Fernando Rodriquez-Villegas University of Texas DMS-9970109
Professor Rodriquez-Villegas will continue his study of the special values of L-functions. He will explore the recently discovered connection between the Mahler measure of polynomials and special values of L-functions. Mahler measure appears in many fascinating and seemingly unrelated contexts like heights of varieties in Arakelov theory, transcendental number theory, entropy of algebraic dynamical systems, and spectrum of discrete operators. In this project Professor Rodriquez-Villegas will try to use our understanding of Mahler measure of polynomials and this new relation to special values of L-functions to shed light on the conjectures of Bolch and Beilinson.
This is a project in modern Number Theory. The study of the basic properties of the natural numbers has long been a part of mathematics. Our ever-increasing understanding of these basic properties has led to ingenious practical and theoretical applications of the familiar number systems well beyond mathematics. Many of the basic properties of numbers can be summarized into number themselves by counting certain collections of important objects. Unfortunately many times these collections are much too difficult to actually count. Mathematicians have discovered that many of the important counts can be found by encoding information into certain L-functions that can be constructed via calculus. Some special values of these L-functions give the answers they are looking for. By studying these special values, Professor Rodriquez-Villegas hopes to establish estimates of the numbers that will shed light on the important properties they summarize.