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。代数几何研究许多多项式方程在许多变量下的解。理解这个问题对许多科学、工程和其他学科都至关重要。通常很难确定是期望得到任意解、有限多解还是无限多解：在最后一种情况下，人们想知道在描述解时有多少个自由度。人们可以从几何上或代数上研究解，通过研究解空间上形成所谓交换环的某些函数。这种对偶视角在交换代数和代数几何之间建立了非常有价值的紧密联系。提出研究的问题是长期存在的，具有根本的中心重要性。获得的结果将通过期刊和书籍出版、会议和讲习班讲座以及通过互联网传播。有很强的教育成分。首席研究员有39名博士生，包括14名女性（目前还有5名其他博士生，包括4名女性，其中一名是非裔美国人），并担任了14名初级教员的导师，其中6名是女性。这种活动水平将持续下去。PI将把本科生纳入他的研究。PI将研究诺瑟环理论中几个长期存在的问题。一是证明了Tigran Ananyan和Hochster在特征不为2,3的情况下所做的Stillman猜想的边界射影维。另一种是继续发展紧闭理论：Neil Epstein和Hochster开发了一个新版本，它具有原始理论的许多性质，可以给出更小的闭包，在特征p和相等特征0中都有定义，并且可以与局域化交换。这一理论提出了新的问题，同时也为现有问题提供了洞见。Hochster与Bhargav Bhatt共同给出了正则环正特征上Serre交多重性正性的一个新证明，该证明的思想有望解决存在了50多年的混合特征下一般情况的解决问题。其他方向包括相对较新的局部上同调的准长度和内容理论，局部上同调支持的有限性，莱赫猜想和推广，长期存在的直接和猜想的新方法，以及与矩阵相关的某些代数集的几何。其中一些问题是为了与研究生和博士后教员合作解决的。

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.