CAREER: Synergistic activities in automorphic forms and education

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。大量的现代数论与对称性研究有关。对称在数论中出现的一种方式是通过伽罗瓦理论：这是对有理数上多项式方程解集对称性的研究。数论中出现对称性的另一种方式是通过分析：有一类特殊函数，称为自同构形式，满足某一组微分方程，并具有无限组离散对称性。推测的朗兰兹纲领将伽罗瓦理论与自同构形式联系起来，尽管乍一看这两个数学领域彼此没有任何关系。这个项目涉及自同构形式的研究，特别是那些表现出“特殊”对称群的形式。首席研究员将研究自同构形式与算术之间微妙而令人惊讶的联系。他还将研究自同构形式的l函数，它是黎曼ζ函数的推广。该补助金包括为研究生提供的资金，他们将接受自同构形式的培训。在此资助下，首席研究员将撰写一本关于“特殊代数结构”的研究生教材，以填补文献的需求。他还将通过指导本科生和初入职场的同事，帮助培训美国劳动力。该奖项有三个相关的研究项目。在一个正在进行的项目中，首席研究员将开发特殊群的半积分权模形式理论。预计这些半积分权模形式的傅里叶系数将是非常有趣的算术量。第二个项目涉及到具有有理傅立叶系数的例外群G_2的模形式的产生和这种模形式存在的算术结果的发展。这将导致G_2上的模形式的部分数据库。在第三个项目中，首席研究员将构建新的自同构l函数。这些项目涉及的技术包括theta对应和Rankin-Selberg方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). A large swath of modern number theory is concerned with the study of symmetry. One way in which symmetry arises in number theory is through Galois theory: this is the study of the symmetry of solution sets of polynomial equations over the rational numbers. Another way in which symmetry arises in number theory is through analysis: there are a class of special functions, called automorphic forms that satisfy a certain set of differential equations and possess an infinite group of discrete symmetries. The conjectural Langlands Program relates Galois theory to automorphic forms, even though on first appearance the two areas of mathematics have nothing to do with one another. This project concerns the study of automorphic forms, especially those that exhibit "exceptional" groups of symmetries. The Principal Investigator will investigate the subtle and surprising connections that automorphic forms have to arithmetic. He will also study L-functions of automorphic forms, which are generalizations of the Riemann zeta function. The grant includes funding for graduate students, who will receive training in automorphic forms. While supported by this grant, the Principal Investigator will write a graduate textbook on "exceptional algebraic structures", which will fill a need in the literature. He will also help train the US workforce through mentoring of undergraduates and early-career colleagues.This award has three related research projects. In one project, already underway, the Principal Investigator will develop the theory of half-integral weight modular forms on exceptional groups. It is expected that the Fourier coefficients of these half-integral weight modular forms will be highly interesting arithmetic quantities. The second project involves the production of modular forms on the exceptional group G_2 with rational Fourier coefficients and development of arithmetic consequences of the existence of such modular forms. This will lead to a partial database of modular forms on G_2. In a third project, the Principal Investigator will construct new automorphic L-functions. Techniques involved in these projects include the theta correspondence and the Rankin-Selberg method.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.