Percy Deift Distinguished Lecture

Universality in Numerical Computation with Random Data. Case Studies and Analytical Results.

Percy Deift

Courant Institute of Mathematical Sciences

New York University



Tuesday, March 19, 2024

4:00 pm

120 Snow Hall

Reception at 3:30 pm, 406 Snow



We consider various iterative algorithms with random data. For example, let E be an ensemble of nxn real symmetric matrices and let A be an algorithm to compute the eigenvalues of a matrix . We consider the time T=T(M; A,E,e), or equivalently the number of iterations, to compute the eigenvalues of a matrix M chosen from E to an accuracy e, using algorithm A. The main result is that the distribution for T, suitably centered and scaled, is the same for a wide variety of ensembles E.

Similar universality results are found for many different numerical problems of mathematical and physical interest. The distribution will depend on A, but not on E.

This is joint work with  C.Pfrang, G.Menon, S.Olver and mostly, T.Trogdon.

Percy Deift is a Silver Professor at the Courant Institute of Mathematical Sciences.  He received his Ph.D. in mathematical physics from Princeton University in 1977.

He is a fellow of the American Mathematical Society (2012), a member of the American Academy of Arts and Sciences (2003) and the U.S. National Academy of Sciences (2009).  He is a co-winner of the 1998 Polya Prize in Applied Combinatorics, named a Guggenheim Fellow (1999), and was a winner of the 2018 Henri Poincare Prize for mathematical physics honored “for his seminal contributions to Schroedinger operators, inverse scattering theory, nonlinear waves, asymptotic analysis of Fredholm and Toeplitz determinants, universality in random matrix theory and his deep analysis of integrable models”.

Deift’s basic research interest is in integrable systems. This includes not only dynamical integrable systems, such as geodesic flow on an ellipsoid, the Toda lattice, the Korteweg de Vries equation, and the Nonlinear Schroedinger Equation, but also topics such as orthogonal polynomials and random matrix theory. Within these topics he is particularly interested in asymptotic questions, such as the long-time behavior of solutions of the Korteweg de Vries equation, or the behavior of random matrix ensembles when the size of the matrices becomes large (universality questions). An important tool in his work is the Riemann-Hilbert Problem, and the associated nonlinear steepest-descent method. His work merges many fields, such as Geometry and Combinatorics, and he performs multidisciplinary studies into mathematical physics and pure mathematics.