GEOMETRIA REAL (GEOR)

Contracts

Main Researchers

UCM Members of the project

External members of the project

Summary

    Real Geometry studies many of the objects and structures arising from the mathematical modelling of physical and technological processes, that is, it studies those objects defined through equalities and inequalities involving real valued functions (polynomial, Nash, analytic, differentiable, semialgebraic, constructible, etc.). This subject also includes the construction and development of the suitable structures to analyze those objects, as well as the relevant computational methods: algorithms and complexity.
    Real Geometry and its Applications studies many of the objects and structures arising in the mathematical modelling of physical, technological or social phenomena. More precisely, it studies the objects that can be defined using equalities and inequalities of real-valued functions. This field implies the construction and development of the structures that are necessary in order to analyze those objects, as well as the corresponding computational methods, which may have applications to other subjects. This project is the natural continuation of the so-called REAL GEOMETRY (MTM2008-00272, MTM2011-22435) supported from 2009 to 2011 and 2012 to 2014 by MICINN and MINECO. In this new project we have chosen to redraw our previous lines of research, in order to get a better interaction among the researchers of the group and take so an advantage from their variety. We shall approach problems related with those of former projects, but we will also address new goals which have appeared after solving some of the ones proposed in previous projects, or from the interaction with other research groups and even from other subjects, like the Economy.
    By the nature of the objects that we consider, the questions proposed and the techniques that we use, our research is included in the Real Algebraic and Analytic Geometry (MSC: 14Pxx). This subject is being developed by important European and North-American research groups since more than 40 years, and also researchers of other countries (Japan, Brazil, Chile, New Zealand) work in this field. There exists a tight and fruitful collaboration among all these groups. In order to ease this joint work, we include in our group three foreign researchers with whom we have been working from many years, and who were already included in the previous project.

    Our main objectives for the period 2015-2017 are the following, once organized by themes:

    • (A) Real and Complex Analytic Geometry, with two great lines:

      • Real structure of Stein spaces. Riemann and Klein surfaces: Normalization of Stein spaces and underlying real structure. Real forms of a complex algebraic curve, moduli of real algebraic curves, real and imaginary genus of finite groups, actions of finite groups over Belyi's surfaces.
      • Real Analytic Geometry: Real geometry on non-coherent analytic surfaces. Algebraic and topological properties of C-analytic, C-semianalytic and amenable C-semianalytic sets. Irreducibility and irreducible components. Points of non-coherence.
    • (B) Algebraic, Semialgebraic and o-minimal Geometry, also with two lines:

      • Semialgebraic and o-minimal Geometry: Properties of the Zariski and maximal spectra of rings of continuous and C^k semialgebraic functions on a semialgebraic set. Definable groups on o-minimal structures, abelian groups that are definable compact, groups of finite Morley's rank.
      • Polinomial and Nash mappings: Large families of semialgebraic sets that are either polynomial or regular images of R^n. Characterization of the semialgebraic subsets that are Nash images of Euclidean spaces. Study of the open polynomial, regular ans Nash maps between Euclidean spaces.
    • (C) Applications.

      • Computational Algebra and Algorithms in Real Geometry. Interdisciplinary applications of Real Geometry: Constructive Algebra and dynamical methods, elementary recursive bounds, stability of symbolic calculus. Further applications of their techniques and algorithms.