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谐波分析和拓扑结构。他提出的第一个研究方向是将Arthur对经典群的自同构表示的内视分类应用于估计非调和阿基米德表示在给定水平上出现的多样性问题。本文旨在建立Sarnak和Xue关于这些多重性的猜想的若干情形，并给出同余塔中算术流形Betti数增长率的尖锐渐近性。PI的第二个建议研究是高阶特征函数在紧致局部对称空间上的集中，特别是这样一个特征函数的L^p范数或它对各种子空间的限制有多大的问题。第一步是通过结合半经典分析和表示理论的技术，为这些L^p范数推导出正确的“凸界”，在非负曲率的情况下，对于一般情况下的谱簇，这将是尖锐的。第二个部分是在尽可能多的情况下，通过引入Iwaniec和Sarnak的工作中的算术放大来改进这些凸界（特别是，产生指数的节省）。算术流形是数论的中心对象，因为它们编码了最早由希腊人研究的对象的信息，例如素数和整数或有理数多项式方程的解。它们还允许人们使用数论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的量子唯一遍历性猜想的最新进展，它帮助我们理解量子力学在高能量下开始与经典力学相似的方式。PI提出的本征函数集中的研究将提高我们对这一现象的理解。此外，PI将研究的“非调质”自同构形式在许多计数和动力学问题中发挥重要作用，并且该项目可能在这些领域产生有趣的结果。

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.