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。 他的第一个建议线的调查是申请亚瑟的内窥镜分类自守表示的经典团体的问题估计的多重性与nontempered阿基米德表示发生在一个给定的水平。 PI的目标是建立Sarnak和Xue关于这些重数的猜想的一些情况，并给出同余塔中算术流形的Betti数增长率的尖锐渐近性。PI的第二个建议的研究是紧致局部对称空间上的高秩本征函数的集中，特别是这样一个本征函数的L^p范数或它对各种子空间的限制可以有多大的问题。 第一步是通过结合半经典分析和表示论的技巧，推导出这些L^p范数的正确“凸界”，这在非负曲率的情况下和一般情况下的谱簇中是尖锐的。 第二个组成部分是改善这些凸边界（特别是，以产生一个节省的指数）在尽可能多的情况下，通过引入算术放大的工作Iwaniec和Sarnak.算术流形的中心对象在数论，因为他们编码的信息有关的对象，首先是由希腊人研究，如素数和解决方案的多项式方程的整数或有理数。 它们还允许人们使用数论的强大工具来研究几何，分析和数学物理问题的特殊情况。 这方面的一个例子是鲁德尼克和萨纳克的量子唯一遍历性猜想的最新进展，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“非回火”自守形式在计数和动力学的许多问题中发挥着重要作用，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.