Free Resolutions in Commutative Algebra

在数学领域，代数几何和交换代数的一个核心目标是了解多项式方程组的解，可能是在大量变量和大量方程中。这些解形成了一个几何对象。其主要思想是研究它的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决议的概念，它是由大卫·希尔伯特在1890年和1893年的两篇论文中首次提出的。构造自由解相当于重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，并在最近十年中取得了令人瞩目的进展。这个领域非常广泛，与其他数学领域和弦理论有着很强的联系和应用。最近的计算方法使用计算机计算一些自由分辨率成为可能。这个项目的主要研究目标是在理解自由分辨率及其数值不变量的结构方面取得重大进展。主要研究内容是：(1)完全交上的分解，将使用Eisenbud和Peeva最近在他们的研究专著《完全交上的最小自由分解》中介绍的方法来研究；(2)周期无限极小自由分解的Betti数，对于它，计算代数方法将与具有常数Betti数的周期分解的例子相结合；(3)McCullough和Peeva最近引入的产生素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.