Special Values of L-Functions

费尔南多·罗德里克斯-维尔莱加斯德克萨斯大学DMS-9970109教授将继续他对L函数特殊值的研究。他将探索最近发现的多项式的马勒测度与L函数的特殊值之间的联系。马勒测度出现在许多引人入胜和看似无关的背景下，如Arakelov理论中的簇高度、超越数论、代数动力系统的熵和离散算符的谱。在这个项目中，Rodriquez-Villegas教授将试图利用我们对多项式的马勒测度的理解以及与L函数的特定值的这种新的关系来阐明Bolch和Beilinson的猜想。这是现代数论中的一个项目。自然数的基本性质的研究长期以来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，导致了我们熟悉的数制在数学之外的巧妙的实际和理论应用。通过计算某些重要对象的集合，可以将数字的许多基本属性总结为数字本身。不幸的是，很多时候，这些收藏太难了，以至于无法真正统计。数学家已经发现，许多重要的计数可以通过将信息编码到特定的L函数中来找到，这些函数可以通过微积分构造。这些L函数的一些特殊值给出了他们正在寻找的答案。通过研究这些特殊的数值，Rodriquez-Villegas教授希望建立对这些数字的估计，从而阐明它们总结出的重要性质。

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.