序列空间生成的等价关系之间的 Borel 归约

Invariant descriptive set theory is a new branch of descriptive set theory that deals with the complexity of equivalence relations. Usually we use Borel reducibility to compare the complexity of different equivalence relations. Using this method to characterize the relative complexity of equivalence relations from various branches of mathematics, is becoming a new hot issue in descriptive set theory... In this Project, we intend to study the complexity of equivalence relations generated by sequence spaces under Borel reduction. Firstly, we will study the complexity of equivalence relations generated by Orlicz sequence spaces and Lorentz sequence spaces. On the one hand, we will study Borel reducibility between the landmark equivalence relations and equivalence relations generated by these spaces, then determine their specific locations under Borel reduction. On the other hand, we will study mutual Borel reducibility between these equivalence relations, then determine the complexity of their internal structure. Secondly, we will also study the dichotomy of equivalence relations generated by sequence spaces.

不变量描述集合论是描述集合论的一个新的分支，主要研究的是等价关系的复杂度。通常我们利用 Borel 归约的办法去比较等价关系之间的复杂度。利用这一方法去刻画源自数学各个分支的等价关系之间的相对复杂度，是当前描述集合论方向的热点问题。.. 本项目拟研究由序列空间生成的等价关系在 Borel 归约下的复杂度。首先，我们将研究 Orlicz 序列空间、Lorentz 序列空间这两类序列空间生成的等价关系的复杂度。一方面，我们研究这些空间生成的等价关系和一些标志性等价关系之间的 Borel 归约，从而确定其在 Borel 归约结构中的具体位置。另一方面，我们研究这些等价关系之间的相互归约，从而确定其内部结构的复杂度。其次，本项目也将研究序列空间生成的等价关系的二分性等相关内容。