Free Resolutions in Commutative Algebra

代数几何和交换代数数学领域的核心目标是理解多项式方程系统的解，可能涉及大量变量和大量方程。解形成一个几何对象。其主要思想是研究其几何和代数性质之间丰富而美丽的相互作用。与这项研究密切相关的是自由分辨率的概念，这是由大卫·希尔伯特在1890年和1893年的两篇论文中首次提出的。构造一个自由分辨率相当于反复求解多项式方程组。对这些物体的研究在20世纪下半叶蓬勃发展，并在过去十年中取得了惊人的进展。该领域非常广泛，与其他数学领域和弦理论有很强的联系和应用。最近的计算方法已经使计算机计算一些自由分辨率成为可能。本课题的主要研究目标是在理解自由分辨率的结构及其数值不变量方面取得重大进展。主要研究方向为：(1)完全交叉口上的分辨率，采用Eisenbud和Peeva在其研究专著《完全交叉口上的最小自由分辨率》中介绍的方法进行研究；(2)周期无限最小自由分辨率的Betti数，计算代数方法将与具有常数Betti数的周期分辨率模块的例子相结合；(3) McCullough和Peeva最近提出的产生质数理想分解的新方法的应用。

A core goal in the mathematical areas Algebraic Geometry and Commutative Algebra deals with understanding the solutions of a system of polynomial equations, possibly in a large number of variables and with a large number of equations. The solutions form a geometric object. The main idea is to study the rich and beautiful interplay between its geometric and algebraic properties. Closely related to this study is the concept of a free resolution, which was first introduced by David Hilbert in two papers in 1890 and 1893. Constructing a free resolution amounts to repeatedly solving systems of polynomial equations. The study of these objects flourished in the second half of the twentieth century, and has seen spectacular progress in the last ten years. The field is very broad, with strong connections and applications to other mathematical areas and string theory. Recent computational methods have made it possible to compute some free resolutions by computers. The main research goal in this project is to make significant progress in understanding the structure of free resolutions and their numerical invariants. The main research topics are:(1) resolutions over complete intersections, which will be studied using the methods recently introduced by Eisenbud and Peeva in their research monograph "Minimal Free Resolutions over Complete Intersections";(2) Betti numbers of periodic infinite minimal free resolutions, for which computational algebra methods will be combined with insights from the examples of modules with periodic resolutions with constant Betti numbers;(3) applications of the new approach introduced recently by McCullough and Peeva which produces resolutions of prime ideals.