RUI: Renormalized Period Integrals

研究了GL(n)上周期积分的重整化问题。更具体地说，本项目涉及GL(n)爱森斯坦级数截断理论的发展，从而导致重整化环面积分的定义。这样的结果反过来又使计算积分的极因子成为可能，从而找到爱森斯坦级数重整化周期积分的意想不到的泛函方程。这是数论的一个项目，数论是数学最古老的分支之一。数论的基础在于对正整数的研究。数论中出现的一些基本对象是自同构形式，具有惊人对称性的对象。本课题研究的是被称为爱森斯坦级数的特定自同构形式的某些积分。这些积分，如果用通常的方式来解释，是无限的，所以我们提出的研究的一部分是重新解释这些积分，用重整化来给它们一个明确的意义。研究者已经在一个特殊情况下进行了这一工作，它产生了与另一个以不同方式发展的重整化积分的关系。然而，更引人注目的是，在四维空间中出现了一种意想不到的对称性。这个项目的目的是将这种重整化技术扩展到最一般的情况，并寻找这种意想不到的现象的概括。

The investigator proposes to study the renormalization of period integrals on GL(n). More specifically, this project concerns the development of a theory of truncation of GL(n) Eisenstein series, thereby leading to a definition of renormalized torus integrals. Such results in turn make it possible to compute the polar divisor of the integrals and therefore to find unexpected functional equations of renormalized period integrals of Eisenstein series.This is a project in Number Theory, one of the oldest branches of mathematics. The foundations of Number Theory lie in the study of the positive integers. Some basic objects that have emerged in Number Theory are automorphic forms, objects that possess surprising symmetries. This project studies certain integrals of particular automorphic forms called Eisenstein series. These integrals, if interpreted in the usual way, are infinite, so part of the proposed research deals with reinterpreting these integrals, using renormalization to give them a definite meaning. This has already been carried out by the investigator in a special case, which gave rise to a relationship with another renormalized integral that was developed in a different fashion. More striking, however, was the occurrence of an unexpected symmetry in four dimensions. The aim of this project is to extend this technique of renormalization to the most general case, and to look for generalizations of this unexpected phenomenon.