Local Algebra and Local Representation Theory

数学的众多功能之一是，它提供了表述描述物理世界的方程的语言和求解方程的工具。通常，人们遇到的方程本质上是代数的，比如描述直线、圆、抛物线等的方程，与之形成对比的是，涉及三角函数、对数或导数的方程。通常，方程有无限多个解——想想定义一个圆的方程——通常不可能写出一个完整的解列表。相反，目标是找到方法来研究解集集合的结构，这被称为多样性。一个富有成效的方法是考虑（代数）函数上的变化。这些函数形成了一个叫做交换环的数学结构，我的研究一直致力于理解这些结构；不是抽象的，而是各种各样的表现形式，它们在数学和物理学的不同背景中出现。数学在描述和研究与对称有关的现象方面也取得了显著的成功。这就引出了另一种叫做群的数学结构。有趣的是，在某些情况下，有一种方法可以将交换环和一个变种附加到一个群上，在过去的几年里，包括PI在内的各种研究人员已经能够使用研究变种的工具来解决与群相关的问题。当前项目的一部分涉及这些方面。两个主要的主题贯穿了这个项目。一是研究交换诺瑟局部环上有限自由复的不变量；第二部分是有限群和群格式的模表示理论。前者也通过变换群理论中的某些猜想与群联系在一起。这些猜想——由Adem、Avramov、Browder、Buchweitz、Carlsson、Swan、Halperin等人提出——假设了当同调具有非零有限长度时，这些配合物模的长度下界和总秩下界。最近，PI和马克·沃克（Mark Walker）发现了许多这些猜想的反例。本项目概述的一组问题旨在发现和建立这些不变量的“正确”界限。另一组项目试图探索有限群或有限群方案在正特征域上表示的稳定模类别的结构。重点研究了“局部地层”和该地层中各种模块的有限条件；特别是关于对偶性和上同调有限性。交换代数的Hochschild上同调的乘法结构是本提案的第三个主题。这里的目标是用它们的Hochschild上同调来描述局部完全交代数。该奖项将支持学生在一个非常相关的数学领域进行培训，该领域可以应用于多个领域。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

One of the myriad functions of Mathematics is that it provides language to formulate, and tools to solve, equations that describe the physical world. Often the equations that one encounters are algebraic in nature, like those describing lines, circles, parabolas and the like, in contrast with, say, equations involving the trigonometric functions, or logarithms, or derivatives. Typically, the equations have infinitely many solutions---think about the equation defining a circle---and it is usually not possible to write down a complete list of solutions. Rather, the objective is to find ways to study the structure of the collection of the solution set, which is called a variety. A fruitful approach has been to consider the (algebraic) functions on the variety. These functions form a mathematical structure called a commutative ring, and my research has been dedicated to understanding these structures; not in the abstract, but in their various manifestations, which are galore, they arise in quite diverse contexts across mathematics and physics. Mathematics has also been remarkably successful in describing and studying phenomenon related to symmetry. This leads to another mathematical structure called a group. Intriguingly, in certain contexts, there is a way to attach a commutative ring---and a variety---to a group and in the past few years various researchers, including the PI, have been able to solve problems related to groups using tools that had been developed to study varieties. A part of the current project deals with these aspects.Two major themes weave through this project. One is the study of invariants of finite free complexes over commutative noetherian local rings; the second is the modular representation theory of finite groups and group schemes. The former too is connected to groups via certain conjectures in the theory of transformation groups. These conjectures---due to Adem, Avramov, Browder, Buchweitz, Carlsson, Swan, Halperin and others---postulated lower bounds on the length of the homology modules, and on the total rank, of such complexes, when the homology has nonzero finite length. Recently the PI and Mark Walker found counterexamples for many of these conjectures. One set of problems outlined in this project are aimed at discovering and establishing the ``correct" bounds for these invariants. Another set of projects seek to probe the structure of the stable module category of representations of a finite group, or finite group scheme, over a field of positive characteristic. The focus is on ``local strata" and various finiteness conditions for modules in this strata; in particular, on dualizability and cohomological finiteness. The multiplicative structure of Hochschild cohomology of commutative algebras is a third main topic of this proposal. The goal here is to characterize locally complete intersection algebras in terms of their Hochschild cohomology. This award will support the training of students in a very relevant area of mathematics that has applications to several fields.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.