Extreme value statistics of characteristic polynomials of random matrices and the Riemann zeta-function

其目的是了解随机矩阵的特征多项式的极值的波动，以及Riemann Zeta函数在其临界上的涨落。这是一个非常活跃的研究领域，有许多深刻而有趣的悬而未决的问题。目的是利用对称多项式理论的有关技巧，推导出随机酉阵的特征多项式的矩的精确公式。我们还将探索基于可积系统的方法。最终，我们将试图理解在大矩阵限制下的行为。我们将使用随机矩阵结果来探索Riemann Zeta函数在其临界线上的相应统计，并将结果与来自随机欧拉乘积模型的结果进行比较。

The goal is to understand fluctuations in the extreme values of the characteristicpolynomials of random matrices, and of the Riemann zeta-function on its criticalline. This is a very active area of research with many deep and interesting unsolvedproblems. The aim will be to use techniques relating to the theory of symmetricpolynomials to deduce exact formula for the moments of the moments of thecharacteristic polynomials of random unitary matrices. We will also exploreapproaches based on integrable systems. Ultimately, we will seek to understand thebehaviour in the large-matrx limit. We will use the random matrix results to explorethe corresponding statistics for the Riemann zeta-function on its critical line and willcompare the results with those coming from a randomised Euler product model.