Higgs Bundles, Surface Groups, and Conformal Limits

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This project is based on recent advancements in geometry which generalize hyperbolic geometry, and lie at the interface of mathematics and physics. There are many distinguished subvarieties of moduli spaces of Higgs bundles and flat connections which this project ties together and analyzes how these spaces change as defining data is varied. The PI uses tools coming from Higgs bundle theory, deformation theory and geometric representation theory to tackle these questions. Since both the objects and the tools used involve the interaction of many different mathematical fields, this works has cross-disciplinary implications. The award also supports out-reach programs aimed at youths in demographics traditionally underrepresented in STEM disciplines.This project centers on the interaction between the geometry of surface group representations, representation theory of Lie groups, and algebraic and analytic techniques in Higgs bundles theory. In these projects, a central role is played by the PI's work on Global Slodowy slices, which globalizes certain Lie theoretic spaces to moduli spaces of Higgs bundles and holomorphic connections. The main goals of the project center on elucidating which aspects of the nonabelian Hodge correspondence change as the Riemann surface is deformed and to identify structures and subvarieties which are constant under such deformations. As a result, this will deepen the understanding of geometries associated to surface groups and their relation to Higgs bundles and representation theory. A special focus is placed on generalizing the Fuchsian locus in the space of Quasi-Fuchsian representations to higher rank and explaining the role of the conformal limit.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.