## ESTRUCTURAS ALGEBRAICAS, ANALÍTICAS Y O-MINIMALES (STRANO)

## Contracts

**Current:** MTM2017-82105-P (period: 2018-2021. IP: J.F. Fernando)
**Previous:** MTM2008-00272 (period: 2009-2011), MTM2011-22435 (period: 2012-2015, extended), MTM2014-55565-P (period: 2015-2018, extended)
**Related (in Spain):** MTM2017-88796-P (Computación simbólica: nuevos retos en álgebra y geometría y sus aplicaciones)

## Main Researcher

## UCM Members of the project

## External members and colaborators of the project

## Summary

**Algebraic, analytic and o-minimal structures** study many of the objects and systems arising in the mathematical modelling of physical, technological, economical or social phenomena. More precisely, it studies the objects that can be defined using equalities and inequalities of real-valued functions. The study of these structures, that are inside the general field of Real Algebraic and Analytic Geometry, implies the construction and development of the techniques and tools that are necessary in order to analyze those objects, which may have applications to other subjects. This project can be understood as an updated version of the so-called projects REAL GEOMETRY (MTM2011-22435) and REAL GEOMETRY AND ITS APPLICATIONS (MTM2014-55565-P) supported from 2012 to 2015 (extended one year) and 2015 to 2018 (extended one year) by MINECO.
In this new project we have chosen to redraw substantially many of our previous lines of research, to make a more compact and productive group, in order to get a better interaction and synergies among the researchers of the group, to take so an advantage from their variety and to have better control that all its lines are productive and obtain tangible results during the period in which the project will be developed. We shall approach problems related with those of former projects and reintroduce some topics of former projects that have not been completed yet, but we will also address new goals (with problems related to differential topology) 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 (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 and of our group with others: mainly with those located in Italy, France, Germany and United Kingdom.

Our main objectives grouped by thematic areas are the following:

**(A) Real and complex analytical structures.**

- 1.
*Riemann and Klein surfaces*: real and imaginary genera, spectra, groups of automorphisms.
- 2.
*Real and complex analytic geometry*: real and complex structures of normalization, 17th Hilbert problem and Pythagoras numbers for analytic rings.

**(B) Semi-algebraic and o-minimal structures.**

- 3.
*o-minimality*: Nash, algebraic and definible groups, commutators, Cartan subgroups.
- 4.
*Semialgebraic functions*: rings of C^k semialgebraic functions, rings of adÃ¨les, differentiable approximation of continuous semialgebraic maps.
- 5.
*Interdisciplinary applications*: construction of economic models based on empirical data, separation of convex polytopes.

**(C) Algebraic Structures.**

- 6.
*Sums of squares*: 17th Hilbert problem and Pythagoras numbers for excellent local henselianos rings.
- 7.
*Polynomial, regular, regulous and Nash maps*: study of the images of these types of maps.
- 8.
*Algebraic models*: algebrization of finite families of differentiable, analytical and Nash manifolds with only monomial singularities.

**KEY WORDS:** Klein surfaces, normalization, sums of squares, definible groups, semialgebraic and Nash functions, algebrization.