Arithmetic of differential operators, K-theory and periods of motives

微分算子算法、K 理论和动机周期

• 批准号: 1502296
• 负责人: Deepam Patel
• 金额: $ 15.6万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1502296&HistoricalAwards=false
• 关键词: Arithmetic; differential; operators; theory; periods

Mathematically, this research project lies in the field of algebraic geometry. The fundamental problem in algebraic geometry is the study and classification of geometric objects given by solutions to polynomial equations. This research project focuses more specifically on the application of the tools and techniques used in algebraic geometry to study equations arising from algebraic number theory. The study of such equations has a rich history and can be traced back to Diophantus, who first studied solutions to what are today called Diophantine equations. Because of their diversity, the study of such equations has led to numerous applications in fields such as biology, chemistry, cosmology, and computer encryption. The general strategy towards classification of such arithmetic objects is to associate various invariants to the given geometric objects. One example of such an invariant is a set of complex numbers, usually known as periods, associated to a given polynomial system. A central focus of this research project to study which complex numbers can arise as periods of various objects appearing in algebraic geometry. The project also contributes to the training of the next generation of researchers by engaging postdoctoral fellows, graduate students, and undergraduate students in research.In this project, the principal investigator intends to study periods arising from the theory of algebraic varieties. The study of periods is in fact a shadow of some foundational questions in the theory of motives and algebraic cycles. The PI intends to pursue three broad projects dealing with such questions. In the first project, The PI studies periods of vector bundles with connections. Periods arise when the rational structures on two different cohomology groups are compared. This project attempts to generalize to higher dimensions results on the periods of irregular connections in the case of curves. The second project deals with giving explicit constructions of mixed Tate motives. In previous work, the PI constructed such motives from the higher homotopy of algebraic varieties. An explicit outcome of this project will be to compute the periods and Galois modules coming from these higher homotopy motives. In addition to these projects, the PI also plans to use K-theory to study the deformation theory of algebraic cycles as well as the behavior of algebraic cycles under extension of base fields. The last project attempts to apply methods in algebraic geometry to the study of infinite dimensional representations of certain p-adic algebraic groups, and is an excellent example of an application of algebraic geometry to other areas of mathematics.

从数学上来说，这个研究项目属于代数几何领域。代数几何的基本问题是通过多项式方程的解给出的几何对象的研究和分类。该研究项目更具体地侧重于应用代数几何中使用的工具和技术来研究代数数论中产生的方程。对此类方程的研究有着丰富的历史，可以追溯到丢番图，他首先研究了今天所谓的丢番图方程的解。由于其多样性，对此类方程的研究在生物学、化学、宇宙学和计算机加密等领域得到了广泛的应用。对此类算术对象进行分类的一般策略是将各种不变量与给定的几何对象相关联。这种不变量的一个例子是与给定多项式系统相关的一组复数，通常称为周期。该研究项目的中心重点是研究哪些复数可以作为代数几何中出现的各种对象的周期而出现。该项目还通过吸引博士后、研究生和本科生参与研究，为下一代研究人员的培训做出贡献。在该项目中，主要研究者打算研究代数簇理论产生的周期。对周期的研究实际上是动机和代数循环理论中一些基本问题的影子。 PI 打算开展三个广泛的项目来解决这些问题。在第一个项目中，PI 研究具有连接的向量丛的周期。当比较两个不同上同调群的有理结构时，就会出现周期。该项目试图将曲线情况下不规则连接周期的结果推广到更高维度。第二个项目涉及对泰特美术馆混合动机的明确构建。在之前的工作中，PI 从代数簇的更高同伦性构建了这样的动机。该项目的一个明确成果将是计算来自这些更高同伦动机的周期和伽罗瓦模。除了这些项目之外，PI还计划利用K理论来研究代数环的变形理论以及基场扩展下代数环的行为。最后一个项目尝试将代数几何方法应用于研究某些 p 进代数群的无限维表示，并且是将代数几何应用于其他数学领域的一个很好的例子。