Abstract: Orthogonal polynomials occupy a remarkable position at the intersection of analysis, algebra, approximation theory, mathematical physics, and spectral theory. Beginning with the classical families of Hermite, Laguerre, Jacobi, and Legendre, the subject has evolved into a broad framework encompassing discrete and semiclassical orthogonal polynomials, orthogonality on the unit circle, multiple orthogonality, and matrix-valued orthogonal polynomials.

The theory is driven by a variety of mathematical questions. Classical orthogonal polynomials arise naturally in differential equations and quantum mechanics, orthogonal polynomials on the unit circle are closely related to unitary operators and complex analysis, multiple orthogonal polynomials appear in rational approximation and number theory, while matrix-valued orthogonal polynomials emerge from representation theory and non-commutative spectral problems. Despite their diversity, these objects share many structural features, including recurrence relations, spectral interpretations, and remarkable algebraic properties.

This talk presents an overview of several of the main branches of the subject, emphasizing the motivations behind their introduction, the connections between them, and some of the problems that continue to drive current research. Particular attention will be given to inverse spectral problems and recent developments concerning classical and semiclassical orthogonal polynomials.

Abstract: We present a numerical scheme for Hamilton-Jacobi equations driven by a nonlocal and nonsymmetric operator. The proposed method combines a Lax-Friedrichs discretization of the Hamiltonian with a decomposition of the nonlocal operator into small and large jump contributions. Small jumps are approximated by a local second-order operator obtained through a spectral representation, while large jumps are discretized by suitable quadrature rules combined with monotone multilinear interpolation. The lack of symmetry of the kernel generates an additional drift term, which is approximated using an upwind discretization. We prove that this scheme is consistent, monotone, and, as expected,  is conditionally stable.

Abstract: Spatial pattern formation in nature is neither random nor uniform. Instead it follows a combination of factors which determine its distribution. In this talk we will present a model from population dynamics which consists of a system of two strongly coupled nonlinear and nonlocal PDEs with cross-diffusion. These equations lack a general existence or regularity theory, making their analysis challenging. On the other hand, in the last 20 years optimal transport theory has become a standard tool to study the qualitative behaviour of nonlinear PDEs. In the same spirit we will show that the qualitative behaviour of our model admits a variational formulation, i.e. we will show that its solutions converge exponentially fast towards its equilibirum by showing that they satisfy a functional inequality which implies a Polyak-Łojasiewicz type inequality for an entropy functional on the space of probability measures endowed with the spherical Hellinger-Kantorovich metric. This provides an optimal transport interpretation of the convergence to equilibrium, where the dynamics itself prescribes the optimal plan.