NSF Award
0203353
The investigator proposes to study the renormalization of period integrals <br/>on GL(n). More specifically, this project concerns the development of a <br/>theory of truncation of GL(n) Eisenstein series, thereby leading to a <br/>definition of renormalized torus integrals. Such results in turn make it <br/>possible to compute the polar divisor of the integrals and therefore to <br/>find unexpected functional equations of renormalized period integrals of <br/>Eisenstein series.<br/><br/>This is a project in Number Theory, one of the oldest branches of <br/>mathematics. The foundations of Number Theory lie in the study of the <br/>positive integers. Some basic objects that have emerged in Number Theory <br/>are automorphic forms, objects that possess surprising symmetries. This <br/>project studies certain integrals of particular automorphic forms called <br/>Eisenstein series. These integrals, if interpreted in the usual way, are <br/>infinite, so part of the proposed research deals with reinterpreting these <br/>integrals, using renormalization to give them a definite meaning. This has <br/>already been carried out by the investigator in a special case, which gave <br/>rise to a relationship with another renormalized integral that was <br/>developed in a different fashion. More striking, however, was the <br/>occurrence of an unexpected symmetry in four dimensions. The aim of this <br/>project is to extend this technique of renormalization to the most general <br/>case, and to look for generalizations of this unexpected phenomenon.
