Energy of knots and conformal geometry

曲线与曲面的共形几何（与国外研究者Remi Langevin合作）：设x和y为三维球面上曲线C上的点。我们可以在ac-t上定义一个复值2型，首先用黎曼球CU{∞}通过四个点x， x+dx， y和y+dy来确定球体，然后通过一个立体投影取对应于Jour点的四个复数的交叉比。我们称它为无限小交叉比。根据定义，它在莫比乌斯变换下是不变的。它的实部和虚部可以解释如下。设S（p,3）表示3球中有向p维球的集合。我们可以用Pluecker坐标给出它的伪黎曼结构。空间CxCΔ可以看作是S（0,3）中的曲面。这个极小交叉比的实部等于这个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球的余切束。无限小交叉比的实部也等于S（0,3）的正则辛形式。平面机构的拓扑结构：研究了具有n个肛门的机器人平面机构的位形空间，其中每个肛门都有一个转动关节和一个固定的端点。用莫尔斯理论方法和拓扑方法给出了其拓扑类型。

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.