This project concerns moduli spaces of representations of finitely generated groups into Lie groups. Many moduli spaces of interest arise in this fashion: flat fiber bundles over a surface, and (parabolic) Higgs bundles are examples. These spaces are sometimes called character varieties. The study of character varieties intersects the invariant theory of products of matrix groups, and so owes much of its structure to the non-commutative ring of generic matrices (matrices over polynomial indeterminates). Since character varieties may be considered spaces of equivalence classes of representations of the fundamental group of a base space (topology) into a Lie group (geometry), they find applications throughout differential geometry and mathematical physics. The central goal of the proposed work is to use the strong relationship to non-commutative ring theory to answer questions about the homotopic, arithmetic, and dynamical structure of character varieties.
Mathematicians' endeavor to classify mathematical objects is not unlike, for example, the biologists' endeavor to classify species. The purpose of this project is to classify and advance the understanding of certain mathematical objects called "character varieties." Informally, character varieties provide data which encodes equivalent ways one can associate the flexible shape of a object to a rigid shape of that same object. For example, a rubber band is a flexible "circle," but once one stipulates a radius a rigid circle is determined. The study of the geometry of character varieties in some sense can be thought of as the "geometry of geometry." In part because of this, character varieties find applications throughout differential geometry and in mathematical physics. In this project, the PI and his collaborators and students will use certain non-symmetrical structures associated with character varieties as a tool to explore and answer novel questions about their shape, and how their elements relate to each other by certain motions. Another important part of this project is the training of undergraduate research assistants in the PI's Experimental Algebra and Geometry Lab, and the conducting of community outreach activities which communicate the importance and beauty of pure mathematics.
