Heat kernel methods applied to zeta functions of Ihara, Rankin-Selberg, and Selberg

应用于 Ihara、Rankin-Selberg 和 Selberg zeta 函数的热核方法

• 批准号: 1104115
• 负责人: Jay Jorgenson
• 金额: $ 16.24万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104115&HistoricalAwards=false
• 关键词: Heat; kernel; methods; applied; zeta

The Principal Investigator proposes to use techniques from analysis, in particular from the study of heat and wave kernels and distribution theory, to study zeta functions of number theory. In the field of regular graphs, both finite and infinite, the project will systematically develop a heat kernel approach to number theory on graphs, including the Ihara zeta function. In previous work, the PI has developed a succinct relation between Fourier coefficients of weight zero Maass forms and weight two holomorphic forms, both associated to any finite volume hyperbolic Riemann surface. The proposal involves a continuation of this work, striving toward bounds of the Fourier coefficients.Both the heat kernel and the wave kernel are mathematical objects which admit interpretations from many mathematical fields. The heat kernel, in particular, can be viewed as a solution of a partial differential equation, as a one-parameter family of positive integrable functions, and as a function associated to probability theory and random walks. As a result, the heat kernel provides a means by which one can employ ideas and results from one mathematical discipline in order to approach problems in another field. The PI plans to continue this approach to study problems in number theory, analysis, and graph theory. In addition to research interests, the PI has undertaken a number of educational endeavors striving to enhance opportunities for students. As a faculty member at The City College of New York, the PI has developed courses in statistics and in financial mathematics for the Mathematics Department, and he works closely with the School of Education in their course offerings in teacher training programs. In October 2010, the PI, together with J. Kramer, taught a graduate level course at the University of Sarajevo. The proposed research includes a component of effort by the PI to enhance and further develop these teaching aspects of his mathematical interests.

首席研究员建议使用分析技术，特别是从热波核和分布理论的研究中，来研究数论的zeta函数。在正则图领域，无论是有限的还是无限的，该项目将系统地发展热核方法的数论图，包括Ihara zeta函数。在之前的工作中，PI已经开发出了重量零质量形式和重量二全纯形式的傅里叶系数之间的简洁关系，两者都与任何有限体积双曲黎曼曲面相关。这个建议包括对这项工作的延续，努力求出傅里叶系数的边界。热核和波核都是数学对象，可以从许多数学领域进行解释。特别是热核，可以被看作是一个偏微分方程的解，一个单参数的正可积函数族，以及一个与概率论和随机游走相关的函数。因此，热核提供了一种方法，人们可以利用一个数学学科的思想和结果来解决另一个领域的问题。PI计划继续使用这种方法来研究数论、分析和图论方面的问题。除了研究兴趣，PI还承担了许多教育方面的努力，努力增加学生的机会。作为纽约城市学院的一名教员，PI为数学系开发了统计学和金融数学课程，并与教育学院在教师培训项目的课程设置中密切合作。2010年10月，PI和J. Kramer一起在萨拉热窝大学教授研究生课程。提议的研究包括PI努力的一个组成部分，以加强和进一步发展他的数学兴趣的这些教学方面。