关于NIP域和NIP整环的分类问题的研究

NIP结构在当代模型论中处于核心地位，而dp-秩是对NIP结构复杂度的一种度量。我们称一个结构为dp-有穷，是指其dp-秩是有穷的。申请人在2020年已经完成了dp-有穷域的分类。本项目将研究一些新的分类问题。第一个问题是，我们能否对申请人论文中首次提出的W-拓扑进行分类？该拓扑决定着dp-有穷域中可定义集合的性质。第二个问题是，我们能否对dp-有穷的拓扑域进行分类，或者等价地说，能否对dp-有穷整环给出一个相对弱的分类？第三个问题是，我们能否对NIP诺特环进行分类？最后一个问题是，对dp-有穷域的分类方法能否推广到dp-秩更高的NIP域，比如强NIP域？对于这些问题的研究有望拓展我们关于NIP域和环的知识，同时也能为我们提供新的技术手段用以研究一般的NIP结构。

NIP structures and theories play a central role in contemporary model theory. Dp-rank is a measure of complexity for NIP structures. A structure is "dp-finite" if it has finite dp-rank. Dp-finite fields were classified by the applicant in 2020. This project will consider four new problems which arise from the classification. First, can we classify the "W-topologies" which appear in the proof? These topologies govern the behavior of definable sets in dp-finite fields. Second, can we obtain a classification of dp-finite topological fields, or equivalently, a weak classification of dp-finite integral domains? Third, can we classify NIP Noetherian integral domains? Lastly, can the classification of dp-finite fields be generalized to NIP fields of higher dp-rank, specifically to the "strongly NIP" (or "strongly dependent") fields? Research on these problems will advance the frontiers of knowledge on NIP fields and rings, and will help develop new techniques for working with NIP structures.