Sheaves, Representations, and Dualities

数学或理论物理学中的二重性是两个理论之间的对应关系，它允许一个理论将一个理论中的关键现象与另一个理论中的关键现象联系起来，尽管它们在某种意义上是不同的。一个粗略的类比是视觉图像和它在镜子中的反射之间的关系。这个项目是在表示论，即研究代数结构的对称性。基本的二元性在这一主题中发挥着越来越重要的作用。其中一个对偶是几何朗兰兹对偶，它是数论中互反律的产物。另一个是镜像对称，这是一种起源于物理学的代数几何现象。目前的项目将进一步推导出这些基本思想的后果表示理论。另一个目标是研究表征理论中出现的不同对偶的共同特征，以便找到一种统一的方法，从而更好地理解所观察到的现象的性质。首席研究员将涉及研究生和本科生的项目，从而促进各级数学教育和研究。该项目有几个目标。一是推进模表示理论中迅速发展的拓扑方法。二是进一步发展了p-adic群和有限Lie型群上调和分析的标准层方法。在这里，我们计划在p-adic群的调和分析中揭示内窥镜理论的几何现象。最后，PI将继续致力于量子化辛解析理论的研究，这是表征理论中令人兴奋的最新篇章，与物理学和代数几何有着惊人的联系。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

A duality in mathematics or theoretical physics is a correspondence between two theories which allows one to relate key phenomena in one theory to those in the other, despite their somewhat different, in a sense opposite, nature. A rough analogy is to the relation between a visual image and its reflection in a mirror. This project is in representation theory, that is, the study of algebraic structure of symmetries. Fundamental dualities have been playing an increasingly central role in the subject. One such duality is geometric Langlands duality, an outgrowth of reciprocity laws in number theory. Another is mirror symmetry, a phenomenon in algebraic geometry with origins in physics. The present project will derive further consequences of these fundamental ideas to representation theory. An additional goal is to study the common features of different dualities that appear in representation theory in order to find a unified approach resulting in a better understanding of the nature of the observed phenomena. The Principal Investigator will involve graduate and undergraduate students in projects thus promoting mathematical education and research at various levels. The project has several aims. One is to advance rapidly developing topological methods in the theory of modular representations. Another is to further develop etale sheaf methods in harmonic analysis on p-adic groups and finite groups of Lie type. Here we plan to work on uncovering the geometric phenomena underlying the theory of endoscopy in harmonic analysis on p-adic groups. Finally, the PI will continue to work on the theory of quantized symplectic resolutions, an exciting recent chapter in representation theory with surprising connections to physics and algebraic geometry.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.