Representation Theory and its Interactions with Topology and Geometry

首席调查者(PI)建议解决代数(超)群、李(超)代数、量子(超)群、有限群和相关代数的表示理论中的问题。这一建议的目的是将对这些代数对象的研究与低维拓扑学、上同调、代数几何和代数组合学的工具相结合。在PI和合作者已经取得的结果的基础上，PI将使用支撑簇和上同调来研究复李超代数的表示。特别是，PI将调查复杂性、巴尔默光谱和相关问题，以揭示这一鲜为人知的领域。在另一个单独的项目中，PI将使用低维拓扑中出现的修改的迹和维度函数来证明经典李超代数的广义Kac-Wakimoto猜想，并研究代数和量子群的倾斜模。尽管这些工具已被单独证明是该领域新发展的基础，但结合使用它们所做的工作相对较少。这个项目是在被称为表示理论的数学领域。代数结构，如群和李代数，在自然界中作为某些对象的对称性而产生。这些“超级”版本是既涉及对称又涉及反对称的版本。这些所谓的超对称性在物理学和数学中扮演着重要角色。表征理论致力于理解这些结构是如何与其他物体相互作用的。由于这些对象的错综复杂的性质，使用几何和拓扑工具来提取关于这些系统的新信息是有成效的。这就是这项提案所采取的方法。预计这项提案中进行的研究将为这些领域提供新的线索。由于洞察力表示理论提供了关于对象的基本结构和对称性的洞察，它在数学、物理、化学、生物学、密码学、量子计算、计算机图形学和艺术等领域被证明是有价值的。在理解代表性理论方面的进步有望在这些其他领域带来红利。在更广泛的影响方面，国际和平研究所积极促进研究和教育的融合。PI将继续领导一个由本科生和职业生涯早期研究生组成的垂直整合小组，从事与这项提议有关的研究。他还将继续在吸引年轻学生学习数学方面发挥领导作用。这包括共同组织俄克拉荷马大学数学日，领导系级数学俱乐部，以及OU数学俱乐部博客的作者。值得注意的是，该博客每天有来自世界各地的100多名访问者。他还将继续努力通过正式和非正式的合作来指导初级数学家。作为在美国和国外的特邀演讲者和该领域的会议组织者，PI将促进表示理论中的代数、几何和拓扑工具的发展。

The Principal Investigator (PI) proposes to address questions in the representation theory of algebraic (super)groups, Lie (super)algebras, quantum (super)groups, finite groups, and related algebras. The purpose of this proposal is to integrate the study of these algebraic objects with tools from low-dimensional topology, cohomology, algebraic geometry, and algebraic combinatorics. Building on results already obtained by the PI and collaborators, the PI will use support varieties and cohomology to study the representations of complex Lie superalgebras. In particular, the PI will investigate complexity, the Balmer spectrum, and related questions to shed light in this poorly understood area. In a separate project the PI will use modified trace and dimension functions arising in low-dimensional topology to prove the generalized Kac-Wakimoto conjecture for classical Lie superalgebras, and to investigate the tilting modules of algebraic and quantum groups. Although these tools have individually proven to be fundamental to new developments in the field, relatively little work has been done using them in conjunction. This project is in the area of mathematics known as representation theory.Algebraic structures such as groups and Lie algebras arise in nature as the symmetries of some object. The ``super'' versions of these are the ones which involve both symmetries and anti-symmetries. These so-called supersymmetries play a fundamental role in physics and mathematics. Representation theory is devoted to understanding how these structures interact with other objects. Because of the intricate nature of these objects, it is productive to use geometric and topological tools to extract new information about these systems. This is the approach taken in this proposal. It is expected that the research conducted in this proposal will shed new light on these areas. Because of the insight representation theory provides regarding the underlying structure and symmetries of an object, it has proven valuable in other areas of mathematics, physics, chemistry, biology, cryptography, quantum computing, computer graphics, and art. Gains in understanding in representation theory can be expected to pay dividends in these other fields. In terms of broader impacts, the PI has been active in the promotion of integrating research and education. The PI will continue to lead a vertically integrated group of undergraduate and early-career graduate students on research related to this proposal. He will also continue to take a leadership role in engaging younger students in mathematics. This includes co-organizing the University of Oklahoma Math Day, leading the departmental Math Club, and authorship of the OU Math Club Blog. It is noteworthy that the Blog has over 100 visitors per day from around the world. He will also continue his efforts to mentor junior mathematicians through formal and informal collaborations. The PI will promote the development of algebraic, geometric, and topological tools in representation theory as an invited speaker in the U.S. and abroad and as an organizer of conferences in the area.