The grant addresses several research projects in geometric analysis. The first part continues work on the Yang-Mills flow on higher dimensional Kaehler manifolds. A special focus is a comparison of the analytic singularities that occur along the flow with the algebraic singularities associated to Harder-Narasimhan filtrations of holomorphic vector bundles. The second project analyzes the topology of moduli spaces of coherent systems on Riemann surfaces. Coherent systems are related to a variety of geometric objects such as higher rank Brill-Noether loci and holomorphic maps to Grassmannians and other homogeneous varieties. New methods of Morse theory in the setting of singular spaces provide a framework for these computations. An ongoing project studies energy minimizing maps to the Weil-Petersson completion of Teichmueller space. Such maps are associated to homomorphisms of fundamental groups of Riemannian manifolds to the mapping class group of a compact oriented surface. Regularity of harmonic maps is the main issue, as this is related to rigidity questions. The fourth project consists of further topics related to representation varieties of surface groups. The PI will continue to investigate the functional on Teichmueller space defined by the energy of equivariant harmonic maps associated to surface group representations. A particular aim will be to develop new criteria for the properness and uniqueness of minima of this functional. Results will have implications for the dynamics of the mapping class group action on the moduli space of representations. Representations into the isometry group of the complex ball are related to spherical CR-structures, and there are many open questions as to how these relate to hyperbolic structures. The project also proposes to establish new existence and rigidity results.
A significant branch of mathematical inquiry has been the relationship between the geometric, analytic, and algebraic properties of manifolds. Manifolds are higher dimensional generalizations of curves and surfaces, and they appear in a variety of situations in pure and applied mathematics. Symmetries are also a natural and fundamental part of physical systems, and the dynamics of these symmetries carries important information. The research projects in this proposal will further our understanding of some of these objects. The equations studied -- energy minimizing maps and the Yang-Mills flow -- have their origins in the mathematical description of the physical world and are therefore are of great importance to both mathematicians and physicists.
The projects considered under this grant have furthered our understanding of the relationship between the geometric, analytic, and algebraic properties of stability and moduli spaces. Stability of algebraic geometric spaces has recently become a notion of profound importance in the study of special metric structures. Moduli spaces of Higgs bundles have been used to study the space of representations of surface groups into complex Lie groups and their noncompact real forms. They are related to deep conjectures stemming from number theory in the guise of the Geometric Langlands program. Higgs bundles also appear in theoretical physics through the work of Seiberg-Witten. The PI's work on determinants of Laplace operators is relevant to the computations in perturbitative string theory. Geometric analysis continues to be a popular field among graduate students, and the Mathematics Department at the University of Maryland has a large group of students working in Geometry/Topology. The PI currently supervises one doctoral candidate working on problems related to the current proposal, and he plans to take on more students soon. The PI continues to lecture on these subjects to students at all levels at his host institution and in seminars at other universities. He is also active in organizing workshops and conferences, as well as weekly seminars. Results obtained from this research are disseminated via preprint servers, submitted for publication in scholarly journals, and presented at international conferences.
