Representation Theory, Geometry, and Cohomology in Tensor Triangulated Categories

像群、李代数和李超代数这样的代数结构具有内部对称性，这些内部对称性已经在生物学、化学和物理学中的许多不同物理应用中使用。表示论是一种通过矩阵实现来编码群和代数信息的方法，在过去的一百年里在数学中发挥了重要作用。表示论的现代趋势涉及使用拓扑学的工具（即，上同调）和代数几何，以通过张量积结构来封装代数对象的表示，从而将表示理论连接到周围的几何结构。该项目将使用这些表示，几何和上同调之间的深层相互作用作为一个强大的设备来证明有关这些代数结构的新结果，并回答有关这些表示的几何和同调不变量（如复杂性和一致性）的重要问题。PI将组织代数会议，重点是初级数学家的发展，他将通过研讨会，讲习班和暑期学校的讲座传播表示论现代方法的工作知识。该项目探讨了约化代数群，李代数和李超代数的表示论中感兴趣的中心问题。本文的研究内容包括发现正特征约化代数群和特征为零的经典李超代数的单模的性质。也将探讨有限Chevalley群和循环代数的不可约表示之间的扩展公式的确定。在这项研究中，PI将利用张量三角几何和三角范畴中的其他几何结构来计算上述代数对象的表示的上同调不变量（例如上同调群，模簇）。

Algebraic structures like groups, Lie algebras, and Lie superalgebras have internal symmetries that have been used in many different physical applications in biology, chemistry, and physics. Representation theory, a method to codify information about groups and algebras through matrix realizations, has played a fundamental role in mathematics over the past one hundred years. Modern trends in representation theory have involved using tools from topology (i.e., cohomology) and algebraic geometry to package the representations of an algebraic object via the tensor product structure to connect the representation theory to ambient geometric structures. This project will use these deep interactions between the representations, the geometry, and the cohomology as a powerful device to prove new results about these algebraic structures and to answer important questions pertaining to geometric and homological invariants (like the complexity and atypicality) for these representations. The PI will organize conferences in algebra with an emphasis toward the development of junior mathematicians, and he will disseminate working knowledge of modern methods in representation theory through lectures at seminars, workshops, and summer schools.This project explores central problems of interest in the representation theory of reductive algebraic groups, Lie algebras, and Lie superalgebras. This study includes discovering new conjectures about the characters of simple modules for reductive algebraic groups in positive characteristic, and classical Lie superalgebras in characteristic zero. The determination of formulas for extensions between irreducible representations for finite Chevalley groups and loop algebras will also be explored. In this investigation, the PI will make use of tensor triangulated geometry and other geometric constructions in triangulated categories to compute cohomological invariants (e.g. cohomology groups, module varieties) of representations for the aforementioned algebraic objects.