Energy of knots and conformal geometry

曲线和曲面的共形几何（与国外研究者Remi Langevin合作）：设x和y是三维球面上曲线C上的点。我们可以定义一个ac-t上的复值2-形式，首先通过四个点x，x+dx，y和y+dy用黎曼球面CU {∞}来标识球面，然后通过球极投影来获取对应于Jour点的四个复数的交比。我们称之为无穷小交比。它的真实的实部和虚部可以解释如下：设S（p，3）表示3-球面中定向p维球面的集合。利用Pluecker坐标可以给出它的伪黎曼结构。空间CxCΔ可以被认为是S（0，3）中的曲面。无穷小交比的真实的部分等于该曲面的一个带符号的面积元.空间S（0，3）也允许一个辛结构作为3-球面的余切丛.无穷小交比的真实的部分也等于S（0，3）的标准辛形式。平面连杆机构的拓扑：研究了具有$n$ anus的机器人平面机构的位形空间，每个机构都有一个转动关节和一个固定端点。用莫尔斯理论方法和拓扑方法给出了它的拓扑类型。

Conformal Geometry of Curves and Surfaces (Joint work with Remi Langevin, who is an investigator abroad):Let x and y be points on a curve C in the 3 dimensional sphere. We can define a complex valued 2-form on ac-t by first identifying the sphere through four points x, x+dx, y, and y+dy with the Riemann sphere CU {∞} and then by taking the cross ratio of the four complex numbers corresponding to the Jour points through a stereographic projection. Let us call it the infinitesimal cross ratio. It is, by definition, invariant under Moebius transformations.The real and the imaginary parts of it can be interpreted as follows.Let S(p,3) denote the set of oriented p dimensional spheres in the 3-sphere. We can give a pseudo-Riemannian structure on it by using Pluecker coordinates. The space CxCΔ can be considered a surface in S(0,3). The real part of the infinitesimal cross ratio is equal to a signed area element of this surface.The space S(0,3) also admits a symplectic structure as the cotangent bundle of 3-sphere. The real part of the infinitesimal cross ratio is also equal to the canonical symplectic form of S(0,3).Topology of planar linkages :The configuration space of the planar mechanism of a robot with $n$ anus each of which has a rotational joint and a fixed end point is studied. Its topological type is given by a Morse theoretical way and a topological way.