Periods and Special Values of L-functions

University Of Texas At Austin, Austin TX

Investigators

Abstract

The PI and his collaborators, will continue their investigation of the relations between periods and special values of L-functions. Specifically, those pertaining to the conjecture of Bloch and Beilinson for elliptic curves and those implied by Borel's theorem for a number field and their connection to the Mahler measure of Laurent polynomials and the geometry of hyperbolic 3-manifolds. The PI's research area is in Number Theory, a very old subject with connections to every other conceivable area of Mathematics as well as Physics. It is remarkable how the type of questions one would like to answer in Number Theory, for example, can we write the number 1 as the sum of the cubes of two rational numbers?, which at the time of Fermat (1600's) say, most likely had no direct implications to everyday life, have evolved into highly relevant issues today. In our current technology age, Number Theory is behind much of the safety of common activities like, for example, shopping on the internet. The PI's specific research concerns L-functions, certain analytic objects one associates to questions like the one above, which are central to modern Number Theory and from which one expects to, and often does, retrieve answers.