权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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

基本信息

批准号：
1702372
负责人：
Yuri Berest
金额：
$ 19.6万
依托单位：
Cornell 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 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1702372&HistoricalAwards=false
关键词：
Collaborative Research Representation Varieties Homology

项目摘要

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.

量子模型在物理学和其他自然科学中发挥着越来越重要的作用。因此，通过矩阵对量子对象的近似进行参数化的几何空间是研究多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的多个领域中发挥着至关重要的作用。不幸的是，这样的空间通常很难研究，因为它们通常不够光滑。直观地说，这意味着它们有太多的棱角。该项目的核心是一种工具，该工具可以克服许多困难，从而完善这些空间。这项工作预计将为此类空间发挥作用的几个问题带来新的见解。除了进一步开发该工具本身之外，它还通过专注于该工具在数学不同部分的应用来统一多个研究领域。在早期的工作中，研究人员通过将表示函子扩展到微分分级（DG）代数并在非交换同调代数的意义上推导它，构建了关联代数表示簇的派生版本。这为代数提供了一种新的同调理论，称为表示同调。该项目旨在根据经典（阿贝尔）同调代数给出关联代数表示同调的新构造，并将其扩展到其他拓扑性质的结构。这应该会带来几何和拓扑方面的各种应用，并为高效计算开辟道路。将研究关于经典空间的表示同源性结构的许多精确猜想。此外，研究人员还将利用新的拓扑方法来解决表示论中一些众所周知的难题（例如强麦克唐纳猜想）。