Families of L-functions and automorphic forms

PI计划研究自同构形式解析理论中的各种问题，特别是有关高阶群的问题，如3次及更高次的一般线性群。特别地，PI将研究一些关于局限于小子集的自同构形式的解析研究的问题，特别是周期积分。该研究的其他组成部分包括对包括黎曼ζ函数在内的l函数的平均值的研究。解析数论的目标之一是理解代数对象的统计性质。例如，你可以问一个随机选择的大数是素数的概率是多少。对于现代密码学来说，很容易找到大素数是非常幸运的。密码学是银行、商业，当然还有国家安全领域的关键工具。我们关于素数的一些最好的知识来自黎曼函数的性质，最简单的l函数。许多其他有趣的代数和几何对象被编码在其他类型的l函数中。PI计划研究这些l函数的许多统计性质。

The PI plans to investigate a variety of problems in theanalytic theory of automorphic forms, especially problemsconcerning higher rank groups such as the general lineargroup of degree 3 and higher. In particular, the PI willstudy some problems concerning the analytic study ofautomorphic forms restricted to small subsets, especiallyperiod integrals. Other components of the research include the study of mean values of L-functions including the Riemann zeta function.One of the goals of analytic number theory is to understandthe statistical properties of algebraic objects. For instance,one can ask what is the probability that a randomly chosenlarge number is a prime. It is very fortunate for moderncryptography that it is easy to find large primes. Cryptographyis a crucial tool in banking, commerce, and of course national security. Some of our best knowledge on the prime numberscomes from properties of the Riemann zeta function, thesimplest L-function. Many other fascinating algebraicand geometric objects are encoded in other types of L-functions.The PI plans to study many of the statistical properties ofthese L-functions.