Symplectic topology, generalized geometry and their applications

我的研究项目在数学物理的广阔领域。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，要么在物理学中有潜在的应用。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。从数学上讲，某些属性也可以通过这些形状的额外结构或对称性，或不同类型形状之间的关系来捕获。当两种不同的数学描述从物理考虑产生相同的预测时，这是非常有趣的，在这种情况下，这两种数学描述被称为表现出对偶性。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领域转化为另一个领域的简单问题。镜像对称是一个比较著名的例子，它把一边的辛几何和另一边的复几何联系起来。因此，这些对偶性的数学证明是非常重要的。广义几何（如希钦）和玻恩几何的提出部分是为了帮助从数学上理解某些对偶性。涉及稳定和扭曲向量束的构造，例如它们的派生范畴和模空间；以及考虑从黎曼曲面到相关目标（如辛流形）的映射而产生的量子拓扑，为这种理解提供了几何框架。这些通常是复杂的结构，在诸如李群和齐次空间等明确的例子上进行计算有助于揭示可能的一般结构。更具体地说，我的研究计划涉及广义几何，李理论，稳定束，它们的模空间，量子拓扑和具体的动态系统。

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.