
Algebra I (Divisibility and Principal Ideal Rings), 1985/86. Total of 30 teaching hours.
Algebra II (Galois Theory). 1993/94, 1984/85. Total of 135 hours
Algebra III (Introduction to Commutative Algebra). 19998/99, 1993/94, 1991/92, 1990/91. Total of 190 teaching hours
Algebra IV (Commutative Algebra). 1995/96, 1992/93, 1991/92, 1990/91, 1988/89. Total of 420 teaching hours.
Algebra V (Computer Algebra and Symbolic Computation). 2003/04, 2002/03, 2001/02, 2000/01, 1999/2000, 1998/99, 1997/98, 1996/97, 1993/94. Total of 1148 teaching hours.
Geometry II (Affine, projective and spheric geometry). 1990/91, 1989/80, 1988/89. Total of 195 teaching hours
Linear Algebra and Geometry (Linear Algebra and Affine Geometry). 1992/93, 1985/86. Total of 120 teaching hours
Topology II (Advanced topics in Topological and metric spaces). 1986/87. Total of 60 teaching hours
Topology I (Introduction to topological spaces). 1986/87. Total of 60 teaching hours.
Computer Algebra I. 2006/07, 2005/06, 2004/05. Total of 145 hours.
Lineal Algebra II (Endomorphism Theory and Bilinear Algebra). 2006/07, 2005/06, 2004/05. Total of 120 teaching hours
Advanced Algebra (Introduction to Commutative Algebra). 2011/12, 2010/11, 2009/10, 2008/09, 2007/08. Total of 300 teaching hours.
Algebra Seminar (Introduction to Number Theory). 2012/13. Total of 30 teaching hours.
Commutative Algebra. 2023/24, 2022/23, 2021/22, 2020/21, 2019/20, 2018/19, 2017/18, 2016/17, 2015/16, 2014/15, 2013/14, 2012/13. Total of 72 ECTS credits (i.e. 720 teaching hours).
Introduction to Mathematical Language. 2018/19. Total of 30 teaching hours.
Ampliación de Álgebra (Seminar on Advanced Commutative Algebra, optional for students). 2023/24, 2022/23. Total of 60 teaching hours.
Member of the Commission to establish studies in Computer Science Engineering at the Universuty of Cantabria, I stated and configurated several courses on Foundations of Computer Science intheoretical lectures. Later on, other members of the department continued under the same pre-established guidelines. With a total of 36.0 credits ECTS (i.e. 360 teaching hours)
Introduction to Logic for Computer Scientists. 2007/08, 2006/07, 2005/06. Total of 7.5 ECTS credits (i.e. 75 teaching hours).
Automata Theory and Formal Languages. 2011/12, 2010/11, 2009/10, 2008/09, 2007/08. Total of 28.5 ECTS credits (i.e. 285 teaching hours).
Ocassionally, I taught in mathematical courses for students of Physics and Nautical Studies with a total of 480 hours on a blackboard.
Linear Algebra and Geometry (Elementary course on Linear Algebra for Phisicists). 1998/99, 1995/96, 1988/89, 1987/88. Total of 330 teaching hours.
Mathematics Lab (Introduction to Mathematics for Physicists). 2004/05. Total of 30 teaching hours.
Linear Algebra and Spherical Trigonometry. 1994/95. Total of 120 teaching hours.
The Master of Mathematics and Computer Science arose in 2008 due to a national legal modification of the PhD Studies. I was de Director of this Master since its beginning until its end in 2013. It was modified (again due to national legal modifications) in 2013 and I took back the direction of this Master from 20187 until 2021. Thus, the teaching activitied reflected here correspond to two different study plans with the same name and differente structure. At the University of Cantabria 1 ECTS credit contained 10 hours teaching on the blackboard.
Computational Complexity. 2021/22, 2020/21, 2019/20, 2018/19, 2017/18, 2016/17, 2015/16, 2014/15, 2013/14, 2008/09. Total of 32.3 ECTS credits (i.e. 323 teaching hours).
Effective Methods in Algebraic Geometry. 2021/22, 2020/21, 2018/19, 2016/17, 2015/16, 2014/15, 2013/14, 2011/12, 2010/11. Total of 21.1 ECTS credits (i.e. 211 teaching hours).
With the national law on the modification in 2008 of University degrees, the title of Degree in Mathematics obliged the students to write a Graduation Thesis (called Trabajo Fin de Grado) where the students must write some professional text in Mathematics of a length not exceeding 50 pages. It was unrealistic to require every undergraduate mathematics student to be able to write a "professional" mathematical text (i.e., an original contribution) after such a light and superficial formation. The advisor was responsible for ensuring that the legal objective was met, helping the student to fulfill this requirement. I have advised 15 TFG's on different topics in Mathematics, linked to some of my research in different periods of time:
Darío Martín Cobo, "The Implicit Function Theorem in Local (Commutative) Algebra: Hensel’s Lemma and Non-Archimedean Newton Operator". Dec. , 2025. Co-advised with L.F. Tabera.
Yaize Zhang, "A bit of order, structure and new ideas in the cluster of invariants associated to OIG algorithm in multi-class learning". Sept., 2024. Co-advised with D. Sebastián.
Pablo Muñoz Lara, "Real Nullstellensatz Real, Positivstellensatz, Nichnegativstellensatz: Hilbert's 17th Problem" . Sept., 2024.
Javier García Arias, "On Nash functions", June, 2024.
David Guriérrez Cambra, "A purely algebraic proof of the Sauer-Shelah-Perles Lemma", July, 2023.
Unai Zabaleta Gañan, "Neural Networks with rational activation function: growth function and Erzeugungsgrad". Sept., 2023.
Luis Miguel Soto Sánchez, "Condition number on Singular Matrices and the normal Jacobian of Parameterization from Singular Value Decomposition". July, 2022.
Ignacio González Mantecón. "Serre-Auslander-Buchsbaum’s Theorem". April, 2022.
Rubén López Ruiz, "Serre's Ex-Conjecture: Quillen-Suslin Theorem". Sept., 2021.
Toraya Fernández Ruiz, "J. Heintz’s Geometrical Bézout Inequality: Deconstruction, a self-contained proof ". March, 2021.
Pablo Echegoyen Ruiz, "A self-contained proof of Jelonek’s Effective Nullstellensatz". Oct., 2020.
Víctor Martínez Crespo, "Reproducing Kernel Hilbert Spaces: the Representer Theorem. An interpretation of the representer as an approximant in Learning". Oct., 2020.
Antonio Fernández González, "Ideology of Correct Test Sequences: Kakeya sets, Combinatorial Nullstellensatz and Generalizations to Secant Sequences". June, 2019.
Sara García Hevia, "On the finiteness of the Nash equilibria for generic games". June, 2018.
Miguel Traspuesto Abascal, "Foundations of Cutting Cakes: Lyapunov Theorem and extensions". Dec., 2017.
Prior to the national modification of PhD studies, instead of a Master, Spanish Universities had what was called PhD Courses (Cursos de Doctorado) which were conceived to intiate PhD students into some scientific topics close to their interest. There were at least two plans of study (1987-2003 and 2003-2008). I was the Director of the Ph.D. degree Program in Mathematics between 1995 and 2002. In these period I also taught several topics devoting some 495 hours on the black board. Some of these topics were:
Efficient Algorithms in Computer Algebra
Algebra & Algorithms
Algebra Seminar
Panorama of Mathematics (with invited speakers)
Arithmetic and Diophantine Geometry
Number Theory and Complexity
NumericalAgebraic Geometry
Models of Complexity in Algebra and Geometry
Introduction to Real Algebraic Geometry
etc.