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

从数学上讲，这个研究项目属于代数几何领域。代数几何中的基本问题是对由多项式方程的解给出的几何对象的研究和分类。这项研究项目更具体地侧重于代数几何中使用的工具和技术的应用，以研究由代数数论产生的方程。对这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称的丢番图方程的解。由于它们的多样性，对这类方程的研究在生物、化学、宇宙学和计算机加密等领域得到了广泛的应用。对这样的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联的一组复数，通常称为周期。这个研究项目的一个中心焦点是研究代数几何中出现的各种对象的周期可以出现哪些复数。该项目还通过邀请博士后研究员、研究生和本科生参与研究，为培养下一代研究人员做出了贡献。在这个项目中，主要研究人员打算学习由代数变种理论产生的周期。对周期的研究实际上是动机理论和代数循环理论中一些基本问题的影子。国际和平研究所打算开展三个广泛的项目来处理这些问题。在第一个项目中，PI研究带有连接的向量丛的周期。周期出现在比较两个不同上同调群上的有理结构时。这个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的维度。第二个项目涉及给出混合动机的明确结构。在以前的工作中，PI利用代数簇的高次同伦来构造这样的动机。这个项目的一个明确结果将是计算来自这些更高同伦动机的周期和Galois模。除了这些项目外，PI还计划使用K-理论来研究代数圈的形变理论以及基场扩张下的代数圈的行为。最后一个项目尝试将代数几何中的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。

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.