CAREER: Synergistic activities in automorphic forms and education

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。现代数论的一大部分都与对称性的研究有关。 其中一种方法，对称性出现在数论是通过伽罗瓦理论：这是研究的解决方案集的多项式方程的有理数的对称性。 数论中对称性的另一种产生方式是通过分析：有一类特殊的函数，称为自守形式，它们满足一定的微分方程组，并拥有无限的离散对称群。 数学朗兰兹纲领将伽罗瓦理论与自守形式联系起来，尽管乍看起来这两个数学领域没有任何关系。 这个项目关注自守形式的研究，特别是那些表现出“特殊”对称群的自守形式。 首席研究员将研究自守形式与算术之间微妙而令人惊讶的联系。 他还将研究自守形式的L-函数，这是黎曼zeta函数的推广。 该赠款包括为研究生提供资金，他们将接受自守形式的培训。在这项资助的支持下，首席研究员将编写一本关于“特殊代数结构”的研究生教科书，这将填补文献中的需求。 他还将通过指导本科生和早期职业同事来帮助培训美国劳动力。 在一个项目中，已经在进行中，主要研究者将开发特殊群体的半整数权重模形式理论。 预计这些半整数权模形式的傅立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上模形式的产生，以及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守L函数。 该奖项反映了NSF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.