about us

 
This team was created in 1995 merging two groups of researchers from the universities of Almería and Granada. In June 2015, the group continued with members of the University of Almeria by the constitution of the new Grupo en Ortogonalidad y Aplicaciones (GOYA), affiliated with the University of Granada. Currently, our main lines of research are:

• theory of special functions and orthogonal polynomials,
• approximation theory,
• applications, in particular, in Physics and Medicine.

It is our goal also the computational implementation of our results, creating numerical and symbolic algorithms. Several team members belong simultaneously to the Institute Carlos I for Theoretical and Computational Physics of the Granada University.

Some of the topics we are currently working on are:

•     Electrostatic interpretation and density of the zeros of families of special functions. We are interested in the study of the complex zeros of families of both classical special functions (where we achieved already several goals) and nonlinear special functions (such as Painlevé transcendents), field that remains almost virgin.
•     Modern methods of asymptotic analysis of special functions. A relevant advance experienced very recently is the matrix Riemann-Hilbert method, that allows to find uniform asymptotics for families of orthogonal polynomials, and that uses advanced techniques of the potential theory. Among goals of this team are the analysis of several open cases, such as matrix and Sobolev polynomials, and those not satisfying the Szegö condition.
•     Applications to random matrices and stochastic processes. Analysis of the asymptotic behavior of models reducible to the so called determinantal point processes and their generalizations.
•     Analysis from the algebraic and asymptotic points of view of families of orthogonal polynomials with respect to non-standard inner products (in particular, of Sobolev). Asymptotic analysis of Sobolev orthogonal polynomials for discrete measures or of those with unbounded support, as well as balanced polynomials.
•     Computation and applications of information measures of special functions. Among the information measures, the best known is the Boltzmann-Shannon entropy, for which this team has obtained several results of both numerical and analytic nature. These results, beside their intrinsic interest for the theory of special functions, have applications in Quantum Mechanics. Next goals: discrete entropies, other information measures, and their applications.
•     Study of orthogonal polynomials of several variables. Presently, no complete and satisfactory theory of multivariate orthogonal polynomials is available. This is a new line of research, that has produced its first results, such as classification of classical families, and that also has direct applications in modern Physics.
•     Methods of modelling and diagnosis in Ophthalmology, in particular, techniques of surface reconstruction of the human cornea from topographic data, and mechanisms of early detection of related pathologies.

If you want to know more about the activities of the team, follow the navigation links you will find on the left, as well as visit the webpage of the Institute Carlos I of Theoretical and Computational Physics.
 

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PHotos

1997: At the very beginning... Working at theEscuela Politécnica Superior in Almería on the asymptotics of Sobolev orthogonal polynomials for the coherent pairs: