Analysis

The Analysis group conducts research in a wide variety of topics, including integral equations and operator theory, special functions and orthogonal or multiple orthogonal polynomials for the analysis and solution of differential problems. The research covers a wealth of different perspectives on mathematical analysis from theoretical aspects, numerical analysis, scientific computing and applications.

Pure Analysis: Development of new methods to solve integral equations of different classes, in closed form, to invert and study mapping properties of different integral transformations with special kernel functions. Operational formulations for integral, differential and fractional calculus are developed together with the use of transform methods to obtain new systems of polynomials. The development of analytic methods of number theory, involving the theory of Riemann's zeta-function, and proving equivalences to the Riemann hypothesis are part of the group's research.

Numerical Analysis: Development of new numerical algorithms, as well as adaptation of classical ones, tuned to novel computer architectures. In-depth research to efficiently solve certain problems involve the exploitation of numerical linear algebra kernels, such those related with large dimensional eigenvalue computations. Furthermore, the exploitation of the connection between stochastic PDEs and forward-backward doubly stochastic differential equations to obtain existence results for the multidimensional stochastic Burgers equation are being investigated.

Computational Mathematics: Deduction of new closed formulas for the connection coefficients of perturbed Chebyshev polynomials with arbitrary order of perturbation. Study of some properties of common points and zeros of those polynomials, including their Gershgorin location. Development of spectral methods for the solution of integro-differential problems. These topics are being offered via the implementation of efficient and robust mathematical software.

Mathematical Models and Applications: Applications to biomedicine, engineering and economics are being developed. The biomedical mathematics application work tackles (i)  epidemiology modelling and (ii) models of central pattern generators for animal and robot locomotion. Among the engineering problems tackled stand out (i) numerical linear algebra in large scale problems arising in mechanical models, (ii) methods of graph theory and of optimal control for fluid dynamic problems, as well as numerical treatment of partial differential equations using multi-scale techniques. The problems addressed in economics include (i) conditions for existence of multiple equilibria on macroeconomic growth models, (ii) hysteresis in labour economics, (iii) option pricing (study  of a market with jumps in the presence of a large investor), (iv) financial markets (study of persistence characteristics and long-memory in financial prices, using novel fractal dimension estimation algorithms).