本项目内容属BANACH空间的非线性几何及其应用范畴，旨在以BANACH空间及其子集的同胚嵌入问题包括（1）“线性”同胚；（2）Lipschitz同胚；（3）一致连续同胚；（4）粗同胚嵌入或等价性, 以及它们在Fourier乘子理论和在梯度流构造上的应用研究为主线，重点研究（a）保度量映射, 扰动保度量映射, 粗保度量映射的基本性质；（b）有界几何空间粗嵌入到Hilbert空间，粗嵌入到超自反空间（或一致凸空间）；（c）一类特殊度量空间同胚嵌入到Hilbert空间；（d）线性同胚理论在Fourier乘子理论、系统解决几类向量值边值问题中的最大正则性问题上的应用，以及（e）Lipschitz同胚理论在梯度流构造和收敛性上的应用。

This project belongs to the research of nonlinear geometry of Banach space and its application, we will investigate the problems about isomorphic embedding of Banach space and its subset as following: (1) "linear" isomorphism; (2) Lipschitz isomorphism; (3) uniformly continuous isomorphism; (4) coarsely isomorphic embedding or equivalence, and their applications in Fourier multiplier theory, and in construction of gradient flow; especially, we want to research (a) properties of isometric mapping, disturbed isometric mappings and coarsely isometric mapping; (b) the coarse embedding of bounded geometry into Hilbert space, superreflexive space (or uniform convex space); (c) coarse embedding of a class of special metric spaces into Hilbert spaces; (d) applications of above theories to Fourier multiplier theory and to maximal regularity of several typical vector-valued boundary problems; and （e) applications of Lipschitz isomorphism theory on the constructions of gradient flows and their convergence.