格拉斯曼空间上的等距

算子代数全体投影集合中的某些子集被称为格拉斯曼空间。作为一类在多个现代数学分支中被广泛研究的数学对象，格拉斯曼空间在量子物理，机器学习，信息科学等许多学科中都有重要的应用。. 在本项目中，我们主要计划针对一些和格拉斯曼空间之间等距映射有关的问题展开研究。对此类问题的讨论源于Wigner定理。该定理于1931年被证明。此后，为了理解格拉斯曼空间之间的等距映射，众多学者付出了相当的努力，也取得了很多关键性的进展。可即便如此，依然有许多问题悬而未决。我们准备从探索格拉斯曼空间的几何结构入手，尝试为其中的一些问题提供解答。具体来说，我们将研究格拉斯曼空间之间的算子范数等距映射和2-范数等距映射。此外，对于格拉斯曼空间上保正交性的映射，我们也将尝试进行相应的刻画。

The Grassmann spaces can be identified as certain subsets of projection lattices of operator algebras. As a fundamental object of study across various subdisciplines of modern mathematics, the Grassmann space finds serious applications in many fields such as quantum physics, machine learning and information science, etc...In the proposed research, we mainly focus on several problems related to the isometries between Grassmann spaces. The research of isometries between Grassmann spaces is motivated by Wigner’s unitary–antiunitary theorem which is proved in 1931. Since that time, considerable effort has been expanded in attempts to understand these isometries. Although substantial progress has been achieved over the years, there are still many interesting questions unanswered. We will try to answer some of these questions by exploring the geometry structures of Grassmann spaces. More specifically, we will study the isometries between Grassmann spaces with respect to 2-norm and operator norm. We will also try to give a characterization of orthogonality preserving maps on Grassmann spaces.