Geometría algebraica y analítica real (Real algebraic and analytic geometry)

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. Our research lines include "Rings of Semialgebraic Functions", "O-minimality", "Polynomial, regular and Nash images of Rn", "Real Analytic Geometry", "Real algebraic curves", "Sums of squares and Knot Theory", "Constructibility and algorithms".<br /><br />By the nature of the studied objects, the approached problems and the used techniques, all our research lines can be included in the so-called Real Algebraic and Analytic Geometry (Mathematical Subject Classification: 14Pxx). Important research European and North-American groups and independent researchers of other countries (for instance: Japan, Brasil, Chile, etc.) have worked in this field for more than 40 years. It is worthwhile mentioning that our group collaborate with several of these international research groups.<br />

Our main goals are the following:<br /><br />1) Rings of semialgebraic functions. Classification of semialgebraic sets, semialgebraic Stone-Cech compactification, semialgebraic Zariski spectrum, Lojasiewicz inequalities.<br /><br />2) o-minimality. Definable groups on o-minimal structures, definiblely compact abelian groups, groups of finite Morley"s rank.<br />Real Analytic Geometry. Real Geometry on non-coherent analytic surfaces. 17th Hilbert"s problem for Rn: multiplicative formulae and reduction to the irreducible case.<br /><br />3) Polynomial, regular and Nash images of Rn. Conditions concerning the boundary and the set of infinite points of the polynomial and regular images of Rn. Characterization of those semialgebraic sets which are Nash images of Rn.<br /><br />4) Applications of sums of squares:<br />(A) Integral sums of squares and their relationship with the theory of knots and coverings.<br />(B) Optimization of polynomials on semialgebraic sets.<br /><br />5) Real algebraic curves. Real forms of a complex curve, moduli of real curves, real and complex genera of a finite group, actions of finite groups on Belyi surfaces.<br /><br />6) Constructibility and Algorithms. Constructive algebra and dynamic methods, elementarily recursive bounds, stability in symbolic computation<br />

The interested fellows must submit a CV and a recommendation letter. They also have to attach a brief presentation letter with a description of their research work, and how it is linked with the research lines of our group.