Representation Theory and Geometry in Monoidal Categories

首席研究员（PI）将研究各种代数对象的表示理论。抽象代数对象的表示是通过数字矩阵实现该对象。通常，将一个代数对象的表示的整个集合看作一个被称为张量范畴的结构是有利的。张量类别由具有加法和乘法运算的对象组成，如整数或方阵。使用乘法运算，可以引入张量范畴的谱，它是一个几何对象（如圆锥、球体或环面）。PI将利用张量范畴的代数和几何性质之间的重要联系，在表示理论方面取得进展。PI将继续让本科生和研究生参与这些项目。他将继续作为数学界的活跃成员，在美国数学学会（AMS）的国家委员会任职，并担任一家主要数学期刊的编辑。PI将开发新的方法来研究一元三角形几何。几个中心问题将利用在一般单轴情况下同调素数的构造和MTCs的表示理论的介绍。这种表示理论有望提供关于MTC的巴尔默谱的新信息。特别地，一般的MTC理论将被应用于李超代数的表示。PI还将探索研究经典单李超代数表示的新思路。这涉及到系统地研究O类的各种版本以及相关的拟约超群的有理表示。其中一个主要思想是利用探测和BBW抛物子群/子代数。此外，PI将研究幂零锥的轨道结构，并构造轨道闭包的奇点分辨率。PI将研究涉及约化代数群表示的重要问题。关键问题将集中在理解诱导表征的结构，以及这些模块是否允许p过滤。这些问题与30多年来通过还原代数群的倾斜模来实现Frobenius核的射影模的问题以及第一Frobenius核的简单模间的扩展结构有关。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。

The Principal Investigator (PI) will investigate the representation theory of various algebraic objects. A representation of an abstract algebraic object is a realization of the object via matrices of numbers. Often times, it is advantageous to view the entire collection of representations of an algebraic object as a structure known as a tensor category. Tensor categories consist of objects with additive and multiplicative operations like the integers or square matrices. Using the multiplicative operation, one can introduce the spectrum of the tensor category which is a geometric object (like a cone, sphere or torus). The PI will utilize the important connections between the algebraic and geometric properties of tensor categories to make advances in representation theory. The PI will continue to involve undergraduate and graduate students in these projects. He will continue to be an active member of the mathematical community by serving on national committees for the American Mathematical Society (AMS), and as an editor of a major mathematical journal.The PI will develop new methods to study monoidal triangular geometry. Several central problems will utilize the construction of homological primes in the general monoidal setting and the introduction of a representation theory for MTCs. This representation theory promises to yield new information about the Balmer spectrum of the MTC. In particular, the general MTC theory will be applied to study representations of Lie superalgebras. The PI will also explore new ideas to study representations of classical simple Lie superalgebras. This involves systematically studying various versions of Category O and the rational representations for the associated quasi-reductive supergroups. One of the main ideas entails the use of the detecting and BBW parabolic subgroups/subalgebras. Furthermore, the PI will study the orbit structure of the nilpotent cone and will construct resolutions of singularities for the orbit closures. The PI will study important questions involving representations of reductive algebraic groups. Key questions will focus on the understanding the structures of induced representations, and whether these modules admit p-filtrations. These questions are interrelated with the 30-year-old problem of realizing projective modules for the Frobenius kernels via tilting modules for the reductive algebraic group, and the structure of extensions between simple modules for the first Frobenius kernel.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.