Collaborative Research: Representation Varieties, Representation Homology, and Applications in Algebra, Geometry, and Topology

合作研究：表示簇、表示同调以及在代数、几何和拓扑中的应用

• 批准号: 1702323
• 负责人: Ajay Candadai Ramadoss
• 金额: $ 15万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1702323&HistoricalAwards=false
• 关键词: Collaborative; Research; Representation; Varieties; Homology; Collaborative; Research; Representation; Varieties; Homology

Quantum models are playing an increasingly important role in physics and other natural sciences. Geometric spaces that parametrize approximations of quantum objects by matrices are therefore an important tool in the study of several models of natural phenomena. Not surprisingly, these spaces play a crucial role in several areas of mathematics and mathematical physics. Unfortunately, such spaces are often difficult to study since they are usually not smooth enough. Intuitively speaking, this means that they have too many edges and corners. This project is centered on a tool that refines these spaces in a way that appears to overcome many of these difficulties. The work is anticipated to lead to new insights into several questions where such spaces play a role. It also unifies several research areas by focusing on applications of this tool in different parts of mathematics, in addition to further developing this tool as an end in itself. In earlier work, the investigators constructed a derived version of representation varieties of associative algebras by extending the representation functor to differential graded (DG) algebras and deriving it in the sense of non-abelian homological algebra. This gives a new homology theory for algebras, called representation homology. This project aims to give a new construction of representation homology of associative algebras in terms of classical (abelian) homological algebra and also extend it to other structures of topological nature. This should lead to various applications in geometry and topology and open the way to efficient computations. A number of precise conjectures regarding the structure of representation homology of classical spaces will be investigated. In addition, the investigators will attack some well-known hard problems in representation theory (such as the strong MacDonald conjecture) using new topological methods.

量子模型在物理和其他自然科学中起着越来越重要的作用。因此，通过矩阵参数量子对象近似的几何空间是研究几种自然现象模型的重要工具。毫不奇怪，这些空间在数学和数学物理学的多个领域中起着至关重要的作用。不幸的是，这种空间通常很难学习，因为它们通常不够光滑。从直觉上讲，这意味着它们的边缘和角落太多。 该项目集中在一个工具上，该工具以似乎克服了许多困难的方式来完善这些空间。预计这项工作将导致对此类空间发挥作用的几个问题的新见解。除了进一步开发该工具本身之外，它还专注于该工具在数学的不同部分中的应用，从而统一了几个研究领域。在较早的工作中，研究人员通过将表示函数扩展到差分（DG）代数并以非亚伯同源代数的意义来构建了联想代数的表示形式。这给出了代数的新同源理论，称为代表同源性。该项目的目的是在古典（Abelian）同源代数方面对联想代数的代表同源性进行新的形式结构，并将其扩展到拓扑性质的其他结构。这应该导致几何和拓扑中的各种应用，并为有效的计算开辟道路。将研究许多有关经典空间代表同源性结构的确切猜想。此外，研究人员将使用新的拓扑方法来攻击代表理论（例如强大的麦克唐纳猜想）中的一些众所周知的硬问题。