Lie algebroids and moduli spaces

平面上平面束的模空间一直是物理学和数学中深入研究的课题。在物理方面，这些空间出现在共形场理论、规范理论和弦理论中，而在数学上，它们出现在辛几何、代数几何、结理论和表示理论的各种背景中。从20世纪80年代Atiyah-Bott和Witten的工作开始，模空间已经使用辛几何和动量映射理论的技术进行了研究。（辛流形作为物理学中的经典相空间出现，而动量映射则扮演角动量的角色。）与平面束的模空间密切相关的是复杂结构的模空间，直至同位素（Teichmueller空间）或微分同构（曲线的模空间）。矩映射技术通过Teichmueller空间上的Weil-Petersson共轭结构进入，并被用于Wolpert、Goldman和Mirzakhani的经典作品中，以及物理学家最近关于Virasoro协轨的研究中。在过去的15年中，狄拉克几何已经成为模空间研究的基本工具。它最初是由Weinstein和Courant设计的，作为Dirac具有约束的机械系统理论的框架，人们发现它为通过非线性动量映射（所谓的准哈密顿空间）接近模空间的有限维方法提供了适当的设置。从物理学的角度来看，这种几何结构进入了弦理论中的d膜理论以及镜像对称的各个方面。本文的主要目标是：探索Virasoro Lie代数的哈密顿空间，拟哈密顿和动量映射技术在Teichmueller空间中的应用，模空间上Kaehler结构的研究，以及带加权的Lie代数和Lie群的理论。本研究将有助于一般无限维哈密顿系统的理论，特别是与束的模空间有关的理论。该项目包括几个子主题，将适合作为问题的学生在博士和硕士水平。

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.