Symplectic topology, generalized geometry and their applications

点击翻译按钮获取中文摘要

My research program lies in the broad area of mathematical physics. In particular, it concerns geometric structures that carry physical significance --- either originate from physical considerations, or have potential applications in physics. The descriptions of physical systems, both classical and quantum systems, often require geometrical shapes with certain special properties. Mathematically, certain properties can be captured also with extra structures on, or symmetries of, these shapes, or relations among shapes of different types. It is very interesting when two different mathematical descriptions produce the same predictions from physical considerations, in which case, the two mathematical descriptions are said to exhibit duality. Dualities are extremely useful and important, since they often relate quite distinct areas of mathematics and turn complicated problems from one into much simpler ones in another. Mirror symmetry that relates symplectic geometry on the one side to complex geometry on the other is one of the more famous examples. Mathematical justifications of these dualities are thus of great importance. Generalized geometry a la Hitchin as well as Born geometry are proposed partly to help understand certain dualities mathematically. Constructions involving stable and twisted vector bundles, such as their derived categories and moduli spaces; as well as the quantum topology arise from considerations of maps from Riemann surfaces into relevant targets such as symplectic manifolds provide the geometric frameworks for such understanding. These are often complicated constructions, and working them out on explicit examples such as Lie groups and homogeneous spaces helps in uncovering possible general structures. More specifically, my research program concerns generalized geometry, Lie theory, stable bundles, moduli spaces of them, quantum topology and concrete dynamic systems.