Lie algebroids and moduli spaces

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Moduli spaces of flat bundles over surfaces have been the subject of intensive investigation in both physics and mathematics. On the physics side these spaces appear in conformal field theory, gauge theory and string theory, while mathematically they arise in a variety of contexts in symplectic geometry, algebraic geometry, knot theory, and representation theory. Beginning with the work of Atiyah-Bott and Witten in the 1980s, the moduli spaces have been studied using techniques from symplectic geometry and the theory of momentum maps. (Symplectic manifolds arise as classical phase spaces in physics, while momentum maps play the role of angular momentum.) Closely related to the moduli spaces of flat bundles are the moduli spaces of complex structures, up to isotopy (Teichmueller spaces) or up to diffeomorphism (moduli spaces of curves). The moment map techniques enters through the Weil-Petersson symplectic structure on Teichmueller spaces, and were used in classical work of Wolpert, Goldman, and Mirzakhani, as well as in recent work by physicists on Virasoro coadjoint orbits. Over the past 15 years, Dirac geometry has emerged as a fundamental tool in the study of moduli spaces. Originally designed by Weinstein and Courant as a framework for Dirac's theory of mechanical systems with constraints, it was found to provide the appropriate setting for the finite-dimensional approach to moduli spaces via non-linear momentum maps, the so-called quasi-Hamiltonian spaces. From the physics perspective, this geometry enters the theory of D-branes in string theory as well as aspects of mirror symmetry. The principal objectives of this proposal are as follows: An exploration of Hamiltonian spaces for Virasoro Lie algebras, the application of quasi-Hamiltonian and moment map techniques to Teichmueller spaces, the investigation of Kaehler structures on moduli spaces,  and the theory of Lie algebroids and Lie groupoids with weightings. This research will contribute to the theory of infinite-dimensional Hamiltonian systems in general, and those related to moduli spaces of bundles in particular. The project includes several subtopics that will be suitable as problems for students at the Ph.D. and M.Sc. levels.