研究生教育创新计划高水平学术前沿讲座

报告人： Nick J. Simm, University of Sussex

报告时间： 3月 26日 19:00-21:00， 3月 27日 9:30-11:30

报告地点：1518

课程简介：对数关联高斯场展现出丰富多彩的性质， 并广泛出现在概率论与物理的众多研究领域中，尤其在随机矩阵的特征多项式与黎曼Zeta函数在临界线上值。基于随机矩阵与黎曼Zeta函数的潜在关联，报告人Simm博士与其合作者分析了GUE随机矩阵特征多项式的一些新性质并提出一个猜想，此猜想已经引起了相关研究者的很大关注。本课程围绕对数关联高斯场、随机矩阵的特征多项式及其极值展开讨论，内容详细介绍如下。

In this mini-course I will provide an introduction and discuss recent results on the mathematical object known as "log-correlated Gaussian field" (LCGF). In two dimensions this is better known as the Gaussian free field and has equally remarkable and beautiful properties. Recentlythere has been an explosion of interest coming from different directions, including statistics of the Riemann zeta function,eigenvalues of random matrices and Gaussian multiplicative chaos. The LCGF is a fundamental building block in physical theories such as Liouville quantum gravity.

This mini-course will focus on the connection to random matrix theory and is comprised of four parts:

Talk 1: [Convergence of the characteristic polynomial and Gaussian processes] 26/03/2018

Part i) Introduction, motivation and the Circular Unitary Ensemble ofrandom matrices.

Part ii) Log-correlated fields in the Gaussian Unitary Ensemble and fractional Brownian motion.

Talk 2: [Random matrix theory and Gaussian multiplicative chaos] 27/03/2018

Part i) Introduction to Gaussian multiplicative chaos and relation to random matrices.

Part ii) Extreme value statistics of log-correlated Gaussian fields -maximum of the characteristic polynomial of a random matrix.