Wednesday, September 26, 2018
09/26/2018 - 7:00pm
Numerical algorithms are based on approximation of functions. Polynomials can only approximate smooth functions effectively, but rational functions can approximate functions with singularities with fast *root-exponential convergence*: convergence at a rate exp(-C*sqrt(n)), C>0. This property has rarely been exploited. We show how powerful it can be, for example, for solving the Laplace equation on a polygon. An important advance along the way has been the “AAA algorithm” developed with Nakatsukasa and Sete.
09/26/2018 - 8:15pm
Hilbert’s twenty-first problem, formulated to generalize Riemann’s work on hypergeometric equations, concerns the existence of linear differential equations of Fuchsian type on the complex plane with specified singular points and monodromic group. Its modern solution, due to Deligne and known as the Riemann-Hilbert correspondence, establishes an equivalence between two different types of data on a complex algebraic manifold X: the representations of the fundamental group of X (topological data) and the linear systems of algebraic differential equations on X with regular singularieties (algebraic data).
I’ll review this classical theory, and discuss some recent progress towards similar problems for p-adic manifolds.