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

在数学上，本研究项目属于代数几何领域。代数几何的基本问题是多项式方程的解所给出的几何对象的研究和分类。本研究项目更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悠久的历史，可以追溯到丢番图，他第一个研究了今天被称为丢番图方程的解。由于它们的多样性，对这些方程的研究已经在生物学、化学、宇宙学和计算机加密等领域得到了许多应用。对这种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关联起来。这种不变量的一个例子是一组复数，通常称为周期，与给定的多项式系统相关联。本研究项目的中心焦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，为培养下一代研究人员做出贡献。在这个项目中，首席研究员打算研究由代数变分理论产生的周期。周期的研究实际上是动机论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处理这些问题。在第一个项目中，PI研究了带连接的向量束的周期。当比较两个不同上同群上的有理结构时，就会产生周期。这个项目试图推广到更高维度的结果在不规则连接的周期曲线的情况下。第二个项目涉及给出混合泰特动机的明确结构。在以前的工作中，PI从代数变体的高同伦构造了这样的动机。这个项目的一个明确的结果将是计算周期和伽罗瓦模来自这些更高的同伦动机。除了这些项目外，PI还计划利用k理论研究代数循环的变形理论以及代数循环在基场扩展下的行为。最后一个项目试图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.