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，用矩阵参数化量子物体近似值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。该项目以一种工具为中心，以一种似乎克服了许多困难的方式来完善这些空间。这项工作有望为这些空间发挥作用的几个问题带来新的见解。它还通过关注该工具在数学不同部分的应用来统一几个研究领域，并进一步发展该工具作为其本身的目的。在早期的工作中，研究者通过将表示函子扩展到微分梯度（DG）代数，并在非阿贝尔同调代数的意义上推导它，构建了关联代数的表示变体的派生版本。这给出了代数的一个新的同调理论，称为表示同调。本课题的目的是在经典（阿贝尔）同调代数的基础上给出结合代数的表示同调的新构造，并将其推广到其他拓扑性质的结构中。这将导致几何和拓扑的各种应用，并为高效计算开辟道路。本文将研究一些关于经典空间的表示同调结构的精确猜想。此外，研究人员将使用新的拓扑方法来解决表征理论中一些众所周知的难题（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.