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.
