This project combines both ambitious goals in fundamental research on orthogonal polynomials, special functions and their analytic and structural properties, with the applications of this knowledge in other branches of mathematics (stochastic processes, combinatorics, numerical analysis), physics (statistical physics, integrable systems, quantum mechanics, quantum random walks, quantum computation), and technology (signal processing and diagnostic tools in ophthalmology, with applications in clinical practice). Some of the problems to be tackled are:

- Further contributions to general orthogonal polynomials (OP) and rational approximation, including the development of the Riemann-Hilbert asymptotic analysis and its applications to the study of critical phenomena related to non-linear special functions or singularities appearing in the description of the electronic structure of graphene; asymptotic analysis of polynomials of non-standard orthogonality (Sobolev, Hermite-Padé, Wronskians), and new approaches to OP on the unit circle.
- Development of electrostatic models for zeros of several classes of polynomials and analysis of different extremal problems in logarithmic potential theory, in particular, saddle points of energy functionals on the plane, connected to several object from the geometric function theory and non-linear phenomena such as laplacian growth.
- Applications of newly developed tools to the study of multiparticle diffusion processes and random matrices, their possible phase transitions, as well as new insights into classical Markov processes.
- Further development of the OP approach to Quantum Random Walks (QRW), started by members of this team, in particular via Schur functions as a tool to study the dynamics and topological phases in QRW, with applications to quantum computing.
- Extension of known connections between bispectral problems, integrable systems, Darboux transformations and signal processing to broader contexts of block-Jacobi matrices, CMV matrices, and others.
- Search for more efficient algorithms for objective measurements of the eye characteristics and diagnostics, such as the cornea shape reconstruction, calculation of optical functions of the eye from the measured wavefront aberrations, as well as analysis of these aberrations from the known PSF (Point Spread Function) corresponding to several defocus parameters.

- Andrei
Martínez-Finkelshtein (Universidad de Almería), principal
investigator

- Juan José Moreno Balcázar (Universidad de Almería), principal investigator
- Pedro Martínez González (Universidad de Almería)
- Darío Ramos López (Universidad de Almería)

- Leandro Moral Ledesema (Universidad de Zaragoza)
- María José Cantero Medina (Universidad de Zaragoza)

- Luis F. Velázquez Campoy (Universidad de Zaragoza)

- Alejandro Zarzo Altarejos (Universidad Politécnica de Madrid)
- F. Alberto Grünbaum (Universidad de California Berkeley, EEUU)
- Ana Belén Castaño Fernández (Universidad de Almería)
- Juan Francisco Mañas Mañas (Universidad de Almería)

- A. Aptekarev, G. López-Lagomasino, A. Martínez-Finkelshtein, On Nikishin systems with discrete components and weak asymptotics of multiple orthogonal polynomials, preprint arXiv math.1403.3729.
- A. Aptekarev, G. López-Lagomasino, A. Martínez-Finkelshtein, Strong asymptotics for the Pollaczek multiple orthogonal polynomials ensembles, preprint arXiv math.1410.1261.
- A. Martínez-Finkelshtein, G. Silva, Critical measures for vector energy: global structure of trajectories of quadratic differentials, accepted in Advances in Mathematics. Also preprint arxiv math.1509.06704.
- A. Martínez-Finkelshtein, A. Sri Ranga, Daniel O. Veronese, Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure, preprint arXiv math.1505.07788.
- M.J.
Cantero, A. Iserles,
*OPUC and explicit determinantal representations*, in preparation. - F.
A. Grünbaum, L. Velázquez,
*The CMV bispectral problem*, to appear in Int. Math. Res. Notices. - F.
A. Grünbaum, L. Velázquez,
*On the generalization of Schur functions: applications to OPRL and random and quantum walks*,

in preparation. - C.
Cedzich, T. Geib, F. A. Grünbaum, C. Stahl, L.
Velázquez, A. H. Werner, R. F. Werner,
*A topological classification of one-dimensional symmetric quantum walks*, in preparation.

- D. Ramos-López, M. A. Sánchez-Granero, M. Fernández-Martínez, and A. Martínez-Finkelshtein, Optimal sampling patterns for Zernike polynomials, Applied Mathematics and Computation 274 (2016) 247-257. Also preprint arXiv math.1511.00449.
- A. Martínez-Finkelshtein, D. Ramos-López, and D. Robert Iskander, Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions, Applied and Computational Harmonic Analysis, doi 10.1016/j.acha.2016.01.007. Also preprint arXiv math.1506.01670.
- A.
Martínez-Finkelshtein, P. Martínez-González, F. Thabet,
Trajectories of quadratic
differentials for Jacobi polynomials with complex
parameters, Comput.
Methods and Function Theory
**16**(3) (2016), 347-364, doi 10.1007/s40315-015-0146-7. Also preprint arXiv math.1506.03434. - A. Martinez-Finkelshtein, E. A. Rakhmanov, Do orthogonal polynomials dream of symmetric curves?, to appear in Foundations of Computational Mathematics, doi: 10.1007/s10208-016-9313-0. Also preprint arXiv math.1511.09175.
- M.
J. Cantero, A. Iserles,
*From ortogonal polynomials on the unit circle to functional equations via generating functions*, Trans. Amer. Math. Soc.**368 (**2016), 4027-4063. - M.
J. Cantero, L. Moral, F. Marcellán, L. Velázquez,
*Darboux transformations for CMV matrices*, Adv. Math.**298**(2016), 122-206. - C.
Cedzich, F. A. Grünbaum, C. Stahl, L. Velázquez, A. H.
Werner, R. F. Werner,
*Bulk-edge correspondence of one-dimensional quantum walks*, J. Phys. A: Math. Theor.**49**(2016), 12pp. - J.
F. Mañas-Mañas, F. Marcellán, J. J. Moreno-Balcázar,
*Asymptotic behavior of varying discrete Jacobi–Sobolev orthogonal polynomials*, Journal of Computational and Applied Mathematics**300**(2016) 341-353.