Minimal Free Resolutions and Syzygies

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。解集形成几何对象。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。与此密切相关的是自由分解的概念，这一概念是由著名数学家大卫·希尔伯特在1890年和1893年的两篇论文中提出的。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，最近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某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。主要研究内容有：(1)多项式环中具有二次初始理想的二项边理想的最小自由分解的结构。这个项目的灵感来自于Ene，Herzog和Hibi的一个猜想，即这样一个理想的分次Betti数与它的二次初始理想的分次Betti数(关于Lex阶)一致。(2)Clements-Lindstrom商环上词典理想的无限极小自由分解的结构，将使用Eisenbud和Peeva最近在他们的研究专著《完全交集上的极小自由分解》中介绍的方法进行研究。(3)PI计划探索一个新的研究领域：外部代数上二项边理想的极小自由分解。PI计划在她的研究领域培训研究生。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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 solution set forms a geometric object. The main idea is to study the rich and beautiful interplay between its geometric and algebraic properties. Closely related to this is the concept of a free resolution, which was introduced by the famous mathematician 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 recently. 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 computer. 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) The structure of minimal free resolutions of binomial edge ideals with a quadratic initial ideal in a polynomial ring. The project is inspired by a conjecture of Ene, Herzog, and Hibi, that the graded Betti numbers of such an ideal coincide with the graded Betti numbers of its quadratic initial ideal (with respect to the lex order). (2) The structure of infinite minimal free resolutions of lexicographic ideals in Clements-Lindstrom quotient rings, which will be studied using the methods recently introduced by Eisenbud and Peeva in their research monograph "Minimal free resolutions over complete intersections”.(3) The PI plans to explore a new area of research: minimal free resolutions of binomial edge ideals over exterior algebras. The PI plans to train graduate students in her research field.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.