- Further contributions to general orthogonal polynomials (OP), multiple OP, and rational approximation, including the properties of polynomials satisfying non-standard orthogonality conditions, their connections with different branches of mathematics and mathematical physics, and new interfaces between OP on the real line (OPRL) and on the unit circle (OPUC).
- Asymptotic methods for OP, including the development of the Riemann-Hilbert asymptotic analysis and its applications to the study of critical phenomena related to non-linear special functions; asymptotic analysis of polynomials of non-standard orthogonality (Sobolev, Hermite-Padé).
- Analysis of extremal problems in logarithmic potential theory, in particular, saddle points of energy functionals on the plane for vector measures, connected to several object from the geometric function theory and non-linear phenomena such as laplacian growth.
- Generalizations of Schur function theory via abstract first return generating functions (FR-functions), and applications to OP, harmonic analisis and operator theory.
- 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 this team, via Schur functions as a tool to study the dynamics and topological phases of interest in quantum computation. Extension to open QRW via FR-functions.
- 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. Possible applications to random systems.
- Development of numerical algorithms for biomedical applications in the framework of the Fourier optics, and exploration of possible new optical surfaces in the design of soft contact lenses.

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

- Juan José Moreno Balcázar (Universidad de Almería), principal investigator
- Pedro Martínez González (Universidad de Almería)
- Leandro Moral Ledesema (Universidad de Zaragoza)
- María José Cantero Medina (Universidad de Zaragoza)

- Luis F. Velázquez Campoy (Universidad de Zaragoza)

- Lance L. Littlejohn (Baylor University, TX, USA)
- Ana Belén Castaño Fernández (Universidad de Almería)
- Juan Francisco Mañas Mañas (Universidad de Almería)

- A.
Martinez-Finkelshtein, G. Silva,
*Spectral Curves, Critical Measures and the Hermitian Random Matrix Model with External Source*, in preparation.

- A. Martínez-Finkelshtein, L.L. Silva Ribeiro, A. S. Ranga, M. Tyaglov, Complementary Romanovski-Routh polynomials: From orthogonal polynomials on the unit circle to Coulomb wave functions, Proceedings of the AMS 147 (6) (2019), 2625–2640, https://doi.org/10.1090/proc/14423. Also preprint arXiv math.1806.02232.
- A.
Martinez-Finkelshtein, G. Silva,
*Critical measures for vector energy: asymptotics of non-diagonal multiple orthogonal polynomials for a cubic weight*, Advances in Mathematics**349**(2019), 246-315. Also preprint ArXiv math:1805.01748. - A.
Martínez-Finkelshtein, Brian Simanek, Barry Simon,
Poncelet's Theorem,
Paraorthogonal Polynomials and the Numerical Range of
Compressed Multiplication Operators, Advances
in Mathematics
**349**(2019), 992-1035. Also preprint arXiv math.1810.13357.