Syzygies

这个项目是在交换代数领域。主要研究目的是了解完全交环上的最小自由分解的结构。其思想是描述一般Eisenbud算子的渐近结构，然后利用它来获得微分。拟议的研究还包括一个涉及将交换代数与组合学和拓扑学领域联系起来的跨学科方法的主题。国际交换学会将共同组织交换代数的两个会议，这将为讨论最近的发展提供一个论坛，并促进新方向的研究。自由分解的研究是交换代数的一个核心和美丽的领域。它包含许多具有挑战性和重要的猜想和未决问题。希尔伯特在1890年和1893年的两篇著名论文中提出了将自由分解与有限生成模联系起来的想法。他证明了在多项式环(域上)上，每个有限生成的模都有一个有限分解。在局部和分次情形下，存在一个最小自由分解；它在同构之前是唯一的，并且包含在分解的模的任何自由分解中。它的性质与分解模的不变量密切相关。多年来，极小自由分解一直是交换代数的中心对象和卓有成效的工具，它们在其他数学领域也有应用。

This project is in the field of commutative algebra. The main research goal is to understand the structure of minimal free resolutions over complete intersection ring. The idea is to describe the asymptotic structure of the generic Eisenbud operators, and then use it in order to obtain the differentials. The proposed research also includes a topic involving interdisciplinary approaches connecting commutative algebra with the fields of combinatorics and topology. The PI will co-organize two conferences in commutative algebra, which will provide a forum for discussion of recent developments and foster research in new directions.Research on free resolutions is a core and beautiful area in commutative algebra. It contains a number of challenging and important conjectures and open problems. The idea to associate a free resolution to a finitely generated module was introduced by Hilbert in two famous papers in 1890 and 1893. He proved that over a polynomial ring (over a field) every finitely generated module has a finite resolution. In the local and the graded cases there exists a minimal free resolution; it is unique up to an isomorphism and is contained in any free resolution of the resolved module. Its properties are closely related to the invariants of the resolved module. For many years, minimal free resolutions have been both central objects and fruitful tools in Commutative Algebra; they also have applications in other mathematical fields.