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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.

该提案涉及复杂函数，由著名挪威数学家、菲尔兹奖得主 Atle Selberg 于 1956 年首次提出，随后被称为 Selberg zeta 函数。它们最初与恒定负曲率的致密曲面相关。它们的定义与著名的黎曼 zeta 函数类似，只不过素数的作用被曲面上闭合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，这与著名的黎曼猜想非常相似，两者都是希尔伯特著名的 23 个问题清单和克莱研究所的千年问题中的问题之一。然而，相比之下，在开放表面或无限面积表面的许多示例中，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”，其作用类似于紧凑情况下拉普拉斯特征值的作用，并且是曲面的重要几何和动力学不变量。随着近年来更好的计算方法和计算机硬件的发展，在一些有趣的情况下出现了这些零点模式的更清晰的图像。有点令人惊讶的是，这些零点的图具有惊人的美丽图案。它们似乎位于非常精致的曲线上，其形状让人想起蕾丝刺绣。当下垫面具有更多对称性时，这些零点图具有最简单的结构。这项工作将有助于理解 zeta 函数零点的这些模式以及它在 zeta 函数和相关表面上提供的信息。