Automorphic Forms for Function Fields and Related Geometry

PI建议使用几何方法来解决表示论和数论中的问题。最近，PI发现了一种利用自同构形式和几何朗兰兹对应来构造具有特殊Galois群的动机的方法。这个想法将在接下来的几年里得到系统的发展，并将应用于经典的伽罗瓦逆问题。PI还将使用Hitchin型模空间来研究有理Cherednik代数的表示、Galois表示的计数以及Lusztig特征标的全局类似。在数学中，最有趣的对称应该以几何形式出现是一个普遍的规则。例如，柏拉图式的古代固体分类如今被视为在三维空间中寻找所有“有限对称性”的实例。随着20世纪下半叶代数几何的革命，我们现在有了强大的几何工具来解决数论和群论中的经典问题和新问题。PI将使用这些几何工具来解决朗兰兹计划中的问题。在70年代，朗兰兹提出了一系列猜想，将算术不变量(例如，丢番图方程的解)与解析不变量(例如，模形式)联系起来。朗兰兹的预言如果得到证实，将在解决经典问题(包括用这种方法解决的费马问题)方面极其强大。通过这个项目，PI希望阐明朗兰兹计划中的一些问题，并发现更多出现在几何学中的对称性。

The PI proposes to use geometric methods to attack problems in representation theory and number theory. Recently the PI has found a way to construct motives with exceptional Galois groups using automorphic forms and the geometric Langlands correspondence. This idea will be systematically developed in the following years and will have applications to the classical inverse Galois problem. The PI will also use moduli spaces of Hitchin type to study representations of rational Cherednik algebras, enumeration of Galois representations, and global analogs of Lusztig's character sheaves.In mathematics, it is a general rule that the most interesting symmetries should arise geometrically. For example, the ancient classification of Platonic solids is seen nowadays as an instance of finding all 'finite symmetries' in three-dimensional space. With the revolution of algebraic geometry in the second half of the 20th century, we now have powerful geometric tools available to solve classical and new problems in number theory and group theory. The PI will use these geometric tools to attack problems in the Langlands program. In the 70s, Langlands made a series of conjectures which links arithmetic invariants (e.g., solutions to Diophantine equations) to analytic invariants (e.g., modular forms). The predictions that Langlands made, if proved, will be extremely powerful in solving classical problems (including the Fermat Problem which was solved in this way). Through this project, the PI hopes to shed light on some problems in the Langlands program and to discover more symmetries that appear in geometry.