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

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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.