有限p群的模不变量理论及Noether问题研究

代数不变量理论是一门历史悠久、结果丰富的研究课题；其思想方法广泛应用于代数几何、代数拓扑、代数数论等学科。许多著名的数学问题，如Hilbert第14问题、Noether问题、反Galois问题，使得代数不变量理论成为当前国内外非常活跃的研究方向之一。..本研究项目内容包括代数不变量理论的两个研究分支：有限p群的模向量不变量环及Noether问题。更准确地，本项目拟完成以下三个研究问题：第一、刻画清楚有限域上一般线性群的1向量及1余向量不变量环的结构；第二、完整地回答阶数为p^5的非阿贝尔群的Noether问题；第三、分类清楚那些带有平凡Bogomolov乘子的p^6阶群。

Algebraic invariant theory (AIT) is a research subject with long history and beautiful results. This subject has been applied extensively to algebraic geometry, algebraic topology and algebraic number theory. A lot of famous problems, such as Hilbert 14th problem, Noether problem and inverse Galois problem, make AIT to be a very active research direction. ..The present project studies two topics in AIT: modular vector invariant theory and Noether problem for finite p-groups. More precisely, we will answer completely the following three questions: first of all, whether or not the ring of modular 1 vector and 1 covector invariant for the general linear group over a finite field is Gorenstein algebra? Secondly, whether the Noether problem holds for all groups of order p^5? Finally, how to classify the groups of order p^6 with trivial Bogomolov multiplier?