Representation Theory and its Interactions with Topology and Geometry

主要研究者（PI）提出解决代数（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。这个建议的目的是将这些代数对象的研究与低维拓扑，上同调，代数几何和代数组合学的工具相结合。在PI和合作者已经获得的结果的基础上，PI将使用支持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，巴耳末光谱和相关问题，以揭示这一知之甚少的领域。 在一个单独的项目中，PI将使用低维拓扑中产生的修改的迹和维数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。虽然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。这个项目是在数学领域被称为表示论。代数结构，如群和李代数出现在自然界中的一些对象的对称性。 这些的"超级“版本是那些既涉及对称性又涉及反对称性的版本。 这些所谓的超对称性在物理学和数学中起着重要作用。 表征理论致力于理解这些结构如何与其他物体相互作用。 由于这些对象的复杂性质，使用几何和拓扑工具来提取关于这些系统的新信息是富有成效的。 这就是本提案所采取的办法。 预计本提案中进行的研究将为这些领域提供新的线索。由于表征理论提供了关于对象的底层结构和对称性的见解，它已被证明在数学，物理，化学，生物学，密码学，量子计算，计算机图形学和艺术等其他领域有价值。在更广泛的影响方面，PI一直积极促进研究与教育的结合。PI将继续领导一个垂直整合的本科生和早期职业研究生小组进行与此提案相关的研究。 他还将继续发挥领导作用，吸引年轻学生在数学。 这包括共同组织的俄克拉荷马州数学日的大学，领导部门的数学俱乐部，并撰写的数学俱乐部博客。 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，指导初级数学家。 PI将促进代表理论中代数，几何和拓扑工具的发展，作为美国和国外的特邀演讲者以及该地区会议的组织者。

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.