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Dynamical zeta functions and resonances for infinite area surfaces

无限面积表面的动态 zeta 函数和共振

基本信息

批准号：
EP/T001674/1
负责人：
Mark Pollicott
金额：
$ 50.27万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2019
资助国家：
英国
起止时间：
2019 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT001674%2F1
关键词：
Dynamical zeta functions resonances infinite

项目摘要

This proposal deals with complex functions first introduced by the famous norwegian mathematician and Fields medalist Atle Selberg in 1956, and subsequently called Selberg zeta functions. These were originally associated to compact surfaces of constant negative curvature.Their definition was by analogy with the famous Riemann zeta function, except that the role of the prime numbers is replaced by the lengths of closed geodesics on the surface. The striking fact is that in this setting the zeros lie on specific lines, which is very similar to the famous Riemann Hypothesis, both one of the problems from Hilbert's famous list of 23 problems and the Clay Institute's Millennium Problems. However, by contrast, in the case of many examples of open surfaces, or infinite area surfaces, the zeros of the associated Selberg zeta functions are much more complicated. These individual zeros are often called "resonances" and play a role similar to that of the eigenvalues of the laplacian for the compact case, and are important geometric and dynamical invariants for the surfacesWith the development of better computational methods and computer hardware over recent years a much clearer picture of the patterns of these zeros has emerged in some interesting cases. Somewhat surprisingly, the plots of these zeros had strikingly beautiful patterns. They appear to lie on very delicately defined curves in shapes reminiscent of lace embroidery. These plots of the zeros have their simplest structures when the underlying surface has more symmetries.This work will help to understand these patterns of zeta function zeros and the information that it gives on both the zeta function and the associated surface.

这个建议涉及复杂的功能首先介绍了著名的诺克斯维尔数学家和菲尔兹奖得主Atle Selberg在1956年，后来被称为Selberg zeta函数。它们最初与常负曲率的紧致曲面相关联，其定义类似于著名的黎曼zeta函数，只是素数的作用被曲面上闭测地线的长度所取代。引人注目的事实是，在这种情况下，零点位于特定的线上，这与著名的黎曼假设非常相似，这两个问题都是希尔伯特著名的23个问题和克莱研究所的千年问题之一。然而，相比之下，在许多开放曲面或无限面积曲面的例子中，相关Selberg zeta函数的零点要复杂得多。这些单独的零点通常被称为“共振”，并发挥类似的作用，在紧凑的情况下的拉普拉斯算子的特征值，是重要的几何和动力学不变量的表面随着更好的计算方法和计算机硬件的发展，近年来，这些零点的模式更清晰的图片已经出现在一些有趣的情况。有些令人惊讶的是，这些零的曲线有着惊人的美丽图案。他们似乎躺在非常微妙的定义曲线的形状让人想起花边刺绣。这些图的零点有其最简单的结构时，底层表面有更多的对称性。这项工作将有助于理解这些模式的zeta函数零点和信息，它给出了两个zeta函数和相关的表面。