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1956年著名的挪威数学家和赛获得者Atle Selberg首次引入的复杂功能，随后被称为Selberg Zeta功能。这些最初与恒定负曲率的紧凑表面相关联。它们的定义与著名的Riemann Zeta函数相比，只是质数的作用被表面上封闭的大地测量学的长度所取代。惊人的事实是，在这种情况下，零位于特定的线条上，这与著名的Riemann假设非常相似，这两个假设都是希尔伯特著名的23个问题列表和粘土学院的千年问题的问题之一。但是，相比之下，在许多开放表面或无限面积表面的例子中，相关的Selberg Zeta功能的零要复杂得多。这些单独的零通常被称为“共振”，并且在紧凑型情况下发挥了与拉普拉斯人的特征值相似的作用，并且是近年来开发更好的计算方法和计算机硬件的重要几何和动态不变的。在某些有趣的情况下，出现了这些零模式的清晰图片。令人惊讶的是，这些零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，形状让人联想到蕾丝刺绣。当基础表面具有更多的对称性时，这些零图具有最简单的结构。这项工作将有助于理解Zeta函数零的这些模式及其在Zeta函数和相关表面上提供的信息。