Moduli Techniques in Graded Ring Theory and Their Applications

环是一种数学结构，它模拟了许多类型的对称。在自然界中遇到的大多数环都是非对易的：操作的顺序很重要。这个项目将研究非对易环理论和几何学之间的深层联系。环通过它们的模来研究：反映环中编码的对称性的对象。环的结构微妙而有力地依赖于环上模族的几何，这种联系导致了许多进展。这个项目将深入探索模族几何和环的代数结构之间的联系。我将扩展现有的方法并开发新的方法，并将我的结果应用于重要的未解决的代数问题。著名的Virasoro代数给出了几何和代数之间这种联系的力量的一个例子。Virasoro代数在数学和物理学中享有盛誉。它可以被看作是统计力学的数学模型，因此对物理学，特别是保形场理论具有深远的重要性。Virasoro代数是李代数，而不是环；它可以通过形成所谓的泛包络代数而变成环。虽然Virasoro代数已被广泛研究多年，但有关其泛包络代数的重要基本问题仍未得到解答。具体地说，至少25年来，数学家们一直在问Virasoro代数的包络代数是否具有Noether性质。(诺瑟环的表现相对较好；非诺瑟环更具异国情调。)在最近与沃尔顿的合作中，我应用几何来解决这个问题：Virasoro代数的包络代数不是Notherian的。我们的工作展示了几何技巧解决纯代数问题的能力。我们证明Virasoro代数的包络代数不是Notherian的一个关键方法是构造一个更简单的模型，称为正则二元交换因子。因为它更简单，所以模型更容易学习；另一方面，传递给模型会丢失大量信息。在这个项目中，我将开发一种通用的方法，它将适用于比Virasoro代数的包络代数更多的环，以构造包含更多信息但仍可修改的其他典型因子。更复杂的正则因子的普遍构建将是一个重大的进步。通过这个项目将开发的新技术，我将回答环论中的许多重要问题。我将使用几何来获得关于Virasoro代数的包络代数的更多信息。我将探索是否可以通过几何学来检测上述Noether性质。我将把几何方法应用于一大类环，其中Virasoro的包络代数只是一个例子：分次无限维李代数的泛包络代数。通过这些方法，我将证明这些环不是诺瑟环。这些环是出了名的棘手，如果没有我将带来的新方法，这个问题是无法解决的。

A ring is a mathematical structure that models many types of symmetry. Most rings encountered "in nature" are noncommutative: the order of operations matters. This project will investigate deep relationships between noncommutative ring theory and geometry.Rings are studied through their modules: objects that echo the symmetry encoded in the ring. The structure of a ring depends subtly and powerfully on the geometry of families of modules over that ring, and this connection has led to many advances. This project will explore this connection between the geometry of families of modules and the algebraic structure of rings in depth. I will extend current methods and develop new ones, and will apply my results to important unsolved algebraic problems. An example of the power of this connection between geometry and algebra is given by the famous Virasoro algebra. The Virasoro algebra is renowned in mathematics and physics. It may be viewed as a mathematical model of statistical mechanics, and so is of deep importance to physics, particularly conformal field theory. The Virasoro algebra is a Lie algebra, rather than a ring; it can be turned into a ring by forming its so-called universal enveloping algebra. Although the Virasoro algebra had been intensively studied for many years, important basic questions about its universal enveloping algebra remained unanswered. Specifically, for at least 25 years mathematicians had been asking if the enveloping algebra of the Virasoro algebra had the noetherian property. (Rings that are noetherian are relatively well-behaved; those that are not noetherian are more exotic.) In recent joint work with Walton, I applied geometry to solve this problem: the enveloping algebra of the Virasoro algebra is not noetherian. Our work shows the power of geometric techniques to address purely algebraic problems.One key method of our proof that the enveloping algebra of the Virasoro algebra is not noetherian was to construct a simpler model, called the canonical birational commutative factor. Because it is simpler, the model is easier to study; on the other hand, passing to the model loses a great deal of information. In this project, I will develop a general method, which will apply to many more rings than the enveloping algebra of the Virasoro algebra, to construct other canonical factors that contain more information but are still amendable to study. A general construction of more complex canonical factors will be a significant advance.Through the new techniques this project will develop, I will answer many important questions in ring theory. I will use geometry to get more information about the enveloping algebra of the Virasoro algebra. I will explore whether the noetherian property described above can be detected through geometry. I will apply geometric methods to a large class of rings, of which the enveloping algebra of the Virasoro is only one example: to universal enveloping algebras of graded infinite-dimensional Lie algebras. Through these methods, I will show these rings are not noetherian. These rings are famously intractable, and this problem is inaccessible without the new methods that I will bring to bear.