Studies in Commutative Algebra and Algebraic Geometry

PI计划研究交换代数和代数几何中的几个问题。代数几何研究多变量多项式方程的解。了解这个问题在许多科学、工程学以及其他学科中都是非常重要的。通常很难确定是期待任何解决方案、有限多个解决方案还是无限多个解决方案：在最后一种情况下，人们想知道在描述解决方案时有多少自由度。人们可以通过研究解空间上的某些函数来几何地或代数地研究这些解，这些函数形成了所谓的交换环。这种双重视角在交换代数和代数几何之间建立了密切的联系，这是非常有价值的。提出要研究的问题由来已久，具有根本和核心的重要性。所取得的成果将通过期刊和书籍出版、在会议和研讨会上的演讲以及通过互联网传播。这里面有很强的教育成分。首席调查员有39名博士生，其中包括14名女性(目前还有5名博士生，包括4名女性，其中1名是非裔美国人)，并曾担任14名初级教职员工的导师，其中6名是女性。这种活动水平将继续下去。PI将把本科生纳入他的研究。PI将研究诺德环理论中的几个长期存在的问题。其一是证明了Stillman关于射影维有界的猜想，Tgran Ananyan和Hochster所做的特征不为2，3为至多4次。另一种是继续发展紧闭包理论：Neil Epstein和Hochster发展了一个新的版本，它具有原理论的许多性质，给出了一个更小的闭包，定义在特征p和等价特征0中，并具有局部化交换。这一理论提出了新的问题，同时也提供了对现有问题的洞察。Hochster与Bhargav Bhatt共同给出了正特征正则环上Serre交重数正性的一个新证明，这种方法有望解决50多年来解决一般混合特征环的问题。其他方向包括相对较新的局部上同调的拟长度和内容理论，局部上同调的支撑性的有限性，Lech猜想和推广，长期存在的直接求和猜想的新方法，以及与矩阵相关的某些代数集的几何。其中许多问题是为了与研究生和博士后教师合作而设计的。

The PI plans to work on several questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of many polynomial equations in many variables. Understanding this problem is of fundamental importance in many sciences, in engineering, and in other disciplines as well. It is often difficult to determine whether to expect any solution, finitely many solutions, or infinitely many: in the last case one wants to know how many degrees of freedom one has in describing the solutions. One can study the solutions geometrically, or algebraically, by investigating certain functions on the solution space that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry that is very valuable. The problems proposed for study are long standing and of fundamental, central importance. The results obtained will be disseminated by journal and book publication, lectures at conferences and workshops, and via the internet. There is a strong educational component. The principal investigator has had thirty-nine Ph.D. students including fourteen women (and has five other Ph.D. students currently, including four women, one of whom is African-American), and has served as mentor to fourteen junior faculty members, of whom six were women. This level of activity will continue. The PI will integrate undergraduate students into his research.The PI will investigate several long standing questions in the theory Noetherian rings. One is to prove Stillman's conjecture bounding projective dimension, which Tigran Ananyan and Hochster have done in characteristic not 2, 3 for degree at most four. Another is to continue the development of tight closure theory: Neil Epstein and Hochster have developed a new version with many of the properties of the original theory that gives a smaller closure, is defined in both characteristic p and equal characteristic 0, and commutes with localization. This theory raises new questions while offering insight into existing ones. Jointly with Bhargav Bhatt, Hochster has given a new proof of the positivity of Serre intersection multiplicities over regular rings in positive characteristic using an idea that has promise for solving the more than fifty year old problem of settling the general case in mixed characteristic. Other directions include the relatively recent theory of quasilength and content of local cohomology, finiteness of support of local cohomology, Lech's conjecture and generalizations, a new approach to the long standing direct summand conjecture, and the geometry of certain algebraic sets associated with matrices. A number of these problems are intended for collaboration with graduate students and postdoctoral faculty.