Enumerative Geometry of Character Varieties

This project explores a class of geometric spaces known as character varieties, which capture how the shapeor topologyof a surface is governed by its symmetries. These symmetries are encoded by mathematical objects called Lie groups, which are fundamental to the study of continuous transformations in both geometry and physics. The focus of the project is on character varieties that arise as moduli spaces of vector bundles and parabolic Higgs bundles. These spaces appear in many areas of modern mathematics and theoretical physics, including enumerative geometry, string theory, and non-abelian gauge theory, providing a framework for investigating how complex geometric and physical structures can vary and interact. The central aim of the project is to deepen our understanding of these moduli spaces by computing key mathematical invariants, such as intersection cohomology Betti numbers and the Poincaré pairing in the intersection cohomology of singular moduli spaces, and Euler characteristics of line bundles over moduli spaces of (parabolic) Higgs bundles. These quantities reveal essential geometric and topological features that illuminate the spaces underlying structure. While such invariants have traditionally been studied using abstract, infinite-dimensional techniques, the project adopts a finite-dimensional, computationally accessible framework that enables direct and explicit calculations. This approach not only addresses longstanding challenges in the field and provides new perspectives on the intricate geometry of Higgs bundle moduli spaces, but also may lead to exciting new discoveries across mathematics and mathematical physics.