Do polynomials dream of symmetric curves?

Job Candidate Talk
Tuesday, March 31, 2015 - 11:05am for 1 hour (actually 50 minutes)
Skiles 005
Andrei Martinez-Finkelshtein – Universidad de Almeria, Spain
Jeff Geronimo
Polynomials defined either by some type of orthogonality or satisfying differential equations are pervasive in approximation theory, random matrix theory, special functions, harmonic analysis, scientific computing and applications. Numerical simulations show that their zeros exhibit a common feature: they align themselves along certain curves on the plane. What are these curves? In some cases we can answer this question, at least asymptotically. The answer connects fascinating mathematical objects, such as extremal problems in electrostatics, Riemann surfaces, trajectories of quadratic differentials, algebraic functions; this list is not complete. This talk is a brief survey of some ideas related to this problem, from the breakthrough developments in the 1980-ies to nowadays, finishing with some recent results and open problems.