本项目的研究工作围绕代数组合专家Richard Stanley的一个著名猜想展开。该猜想提出，对于多项式环上任意有限生成的多重分次模，它的Stanley深度不小于它的（代数）深度。针对该猜想，本项目同时从代数和组合两个方面着手，开展针对若干典型单项式理想的Stanley深度和Hilbert深度的研究。我们将具体关注以下两个方面：（1）、在基本的代数运算下Hilbert深度所具有的性质；（2）、在极值条件下（具有非平方因子的）单项式理想Stanley深度和Hilbert深度所具有的性质。本项目延续了数学天元青年基金项目（批准号11026072）的研究工作。

The current project originates from a well-known conjecture by Richard Stanley. This conjecture asserts that the Stanley depth of a finitely generated multigraded module over a polynomial ring will be greater than or equal to the depth of this module. Regarding this conjecture, we investigate the Stanley depth and Hilbert depth of some typical monomial ideals, from both the algebraic and the combinatorial points of views. To be more precise, we will focus on the following several aspects: (1). Properties of the Hilbert depth under various algebraic operations; (2). Properties of the Stanley depth and Hilbert depth of the (squarefree) monomial ideals under various extremal conditions. This is a continuation of the research project, which was supported by the Tianyuan Youth Foundation of Mathematics (#11026072).