Marco Bertola
Orthogonal polynomials, some of their applications and asymptotic analysis
The course provides an overview to the theory and applications of orthogonal polynomials (OPs). The undergraduate student most likely encounters OPs when discussing separation of variables in solutions of important PDEs, notably the harmonic oscillator in quantum mechanics (i.e. Hermite polynomials). However, their applications cover a much wider range of topics, whose list includes (but is not limited to): elements of combinatorics, number theory (e.g. the proof that the Euler constant is transcendental); integrable systems (e.g. the Toda lattice equations); stochastic models (random matrices); special equations (Painlevé equations).
The topics covered in the course will be:
- Origins, definitions and fundamental properties.
- Asymptotic analysis for large degrees; elements of nonlinear steepest descent analysis
- Some applications to spectral theory of large random matrices (hopefully with mention of Fredholm determinants and Tracy-Widom distribution, time permitting).
The course is aimed at graduate students (or advanced undergraduate) with a solid grasp of complex analysis (contour integration, conformal properties of holomorphic functions, Cauchy theorem(s)), linear algebra and elementary measure theory.
