2016 Mini-Course by Chuck Newman - Statistical Mechanics and the Riemann Hypothesis

Topic: Statistical Mechanics and the Riemann Hypothesis
Date & Time: Tuesday, February 23, 2016 - 13:30 to Tuesday, March 22, 2016 - 15:30
Speaker: Charles Newman, Silver Professor of Mathematics, NYU/Affiliated Professor of Mathematics, NYU Shanghai
Location: Room 264, Geography Building, Zhongbei Campus, ECNU (中山北路校区,地理楼264室)

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The Mini-Course is sponsored by NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai.
© 2016 NYU Shanghai

Every Tuesday, Feb. 23 - Mar. 22, 2016

Abstract:

We review a number of old results concerning certain statistical mechanics models and their possible connections to the Riemann Hypothesis.

A standard reformulation of the Riemann Hypothesis (RH) is: The (two-sided) Laplace transform of a certain specific function Ψ​ on the real line is automatically an entire function on the complex plane; the RH is equivalent to this transform having only pure imaginary zeros. Also Ψ​ is a positive integrable function, so (modulo a multiplicative constant C) is a probability density function.

A (finite) Ising model is a specific type of probability measure P​ on the points S=(S1,...,SN)​ with each Sj = +1​ or -1​. The Lee-Yang theorem (of T. D. Lee and C. N. Yang) implies that for non-negative a1, ..., aN​, the Laplace transform of the induced probability distribution of a1 S1 + ... + aN SN​ has only pure imaginary zeros. There are also other models, where the variables are real-valued or vector-valued which have moment generating functions with only pure imaginary zeros.

An intriguing question is whether it's possible to find a sequence in N​ of models and generating functions  so that the limit as N → ∞​ of such distributions has density exactly C Ψ​. We'll discuss some of the cases where one can study the limiting distribution and some hints as to how one might try to find the "right'' choice.

 

Biography:

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.

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