Tomás Recio , defended in February 1987. The same year I became Associate Professor (Profesor Titular) of "Algebra" (25 years old). In January 1988, I had to leave the the University, returning to the academic system in February 1989. I returned with the firm conviction of devoting most of my time and effort to the scientific formation of young mathematicians through a high quality doctoral program in Mathematics at the University of Cantabria (the program does not exist anymore). I was advising my first PhD Student (J. L. Montaña ) when I was 27 years old. Since then I have tried to interpret my research as a pathfinder, seeking less traveled roads in the frontiers of mathematical knowledge, accompanied by a PhD Student who gets thus a formation as researcher.
Computational Complexity ,
Foundations of Computational
Mathematics, either Symbolic or Numeric Effective
Methods in Algebraic Geometry. Obviously, these interests also include
related fields as Commutative Algebra,
Algebraic, Arithmetic and Diophantine Geometry, Numerical Analysis, Linear Algebra,
Theoretical Computer Science,
Integral Geometry and many others.
In my poor opinion, there is no isolated field of knowledge in Mathematics, there is just a missed path which
connects it to the others .
and Marc Giusti ) in the nineties
of the past century. The manuscripts we published (together with my former student
in this period yielded the most efficient symbolic
method that solves universally systems of multivariate polynomial equations.
Another milestone was the
solution of Smale's 17th Problem toghether with my former student
C. Beltrán , obtained after several joint publications in the first decade of
the present century. Other milestones were the works on the Arithmetic Nullstellensatz (
firstly developped with Teresa Krick , and,
also with M. Sombra and my former student
K. Hägele ). Last but not least,
I have devoted an special effort to understand
lower complexity bounds of several problems in several contexts as Algebraic Complexity Theory
(including a solution to M. Ben-Or's conjeture on the usage of higher order Betti numbers as
lower complexity bound, published with J.E. Morais and J.L. Montaña) or to prove
exponential lower complexity bounds for universal, geometrically robust algorithms
for eliminating a single block of existencial quantifiers (most of them with
J. Heintz and G. Matera ).
published more than 60 scientific
manuscripts. Some of them have been well treated by my colleagues, yielding an h-index
17 in Thomson Reuters Web of Science (with more than 600 citations), and an h-index 20 in
(with more than 1700 citations in this data basis) and an h-index 21 in
Research Gate . I have completed
30 years of positive evaluation
( 5 periods of 6 years each,
called "sexenios" in the Spanish
academic system). I should apply for next evaluation on December, 2023 if not in retirement.
I also liked Editing, being a
member of the Editorial Board of the
Journal of Complexity since 2004 and of the Editorial Board of
Applicable Algebra in Engineering Communications and Computing since 2014. I have
also edited several books
and Special Issues
I have leaded several national
Research Grants and participated in many since my youngest years. I also
organized (either in the scientific committe or in the local committee)
scientific meetings and conferences . Colleagues found interesting to invite me to give
and invited talks
and I presented a few referred
communications along these years. I have also been invited to
visit several scientific Laboratories and Universities.
many courses either of undergraduate or
I have devoted more than 4800 hours lecturing in a classroom , not counting
the preparation, correction of excercises and projects, exams or tutorial hours, which, obviously,
were much more than these. Among
undergraduate courses, I was committed with courses in Mathematics Studies
Commutative Algebra, Linear Algebra,
Computer Algebra, Algebraic Geometry, Geometry or General Topology), Computer Engineering
(as Introduction to
Logic for Computer
Scientists or Automata and Formal Languages Theory), Physics (Linear Algebra) or
Nautic Studies (Linear Algebra and Spheric Trigonometry).
Among graduate courses I was committed with several courses in the Master Program of
"Mathematics & Computation" as Computational Complexity. Effective Methods in Algebraic Geometry, since its occurence in 2008.
In former Doctoral Program in Mathematics courses I taught many different
courses as Introduction to Computational Complexity,
Real Algebraic Geometry, Algebraic Complety Theory, Algorithmic Algebra and others.
several duties and
services to the Academic Community both in different structures of the University of
Cantabria, in national and internatonal Scientific Committees or as chosen delegate of the union
CC.OO. and member of its
Executive Committee in Cantabria. I'm also the head of the Research Group
M&C:FLAI (Mathematics & Computation: Foundations, Learning and Artificial Intelligence).