The Geometry and Global Analysis of Arithmetic Manifolds

PI将使用解析数论的技术来研究算术流形的全局调和分析和拓扑。他提出的第一条研究路线是将Arthur对经典群的自同构表示的内窥镜分类应用于估计在给定水平上出现的非调和阿基米德表示的多重性的问题。PI的目的是建立关于这些重数的Sarnak和Xue猜想的若干情形，并给出同余塔中算术流形的Betti数的增长率的精确渐近。PI的第二个研究是关于紧致局部对称空间上的高阶特征函数的集中，特别是这样的特征函数的L^p范数能有多大或它对各个子空间的限制。第一步是结合半经典分析和表象理论的技巧，推导出这些L^p范数的正确的‘凸界’，它在非负曲率的情况下是尖锐的，对于谱团簇在一般情况下是尖锐的。第二个组成部分是通过引入算术放大，在尽可能多的情况下改进这些凸界(特别是节省指数)。算术流形是数论中的中心对象，因为它们编码关于希腊人最早研究的对象的信息，如素数和整数或有理数中的多项式的解。它们还允许人们使用数论中的强大工具来研究几何、分析和数学物理中的特殊问题。这方面的一个例子是Rudnick和Sarnak关于量子唯一遍历猜想的最新进展，这有助于我们理解量子力学在高能时开始类似于经典力学的方式。PI提出的关于本征函数集中的研究将加深我们对这一现象的理解。此外，国际和平研究所将研究的“非调和”自同构形式在许多计数和动力学问题中发挥重要作用，该项目可能会在这些领域产生有趣的后果。

The PI will use techniques from analytic number theory to investigate the global harmonic analysis and topology of arithmetic manifolds. His first proposed line of investigation is to apply Arthur's endoscopic classification of automorphic representations of the classical groups to the problem of estimating the multiplicities with which nontempered Archimedean representations occur at a given level. The PI aims to establish a number of cases of a conjecture of Sarnak and Xue on these multiplicities, and give sharp asymptotics for the growth rate of Betti number of arithmetic manifolds in congruence towers.The PI's second proposed investigation is into the concentration of higher rank eigenfunctions on compact locally symmetric spaces, and in particular the question of how large the L^p norms of such an eigenfunction or its restriction to various subspaces can be. The first step of this is to derive the correct 'convex bounds' for these L^p norms, which will be sharp in the case of non-negative curvature and for spectral clusters in the general case, by combining techniques from semiclassical analysis and representation theory. The second component is to improve these convex bounds (in particular, to produce a saving in the exponents) in as many cases as possible by introducing arithmetic amplification as in the work of Iwaniec and Sarnak.Arithmetic manifolds are central objects in number theory, as they encode information about objects which were first studied by the Greeks such as prime numbers and the solutions of polynomial equations in integers or rational numbers. They also allow one to study special cases of questions in geometry, analysis and mathematical physics using powerful tools from number theory. An example of this is the recent progress on the Quantum Unique Ergodicity conjecture of Rudnick and Sarnak, which helps us to understand the way in which quantum mechanics starts to resemble classical mechanics at high energies. The PI's proposed investigation of the concentration of eigenfunctions would improve our understanding of this phenomenon. In addition, the `non-tempered' automorphic forms that the PI will study play an important role in many problems of counting and dynamics, and the project may have interesting consequences in these areas.