Cohomology, Geometry and Representation Theory: Algebraic Groups, Quantum Groups and Lie Superalgebras

主要研究者（PI）将研究涉及代数对象的表示及其底层几何结构之间的连接的问题。PI建议研究的基本代数结构是李超代数，代数/有限群，量子群和Frobenius核。代数对象有具体的（离散的）实现，并且经常在派生层次上出现底层丰富的几何结构。上同调方法对于揭示这种几何是有用的。 PI提出使用新的方法，包括巴耳末谱来描述李超代数的同调性质。他还计划计算代数和量子群的支持簇，作为连接几何对象和表示论的一种方式。PI计划使用几何结构来理解有限和代数群的上同调的行为。众所周知，代数结构如群、环、李代数和李超代数在科学中自然地表现出来。对这些物体的基本理解已经在物理和化学的许多不同应用中使用。这些结构往往是复杂的，通常需要代数和几何方法来提取这些代数对象中的重要编码信息。在更广泛的影响方面，PI在全国范围内积极促进研究与教育的结合。他将继续在格鲁吉亚大学（UGA）指导NSF资助的VIGRE（研究和教育的垂直整合）计划。他还是UGA的VIGRE代数组的共同组织者，该组织为博士后研究员和研究生提供当代数学的实践培训。PI将继续组织代数会议，重点是初级数学家的发展，并将在美国和国外的研讨会，研讨会和暑期学校中作为特邀演讲者促进上同调和表示论的工作知识。

The Principal Investigator (PI) will investigate problems involving the connections between representations of algebraic objects and their underlying geometric structures. The basic algebraic structures that the PI proposes to study are Lie superalgebras, algebraic/finite groups, quantum groups, and Frobenius kernels. The algebraic objects have concrete (discrete) realizations, and often times the underlying rich geometric structures arise at the derived level. Cohomological methods are useful for unveiling this geometry. The PI proposes to use new methods involving the Balmer spectrum to describe homological properties of Lie superalgebras. He also plans to make calculations of support varieties for algebraic and quantum groups as a way to connect geometric objects and representation theory. The PI plans to use geometric structures to understand the behavior of the cohomology of finite and algebraic groups. It is well known that algebraic structures such as groups, rings, Lie algebras, and Lie superalgebras manifest themselves naturally in science. The basic understanding of these objects have been used in many different applications in physics and chemistry. These structures are often complicated.Both algebraic and geometric methods are often necessary to extract the important encoded information within these algebraic objects. In terms of broader impacts, the PI has been active nationally in the promotion of integrating research and education. He will continue to direct the NSF funded VIGRE (Vertical Integration of Research and Education) Program at the University of Georgia (UGA). He is also a co-organizer of the VIGRE Algebra Group at UGA which provides practical training in contemporary mathematics to postdoctoral fellows and graduate students. The PI will continue to organize conferences in algebra with an emphasis toward the development of junior mathematicians, and will promote the working knowledge of cohomology and representation theory as an invited speaker at seminars, workshops, and summer schools in the U.S. and abroad.