Geometry and Analysis of Moduli Spaces of Holomorphic Bundles

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The principal objective of this project is the study of the behavior of solutions to certain nonlinear differential equations arising in mathematical physics in various geometrical settings. There is a rich interplay between the analytic and algebraic aspects of this problem which researchers have found useful for both theoretical and computational purposes. The current project will further the understanding of the long time evolution of geometric flows of this type, and it will provide a precise determination of the singular structure of solutions to these equations. A second important aspect of this research is the global nature of the entire space of solutions (called the moduli space). Understanding the structure of moduli spaces reveals underlying phenomena that unify and explain particular aspects of individual examples. The Yang-Mills equations and moduli spaces of Higgs bundles are the major examples of interest to the PI. They also form an important point of intersection between mathematics and theoretical physics. Progress on the research in this project will have an impact outside of pure mathematics. The project outlines research in three principal areas of complex geometry related to holomorphic bundles and moduli problems. The first continues work of the PI on the Yang-Mills flow on Kaehler manifolds, compactifications of spaces of Hermitian-Einstein metrics, and the structure of singular sets. The second studies the asymptotic and global geometry of moduli spaces of Higgs bundles. This is currently an area of intense activity motivated by constructions from mathematical physics and invariants in low dimensional topology. The third project seeks to extend and apply the PI's work on determinants of Dolbeault laplacians. This will provide a unified framework for Quillen metrics over moduli spaces of bundles with additional parabolic structures. The PI has experience working in this field and has published research articles on topics directly related to the subject matter of the current proposal.