Studies on Integrality of Ideals

这个项目涉及交换代数中的问题。虽然这些问题涉及到各种各样的主题，但有一个基本的共同点将它们统一起来：积分闭包理论。PI和她的合作者一起，首先计划开发产生理想的积分闭包的方法。在与Bernd Ulrich的联合工作中，提议者已经建立了链接是捕获完整交叉上的积分元素的方法。现在，调查人员想将同样的程序扩展到Gorenstein连接和剩余交叉点。其次，PI旨在澄清理想的核与Lipman的伴随（或Ein和Lazarsfeld的乘子理想）之间的联系，找到核的显式公式，并计算单项式理想的核（Eisenbud和Sturmfels提出的问题）。研究者打算利用链环理论、剩余交理论和Groebner基理论。第三，研究者想研究特殊的纤维环。这个物体从几何学的角度来看是非常重要的，因为它编码了爆破的特殊纤维上的代数信息。这里的重点是寻找条件，迫使纤维是unmixed或甚至科恩-麦考利。最后，PI计划研究零维正规理想的希尔伯特函数（理想是正规的，如果它的所有幂都是整数闭的）。提出者的目标之一是证明这类理想的所有Hilbert系数都是非负的，而另一个目标是用理想本身或其幂之一的相关分次环的Cohen-Macaulay性来刻画Hilbert系数的消失.交换代数的基本内容是求解多项式方程组的技巧.交换代数在过去的50年里有了革命性的发展，因为它为理解纯数学和应用数学中的许多问题提供了工具。在许多应用问题中，多项式方程和交换代数起着至关重要的作用.在过去，交换代数的结果已经被应用到的应用领域包括运筹学、计算机科学、机器人学、控制理论、编码理论和密码学等等。

This project deals with questions in commutative algebra. While thequestions address a variety of topics, there is an underlyingcommon thread that unifies them all: the theory of integral closure.The PI, together with her collaborators, first plans to developmethods that produce the integral closure of an ideal. In joint workwith Bernd Ulrich the proposer has established that linkage is amethod for capturing integral elements over a complete intersection. Nowthe investigator wants to extend this same procedure to Gorensteinlinkage and residual intersections. Second, the PI intends to clarify theconnection between the core of an ideal and the adjoint of Lipman (or themultiplier ideal of Ein and Lazarsfeld), find an explicit formula forthe core, and compute the core of monomial ideals (a question posedby Eisenbud and Sturmfels). The investigator intends to use linkagetheory, residual intersection theory and Groebner basis theory.Third, the investigator would like to study the special fiber ring. Thisobject is very important from a geometric point of view because itencodes algebraic information on the special fiber of the blowup. The focus here is on finding conditions that force the fiber to be unmixed oreven Cohen-Macaulay. Last, the PI plans to study the Hilbert function of zero-dimensional normal ideals (an ideal is normal if all its powers are integrally closed). One of the proposer's goals is to show that all the Hilbert coefficients of such ideals are non-negative, while another goal is to characterize the vanishing of the Hilbert coefficients in terms ofthe Cohen-Macaulayness of the associated graded ring of the idealitself or of one of its powers.The proposed research is concerned with problems in commutativealgebra. On a basic level commutative algebra is about techniquesfor solving systems of polynomial equations. Commutative algebrahad a revolutionary growth in the past fifty years as it providedthe tools for understanding many problems in pure and appliedmathematics. In many applied problems, polynomial equations andhence commutative algebra play a crucial role. Applied areas wherecommutative algebra results have been used in the past includeoperations research, computer science, robotics, control theory, codingtheory and cryptography to mention a few.