IRFP: Mass equidistribution on compact arithmetic surfaces and applications

国际研究奖学金计划使美国科学家和工程师能够在国外进行9到24个月的研究。该项目的奖项提供了联合研究的机会，并提供了使用国外独特或互补的设施、专业知识和实验条件的机会。该奖项将资助保罗·纳尔逊博士与瑞士洛桑联邦理工学院菲利普·米歇尔博士合作的为期24个月的研究奖学金。该项目涉及算术量子唯一遍历猜想的变体，该猜想推测地描述了源于数论思考的某些混沌系统的量子和经典动力学之间的相互作用；该领域的许多问题源于黎曼假说的适当推广(数学中的一个核心悬而未决的问题)，但更容易理解，因此为新技术的发展提供了肥沃的土壤。该项目由几个密切相关的部分组成，在这些部分中，PI将继续他论文中开始的研究路线，接触到主持人和他的合著者介绍的新方法，并在一些共同感兴趣的项目上与主持人合作。PI将研究紧致算术曲面上的全纯特征形式的质量分布，他的论文工作为他做了很大的准备；PI将更深入地学习主持人和他的合作者最近开创的一些技术，这应该会帮助他实现这一目标。在这个问题上的进展可能需要一种新的方法来研究紧致算术曲面上的模形式，以取代在非紧致算术曲面的尖点处使用傅立叶展开；PI和东道主希望在其他情况下进一步研究这种方法。PI和东道主将深入研究PI论文中关于大水平尖形的质量均匀分布的方法和结果的算术几何应用，以期了解大型导体的强Weil曲线的模参数化的零因子以及模形式之间的同余；这将涉及到主持人是专家的技术，建立在PI最近的工作基础上，并对模形式分析理论的算术和几何结果有重要的理解。PI将在EPFL教授和指导学生，从而扩大他与之互动的学生群体，而不是那些他已经有特权指导的加州理工学院的学生：作为助教，作为他从头开始设计的课程的讲师，以及作为监督独立研究项目的官方导师。他将通过制作和提供他和其他人教授的课程的书面记录来加强研究的一般基础设施，就像他在加州理工大学过去几年所做的那样。该项目将导致国际和平研究所与海外顶尖研究人员之间建立持久的伙伴关系，从而加强沟通渠道。所取得的成果将通过在线预印服务器和在数学研究期刊上发表而广泛传播。

The International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad. This award will support a twenty-four month research fellowship by Dr. Paul Nelson to work with Dr. Philippe Michel at Ecole Polytechnique Federale de Lausanne in Lausanne, Switzerland.The project concerns variants of the arithmetic quantum unique ergodicity conjecture, which conjecturally describes the interplay between the quantum and classical dynamics of certain chaotic systems that arise from number-theoretic considera¬tions; many of the problems in this field follow from suitable generalizations of the Riemann hypothesis (a central unresolved problem in mathematics) but are considerably more acces¬sible, and so provide fertile grounds for the development of new techniques. The project consists of several closely-related components in which the PI will continue the line of re¬search initiated in his thesis, become exposed to new methods introduced by the host and his coauthors, and collaborate with the host on a number of projects of common interest.The PI will study the mass distribution of holomorphic eigenforms on compact arithmetic surfaces, for which his thesis work has substantially prepared him; the PI will learn in greater depth some of the techniques recently pioneered by the host and his collaborators, which should help him in this pursuit. Progress on this problem will likely require a new method for studying modular forms on compact arithmetic surfaces that substitutes for the use of Fourier expansions at the cusps of non-compact arithmetic surfaces; the PI and host would want to investigate further such a method in other contexts.The PI and the host will study in depth the arithmetic geometric applications of the methods and results developed in the thesis of the PI concerning the mass equidistribution of cusp forms of large level, with a view towards understanding the zero divisors of the modular parameterizations of strong Weil curves of large conductor as well as congruences between modular forms; this will involve techniques in which the host is an expert, build upon recent work of the PI, and contribute significant understanding of the arithmetic and geometric consequences of the analytic theory of modular forms.The PI will teach and mentor students at EPFL, thereby broadening the group of students with which he has interacted beyond those at Caltech who he has already had the privilege to instruct in several capacities: as a teaching assistant, as an instructor for courses that he has designed from the ground up, and as an official mentor on supervised independent research projects. He will enhance the general infrastructure for research by producing and making available written accounts of courses taught by him and others, as he has done for the past several years at Caltech. The project will result in lasting partnerships between the PI and leading researchers overseas, thereby strengthening the channels of communication. The results obtained will be disseminated broadly through online preprint servers and publication in mathematical research journals.