Syzygies

这个项目属于交换代数领域。主要的研究目标是了解完全交环上最小自由分辨率的结构。其思想是描述一般Eisenbud算子的渐近结构，然后利用它来求微分。建议的研究还包括一个涉及跨学科方法的主题，将交换代数与组合学和拓扑学领域联系起来。PI将共同组织两次交换代数会议，这将为讨论最新发展和促进新方向的研究提供论坛。自由分辨率的研究是交换代数中的一个核心和美丽的领域。它包含了许多具有挑战性和重要的猜想和开放的问题。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辨率与有限生成模块联系起来的想法。他证明了在一个多项式环上（在一个域上），每个有限生成的模都有一个有限分辨率。在局部和梯度情况下，存在最小自由分辨率；它是唯一的，直到同构，并且包含在被解析模块的任何自由解析中。它的性质与解析模块的不变量密切相关。多年来，最小自由分辨率一直是交换代数的中心对象和富有成效的工具；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.