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

首席研究人员建议使用分析中的技术，特别是从热核和波核以及分布理论的研究中，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，该项目将系统地开发一种关于图的数论的热核方法，包括Ihara Zeta函数。在以前的工作中，PI建立了权为零的Maass形式和权为2的全纯形式的傅立叶系数之间的简洁关系，这两种形式都与任意有限体积的双曲Riemann曲面有关。这个提议涉及到这项工作的继续，努力达到傅里叶系数的界限。热核和波核都是数学对象，可以接受许多数学领域的解释。特别地，热核可以看作是偏微分方程解、单参数正可积函数族、以及与概率论和随机游动有关的函数。因此，热核提供了一种方法，通过这种方法，人们可以使用一个数学学科的想法和结果来处理另一个领域的问题。PI计划继续采用这种方法来研究数论、分析和图论中的问题。除了研究兴趣外，国际学生联合会还开展了许多教育工作，努力为学生提供更多的机会。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，并与教育学院密切合作，在教师培训计划中提供课程。2010年10月，国际和平协会与J.Kramer一起在萨拉热窝大学教授研究生课程。这项拟议的研究包括一项由国际数学协会努力加强和进一步发展其数学兴趣的这些教学方面的努力。

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.