组合同余式与群上的和集

我们计划研究组合同余式与群上的和集问题。这两个课题都属于组合数学与数论的交叉领域，需要综合运用多个数学分支的知识与工具。我们准备重点着眼于截断超几何级数的算术性质，同余式的组合证明，非交换群上的DeVos-Goddyn-Mohar型定理以及群上的异元和集问题等。我们计划使用组合变换与p-adic分析去研究关于截断超几何级数的同余式，特别是孙智伟教授的一些相关猜想。我们打算系统地对涉及Stirling数，Euler数，Eulerian数等组合序列的同余式研究其基于群作用的组合证明。我们还希望利用Kemperman变换等工具将DeVos-Goddyn-Mohar定理推广到一般群上。

We plan to study the combinatorial congruences and the sumsets in groups. Those two topics belongs to the intersection of combinatoris and number theory. In order to study them, we need to apply the knowledge and tools from several areas of mathematics. We shall focus on the following problems: the arithmetical properties of truncated hypergeometric series, the combinatorial proof of congruences, the DeVos-Goddyn-Mohar type theorem on non-abelian groups and the restricted sumsets on groups. We plan to use combinatorial transformations and p-adic analysis to research the congruences for truncated hypergeometric sereis, especially some conjectures of Prof. Sun. We shall search the combinatorial proofs for the congruences on integer sequences, such as Stirling numbers, Euler numbers, Eulerian numbers. We also wish to use the Kemperman transformation to extend the DeVos-Goddyn-Mohar theorem to general groups.