Dynamical zeta functions and resonances for infinite area surfaces

这个建议处理的是由挪威著名数学家和菲尔兹奖获得者Atle Selberg于1956年首次提出的复函数，后来被称为Selberg zeta函数。它们最初与常负曲率的紧致曲面有关。它们的定义与著名的黎曼ζ函数类似，只不过质数的作用被表面上封闭测地线的长度所取代。令人惊讶的事实是，在这种情况下，零点位于特定的直线上，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个问题列表中的一个问题，也是克莱研究所的千年问题。然而，相比之下，在许多开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。这些单独的零通常被称为“共振”，其作用类似于紧化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态不变量。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，这些零的模式出现了更清晰的图像。有些令人惊讶的是，这些零的图案有着惊人的美丽。它们似乎躺在非常精致的曲线上，形状让人想起蕾丝刺绣。当底层表面具有更多的对称性时，这些零的图具有最简单的结构。这项工作将有助于理解这些函数零点的模式以及它在函数和相关曲面上给出的信息。

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.