Sheaves, Representations, and Dualities

数学或理论物理中的二元性是两种理论之间的对应关系，这两种理论允许一种理论中的关键现象与另一种理论中的关键现象相关联，尽管它们在某种意义上是相反的性质，略有不同。一个粗略的类比就是视觉形象和它在镜子中的反射之间的关系。本课题研究的是表示论，即研究对称的代数结构。基本二元论在这个问题上发挥着越来越重要的作用。一种这样的二元性是几何朗兰兹对偶，它是数论中互易定律的产物。另一种是镜像对称，这是起源于物理学的代数几何中的一种现象。本项目将进一步推演这些基本思想对表象理论的影响。另一个目标是研究表征理论中出现的不同二元性的共同特征，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。首席研究员将让研究生和本科生参与项目，从而促进各级的数学教育和研究。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学方法。二是进一步发展了p-ady群和李型有限群的调和分析中的etale Sheet方法。在这里，我们计划揭开内窥镜理论背后的几何现象，在p-进群上进行调和分析。最后，PI将继续致力于量化辛分解理论，这是表示理论中令人兴奋的最近一章，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优势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.