Geometric Representation Theory: Interface of Algebra, Geometry and Topology.

几何表示理论：代数、几何和拓扑的接口。

• 批准号: 2596547
• 金额: --
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2596547
• 关键词: Geometric; Representation; Theory; Interface; Algebra

The general aim is to study geometry and topology of compact spaces related to simple algebraic groups. The project will revisit some of the classical papers by Atiyah, Borel, Bott, Hirzebruch, etc. trying to use the modern methods to improve our understanding of the spaces. One particular question is to investigate the automorphic Lie algebras related to the sl_2 embeddings into a simple Lie algebra. The project will look into its representation theory and categorification.The second direction is to study spaces of the form H\G/K where G is a locally compact group, K is its compact subgroup, H is a lattice in G. Say G=PU(2,1). What are the geometric properties of H\B^2 where B^2=G/K is the unit ball in C^2. What are commensurability classes of lattices that produce a smooth surface H\B^2 with c2=3 (or 6), c1^2=9 (or 18)? Are they always arithmetic? Can we classify all arithmetic lattices such that the surface H\B^2 has c2=6, c1^2=18?The third set of spaces are K\G/H where all the groups are compact. Can we develop efficient methods of working with topological invariants of these spaces?All these questions have serious potential applications in fundamental science. The benefits to society and economy are not likely in short or medium term.

总体目的是研究与简单代数基团相关的紧凑空间的几何形状和拓扑。该项目将重新审视Atiyah，Borel，Bott，Hirzebruch等的一些古典论文。试图使用现代方法来提高我们对空间的理解。一个特殊的问题是研究与SL_2嵌入有关的自动层谎言代数，以简单的谎言代数。该项目将研究其表示理论和分类。第二个方向是研究h \ g/k的空间，其中g是局部紧凑的群体，k是其紧凑的亚组，h是g = g = g =。 PU（2,1）。 h \ b^2的几何特性是什么，其中b^2 = g/k是c^2中的单位球。哪些可相当性类别的晶格类别产生光滑的表面H \ b^2，c2 = 3（或6），c1^2 = 9（或18）？它们总是算术吗？我们可以对所有算术晶格进行分类，以使表面H \ b^2具有C2 = 6，C1^2 = 18？第三组空间是所有组紧凑的K/H。我们能否开发与这些空间拓扑不变的有效方法？所有这些问题在基本科学中都有严重的潜在应用。在短期或中期，对社会和经济的好处不太可能。