Analysis, Spectra, and Number Theory

该补助金将支持会议“分析，光谱和数论”，将于2014年12月15日至19日在普林斯顿大学和高级研究所举行。将有大约20名发言者和预计200名与会者。会议将侧重于数论，重点是它与分析和谱理论的许多关系。傅立叶分析和频谱理论都起源于对振荡系统的理解，并得到了广泛的解释。令人惊讶的是，它们被发现在数论中起着基本的作用。在他1859年的研究素数，黎曼观察到，素数的数量到一个给定的整数可以简单地表示为一个总和的振荡组件。后来观察到，在黎曼分析中出现的“频率”显示出许多不确定性，并且表现得好像它们是一个大型酉矩阵的本征值-即，对鼓的频谱的抽象。 进一步的联系是发现，开始在工作的马斯和塞尔伯格，高度对称的几何（局部对称空间），其频谱似乎控制许多问题的数论。这些主题已经扩大了许多方向，因为，包括领域的解析数论以及自守形式。会议将探讨这些领域的最新发展，重点讨论的主题包括算术量子混沌、L函数族的分析、算术统计以及与遍历理论的联系。近年来，这些领域出现了一系列的活动，包括：Lindenstrauss对算术曲面的量子唯一遍历性的解决方案; Bhargava和他的同事在数域和椭圆曲线的统计方面取得了惊人的突破;通过与随机矩阵理论的联系，对L函数的零和值分布进行了详细的模型和预测;高效扩展图的构造和分析;基于绿色和陶的工作的加法组合学的发展。会议将包括一个问题会议，为该领域的未来发展方向提出建议。会议的网址是https://sites.google.com/site/asnt2014/。

The grant will support a conference "Analysis, Spectra and Number theory", to be held to be held at Princeton University and the Institute for Advanced Study, December 15 - 19, 2014. There will be approximately 20 speakers and an anticipated 200 participants. The conference will focus on number theory, with emphasis on its many relationships with analysis and spectral theory. Both (Fourier) analysis and spectral theory have their origins in understanding oscillating systems, interpreted broadly. It is very surprising, then, that they have been found to play a basic role in number theory. In his 1859 study of primes, Riemann observed that the number of primes up to a given integer could be expressed simply as a sum of oscillating components. It was later observed that that the "frequencies" occurring in Riemann's analysis show many regularities, and behave as if they were the eigenvalues of a large unitary matrix - i.e., an abstraction of the frequency spectrum of a drum. A further link was the discovery, beginning in the work of Maass and Selberg, of highly symmetric geometries (locally symmetric spaces) whose frequency spectrum appears to control many problems in number theory. These themes have expanded in many directions since, encompassing the field of analytic number theory as well as much of automorphic forms. The conference will examine the latest developments in these areas.Topics to be highlighted include arithmetic quantum chaos, analysis of families of L-functions, arithmetic statistics, and connections with ergodic theory. These areas have seen a flurry of activity in recent years, including: the resolution by Lindenstrauss of quantum unique ergodicity for arithmetic surfaces; spectacular breakthroughs by Bhargava and his colleagues concerning the statistics of number fields and elliptic curves; detailed models and predictions for the zero and value distribution of L-functions that were inspired via connections with random matrix theory; construction and the analysis of highly efficient expander graphs; the development of additive combinatorics based on the work of Green and Tao. The conference will include a problem session to suggest future directions for the field. The conference website can be found at https://sites.google.com/site/asnt2014/.