Studies of Analysis on manifolds

1.发展了无限维马斯洛夫指数理论。在有限维中我们有一个主丛U(N)→Λ(N)=U(N)/O(N)，我们也有一个类似于无限维主丛的主丛，但根据柯伊珀定理它是平凡的。幸运的是，我们有一个略有不同的主丛，总空间不是一个群，我们称之为它的基空间，FredholmLagrangian Grassmanian。特别地，我们找到了一种泛函分析方法，不仅定义了环的Maslov指数，而且定义了关于固定Maslov圈的任意路径的Maslov指数，并在对称算子的自伴扩张理论的框架下，应用这一理论推广了闭流形上一类一阶自伴椭圆算子的谱流的Yoshida-Nicolaescu定理。结合FredholmLagrangian Grassmanian的研究，证明了形式为“U+ID为Fredholm型”的酉算子空间的同伦群与酉群的稳定同伦群同构。我们在Cayley射影平面的穿孔余切丛上构造了一个Kahler结构，并将其嵌入到8×8复矩阵空间中。在此基础上，给出了基于这种嵌入的Cayley射影平面测地线流的显式表示。利用四元数射影空间P^nH的穿孔余切丛上的Kahler结构，构造了由若干类具有再生核的全纯函数组成的希尔伯特空间，并构造了从这个希尔伯特空间到L_2(P^NH)的算子，证明了它用√&lt；Δ+(2n+1)2&gt；生成的单参数椭圆型傅立叶积分算子族来表示测地流.

1. We developed a theory of Maslov index in the infinite dimension. In the finite dimension we have a principal bundle U(n)→Λ(n) = U(n)/O(n), also we have a principal bundle similar to that in the infinite dimension, but it is trivial by Kuiper theorem. Fortunately we have a slightly different principal bundle with the total space not being a group and we call its base space, Fredholm Lagrangian Grassmannian. Especially we found a functional analytic method to define a Maslov index for not only loops but also for arbitrary paths with respect to a fixed Maslov cycle and we applied this theory to generalize Yoshida-Nicolaescu theorem for the spectral flow of a family of first order self-adjoint elliptic operators on a closed manifold on the framework of the theory of self-adjoint extensions of symmetric operators. Relating with the study of Fredholm Lagrangian Grassmannian we prove that the homotopy groups of the space of unitary operators of the form "U + Id is Fredholm" is isomorphic to the stable homotopy groups of unitary groups.2. We constructed a Kahler structure on the punctured cotangent bundle of the Cayley projective plane and embedded it into a space of 8×8 complex matrices. Then we give an explicit representation of the geodesic flow of the Cayley projective plane in terms of this embedding.3. By making use of a Kahler structure on the punctured cotangent bundle of the quaternion projective space P^nH, we construct a Hilbert space consisting of certain classes of holomorphic functions with the reproducing kernel, and also constructed an operator from this Hilbert space to L_2(P^nH), and proved that it represents the geodesic flow in terms of the one parameter family of elliptic Fourier integral operators generated by √<Δ+(2n+1)_2> (quantization of the geodesic flow).