Topics in algebraic geometry

代数几何专题

• 批准号: RGPIN-2016-04730
• 负责人: Dhillon, Ajneet
• 金额: $ 1.31万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=676962
• 关键词: Topics; algebraic; geometry

This research is centered around principal bundles on algebraic varieties. In studying bundles, tangential subjects of independent interest, such as derived categories, K-theory and Gromov-Witten invariants are touched upon. These subjects have found applications in mathematical physics and we intend to both contribute to and be guided by this literature.*** This research makes systematic use of algebraic stacks. These are geometric objects with added symmetries. Their virtue lies in that many problems have their natural solutions as algebraic stacks rather than spaces. Stacks show up in the work as parameter spaces and as curves with added structure.*** The next tool that we make use of is the derived category and its various enhancements. The derived category is a kind of abelianizing object. It takes a complicated object and replaces it with something more accessible which at the same time encapsulates all its important invariants and properties.***Using these tools we aim to study three fundamental questions.***The first question is of basic interest to number theorists. General theory says that certain curves and maps can be defined via equations that have coefficients in a finite extension of the rational numbers. However not much is known about which particular extension. In other words, we have an existential theorem that we wish to make constructive. ***The second question is about counting are calculation of a particular number. We wish to count maximal subbundles of a fixed parabolic bundle. The approach that we intend to use will make use of ideas originating in mathematical physics. We intend to generalise some important combinatorial formulas in the process of the calculation.***The last question that will be addressed is about calculating parameters. We have an important universal object, the moduli stack of principal bundles that appears in many parts of mathematics and physics. A basic question that can be asked about it is, how many parameters does in generally depend on. Given the objects ubiquity, this is an important question that should be addressed.**

这项研究集中在代数品种上的主要捆绑包上。在研究捆绑包时，涉及独立兴趣的切向主题，例如衍生类别，K理论和Gromov-witten不变性。这些受试者在数学物理学中找到了应用，我们打算既贡献并受到此文献的指导。***这项研究使系统地使用代数堆栈。这些是带有对称性的几何对象。他们的美德在于，许多问题将其自然解决方案作为代数堆栈而不是空间。堆栈在工作中以参数空间和添加结构的曲线形式显示。***我们使用的下一个工具是派生类别及其各种增强功能。派生类别是一种阿贝里亚化对象。它采用一个复杂的对象，并用更容易访问的东西代替，同时封装了所有重要的不变和属性。***使用这些工具，我们旨在研究三个基本问题。***第一个问题是数字理论家的基本兴趣。一般理论说，某些曲线和地图可以通过具有有限的理性数字扩展的方程来定义。但是，对哪个特定扩展不了解。换句话说，我们有一个希望使建设性的定理。 ***第二个问题是关于计数的是特定数字的计算。我们希望计算固定抛物线捆绑包的最大子捆绑。我们打算使用的方法将利用源自数学物理学的想法。我们打算在计算过程中概括一些重要的组合公式。***将要解决的最后一个问题是关于计算参数。我们有一个重要的通用对象，即在数学和物理学的许多地方出现的主要捆绑包。可以询问的一个基本问题是，通常有多少个参数可以取决于。鉴于对象无处不在，这是一个应该解决的重要问题。**