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.

主要研究者建议使用分析中的技术，特别是从热，波核和分布理论的研究中来研究数字理论的Zeta函数。 在常规图（有限和无限）的常规图领域中，该项目将系统地开发出图形上的数字理论（包括Ihara Zeta函数）的热核方法。 在以前的工作中，PI在重量零MAASS形式的傅立叶系数与重量两种圆锥形形式之间建立了简洁的关系，这两种形式都与任何有限体积的双曲线riemann表面相关。 该提案涉及这项工作的延续，努力朝着傅立叶系数的界限努力。热核和波核都是数学对象，可以接受许多数学领域的解释。 特别是，可以将热内核视为偏微分方程的解决方案，是一个阳性整合函数的单参数家族，是与概率理论和随机步行相关的函数。 结果，热核提供了一种手段，通过这种方法可以采用一种数学学科的思想和结果，以解决另一个领域的问题。 PI计划继续这种方法来研究数量理论，分析和图理论的问题。 除了研究兴趣外，PI还进行了许多教育努力，以增加学生的机会。 作为纽约城市学院的教职员工，PI已为数学系开发了统计和金融数学的课程，他在教师培训课程中与他们的课程提供了课程。 2010年10月，PI与J. Kramer一起在萨拉热窝大学教授研究生级课程。 拟议的研究包括PI努力的一部分，以增强和进一步发展其数学利益的教学方面。