Special Values of L-Functions

德克萨斯大学DMS-9970109教授Rodriquez-Villegas将继续他对L-函数特殊值的研究。他将探讨最近发现的马勒措施之间的连接多项式和特殊价值的L-函数。 马勒测度出现在许多迷人的和看似无关的背景下，如高度的品种在阿拉克洛夫理论，超越数论，熵的代数动力系统，和频谱的离散算子。在这个项目中，Rodriquez-Villegas教授将尝试使用我们对多项式的Mahler测度的理解以及与L-函数的特殊值的这种新关系来阐明Bolch和Beilinson的理论。这是现代数论中的一个项目。 自然数的基本性质的研究一直是数学的一部分。 我们对这些基本性质的理解不断加深，导致了熟悉的数字系统的巧妙的实际和理论应用，远远超出了数学。数的许多基本性质都可以通过对某些重要对象的集合进行计数来概括为数本身。 不幸的是，很多时候，这些收集太难了，实际上很难计算。 数学家们发现，许多重要的计数可以通过将信息编码到某些可以通过微积分构造的L函数中来找到。 这些L-函数的一些特殊值给出了他们正在寻找的答案。 通过研究这些特殊值，Rodriquez-Villegas教授希望建立对这些数字的估计，从而揭示它们总结的重要性质。

AbstractFernando Rodriquez-VillegasUniversity of TexasDMS-9970109Professor 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.