Symplectic topology, generalized geometry and their applications

我的研究项目涉及数学物理的广泛领域。特别是，它涉及具有物理意义的几何结构-要么源于物理考虑，要么在物理学中具有潜在的应用。对物理系统的描述，无论是经典系统还是量子系统，通常都需要具有某些特殊性质的几何形状。在数学上，某些属性也可以通过这些形状上的额外结构或其对称性，或不同类型形状之间的关系来捕获。非常有趣的是，当两个不同的数学描述从物理考虑产生相同的预测时，在这种情况下，这两个数学描述被认为展示了二元性。对偶性是非常有用和重要的，因为它们经常涉及到非常不同的数学领域，并将复杂的问题从一个问题变成另一个问题的简单得多的问题。镜像对称性是比较著名的例子之一，它将一边的辛几何与另一边的复杂几何联系起来。因此，这些二元性的数学证明是非常重要的。广义几何，如希钦几何和伯恩几何，被提出部分是为了帮助从数学上理解某些对偶。涉及稳定和扭曲向量丛的构造，如它们的派生范畴和模空间，以及量子拓扑，都是从Riemann曲面到相关目标(如辛流形)的映射的考虑而产生的，为这种理解提供了几何框架。这些通常是复杂的结构，在李群和齐次空间等显性例子上解决它们有助于揭示可能的一般结构。更具体地说，我的研究项目涉及广义几何、李理论、稳定丛、它们的模空间、量子拓扑和具体的动力系统。

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.