Abstract: | In both analytic number theory (the Riemann Hypothesis) and mathematical physics (Ising models and Euclidean field theories) the following complex analysis issue arises. For $\rho$ a finite positive measure on the real line $\mathbb{R}$, let $H(z; \rho, \lambda)$ denote the Fourier transform of $\exp\{\lambda u^2\} d\rho (u)$, i.e., the integral over $\mathbb{R}$ of $\exp\{izu + \lambda u^2\} d\rho (u)$ extended from real to complex $z$, for those $\lambda$ (including all $\lambda < 0$) where this is possible. The issue is to determine for various $\rho$'s those $\lambda$'s for which all zeros of $H$ in the complex plane are real. We will discuss some old and new theorems about this issue. 报告人简介: Charles M. Newman, Silver Professor of Mathematics at the Courant Institute and Global Network Professor at NYU-New York and NYU-Shanghai, received B.S. degrees in Mathematics and in Physics from MIT and M.S. and Ph.D. degrees in Physics from Princeton. With 200+ published papers, mainly in probability and statistical physics, he has been a Sloan and Guggenheim fellow and is a member of the U.S. National Academy of Sciences, the American Academy of Arts and Sciences and the Brazilian Academy of Sciences. |