Representations, Cohomology, and Geometry in Tensor Triangulated Categories

表征理论是在100多年前Frobenius和Schur的开创性工作中出现的。表示是一个代数结构（即群、李代数、量子群或李超代数）通过矩阵的实现。这样的实现已经成功地应用于对这些复杂的代数对象提供深刻的见解。在上个世纪的过程中，表示理论已经成为现代数学的一个中心领域，与组合学、代数几何、拓扑学、数论以及物理学的应用有联系。最近的发展包括使用李群的表示理论来更好地理解信号处理，以及通过几何复杂性理论将计算机科学中的P对NP问题转化为关于分解一般线性群的表示的问题。PI将通过分类和几何方法研究代数群，李代数和李超代数的表示理论。近年来，这些代数几何方法在理解各种代数范畴的表示之间的关系方面发挥了主导作用。上同调方法将用于回答有关表征的过滤的特征和存在性的问题。三角范畴中的几何构造和上同调将用于分析上述代数对象的表示行为。PI还将决定张量三角化的范畴与其潜在的范畴谱之间的关系。

Representation theory emerged over 100 years ago with the pioneering work of Frobenius and Schur. A representation is a realization of an algebraic structure (i.e., a group, Lie algebra, quantum group, or Lie superalgebra) by matrices. Such realizations have been successfully applied to provide deep insights into these complicated algebraic objects. Over the course of the last century, representation theory has emerged as a central area of modern mathematics with connections to combinatorics, algebraic geometry, topology, number theory, along with applications to physics. More recent developments have included using the representation theory of Lie groups to better understand signal processing and the translation of the P versus NP problem in computer science via geometric complexity theory into questions about decomposing representations for the general linear group. The PI will investigate the representation theory of algebraic groups, Lie algebras, and Lie superalgebras via categorical and geometric methods. In recent years, these algebro-geometric methods have played a predominant role in understanding the relationships between representations in various algebraic categories. Cohomological methods will be used to answer questions about the characters and existence of filtrations for representations. Geometric constructions and cohomology in triangulated categories will be employed to analyze the behavior of the representations for the aforementioned algebraic objects. The PI will also determine the relationship between a tensor triangulated category and its underlying categorical spectrum.