Mathematical Sciences: Conference on Zeta Functions in Number Theory and Geometric Analysis

Zeta函数在许多数学领域扮演着公认的角色。在1992-93年，约翰霍普金斯大学的日美数学研究所将举办一个关于数论和几何分析中的Zeta函数的特别年。该奖项是为了表彰将于1993年3月29日至4月3日举行的一次会议，目的是将这些学科的研究人员聚集在一起。1993年春季，霍普金斯大学与数论Zeta函数相关领域的访问者包括A.Gyoja、T.Kimura、D.Meuser、F.Sato和T.Yano。与几何分析相关的区域的访问者包括T·Adachi、A·Katsuda、Y·Petridis和T·Sunada。这两个领域之间的一个联系是研究与对称空间的商有关的Zeta函数，例如在自同构型的谱理论中。另一类是研究各种L函数的零点的精细结构，它类似于各种拉普拉斯函数的特征值的精细结构。一个更非正式的联系是，这些领域的主要研究人员往往对他们的工作与没有直接关系的领域之间的强烈类比非常感兴趣。下面我们将分别描述这两个领域，但在第二部分(几何分析中的Zeta函数)中，我们将尝试指出数论Zeta函数和几何分析Zeta函数之间的一些重叠。这个项目提出了一个数论和几何分析的Zeta函数会议，重点讨论以下主题：(I)准齐次向量空间的Zeta函数和Igusa Zeta函数；(Ii)拉普拉斯谱理论中出现的Zeta函数。计划召开这次拟议的会议，是为了利用约翰霍普金斯大学在日美数学研究所举办的集中年期间集中这些领域的专家。

Zeta functions play a well-recognized role in many areas of mathematics. In 1992-93, the Japan-US Mathematical Institute at Johns Hopkins University is hosting a special year on zeta functions in number theory and in geometric analysis. This award is for a conference to be held March 29- April 3,1993, to bring together researchers in these subjects. Visitors to Hopkins during Spring 1993 in areas related to number theoretic zeta functions include A. Gyoja, T. Kimura, D. Meuser, F. Sato, and T. Yano. Visitors in areas related to the geometric analysis aspects include T. Adachi, A. Katsuda, Y. Petridis, and T. Sunada. One connection between these two areas is the study of zeta functions associated to quotients of symmetric spaces, e.g. in the spectral theory of automorphic forms. Another is the study of the fine structure of the zeroes of various L-functions, which resembles that of the eigen-values of various Laplacians. A more informal connections is that the principal researchers in these areas often have an intense interest in the strong analogies between their work and that in areas not directly related. We will describe the two areas separately below, but in the second part (zeta functions in geometric analysis) we will try to indicate some of the overlap between number-theoretic and geometric-analytic zeta functions. This project proposes a conference on zeta functions in number theory and geometric analysis, with emphasis on the following subjects: (i) zeta functions of prehomogeneous vector spaces and Igusa zeta functions; (ii) the zeta functions which arise in the spectral theory of the Laplacian. The proposed conference is planned so as to take advantage of the concentration of experts in these areas present at Johns Hopkins in connection with the concentration year held by the Japan-U.S. Mathematical Institute.