ESTRUCTURAS ALGEBRAICAS, ANALÍTICAS Y O-MINIMALES (STRANO)
Contracts
- Current: PID2021-122752NB-I00 (period: 2022-2026, IP1: J.F. Fernando, IP2: J.J. Etayo)
- Previous: MTM2008-00272 (period: 2009-2011), MTM2011-22435 (period: 2012-2015, extended), MTM2014-55565-P (period: 2015-2018, extended), MTM2017-82105-P (period: 2018-2022, extended)
- Related (in Spain): PID2020-113192GB-I00 (Visualización matemática: fundamentos, algoritmos y aplicaciones)
Main Researcher
UCM Members of the project
External members and colaborators of the project
Summary
Algebraic, analytic and o-minimal structures study objects and systems that arise from mathematically modeling physical, technological, economic or social phenomena that can be described using equalities and inequalities of functions that take real values. The study of these structures, which are encompassed within the field of Real Algebraic and Analytic Geometry, involves the construction and development of adequate techniques and tools to analyze these objects and they may have potential applications to other disciplines. This project is the natural continuation of the one of reference MTM2017-82105-P (STRANO: "Algebraic, analytic and o-minimal structures").
For this new project we have expanded the work team with two researchers (one Spanish and one foreign) who frequently collaborate with us. The objective is to constitute a transversal group, in which there is a strong interaction and synergies between its researchers and to take advantage of our scientific diversity so that the lines of work are more productive.
We will address some topics similar to those of the previous project, but we will also face new challenges that have arisen when solving objectives proposed in previous projects or through interaction with other research groups. In addition, we will analyze the transfer of research associated with the basic concepts of the project to improve teaching aspects.
Considering the objects studied, the problems addressed and the techniques used, our activity is framed in Real Algebraic and Analytic Geometry (MSC: 14Pxx). This discipline has been developed for more than 45 years by consolidated research groups both in Europe and North America and by researchers from other countries (Japan, Brazil, Chile, New Zealand, etc.). It is worth highlighting the close and fruitful collaboration between the different groups and ours with other groups: mainly those located in Italy, France, Germany and the United Kingdom.
Our main objectives grouped by subject areas are the following:
(A) Real and complex analytical structures.
- 1. Riemann and Klein surfaces: real genus; symmetric crosscap number; symmetric crosscap spectrum; groups of automorphisms.
- 2. Real and complex analytic geometry: extension property for global analytic sets and Cartan's Theorem B; Hilbert 17th problem and Pythagoras numbers for analytic rings; tame extension of automorphisms between real analytic spaces.
(B) Semialgebraic, o-minimal and differentiable structures..
- 3. o-minimality: definable groups; type space; spectral spaces; Ellis enveloping semigroup; compactifications.
- 4. Semialgebraic functions: rings of C^k semialgebraic functions; semialgebraic covers.
- 5. Approximation: of maps between sets and manifolds.
- 6. Applications: separation of convex polytopes; transfer of results to the field of education.
(C) Algebraic Structures.
- 7. Sums of squares: Hilbert's 17th problem and Pythagoras numbers for excellent local Henselian rings.
- 8. Polynomial and Nash maps: images of closed balls under these types of applications; optimization problems; construction of polynomial and Nash paths in semialgebraic sets.
- 9. Algebraic models: algebrization of finite families of differentiable, analytic and Nash varieties; real and complex Q-algebraic sets.
KEY WORDS:
Riemann and Klein surfaces, sums of squares, images of semialgebraic maps, manifolds with corners, approximation, algebrization, definable and semi-algebraic functions, optimization, model theory, definable groups.