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

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，通过矩阵参数化量子对象的近似的几何空间是研究自然现象的几种模型的重要工具。毫不奇怪，这些空间在数学和数学物理的几个领域中起着至关重要的作用。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，这意味着他们有太多的棱角。 这个项目是集中在一个工具，细化这些空间的方式，似乎克服了许多这些困难。预计这项工作将导致对这些空间发挥作用的几个问题的新见解。它还统一了几个研究领域，重点是这个工具在数学的不同部分的应用，除了进一步发展这个工具本身作为一个目的。在早期的工作中，研究人员通过将表示函子扩展到微分分次（DG）代数并在非交换同调代数的意义下导出它，构建了结合代数的表示簇的导出版本。这给出了代数的一个新的同调理论，称为表示同调。本项目的目的是在经典（交换）同调代数的基础上给出结合代数的表示同调的一个新的构造，并将其推广到拓扑性质的其他结构。这将导致各种几何和拓扑学的应用，并开辟了高效计算的道路。一些关于经典空间的表示同调结构的精确描述将被研究。此外，研究人员将使用新的拓扑方法来解决表示论中的一些著名难题（如强麦克唐纳猜想）。

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.