CAREER: Integral Points on Varieties and Related Tools and Topics

Michigan State University, East Lansing MI

Investigators

Abstract

Understanding the set of integer solutions to a system of polynomial equations is one of the oldest and most basic problems in mathematics. The subject is also closely related to topics in computer science. Fundamental questions include determining whether or not the set of integral solutions is finite, and if so how to explicitly compute the set of such solutions. The introduction of powerful geometric ideas and concepts into the subject has led to major advances and a wide-ranging and deeper understanding of the basic problems. From this geometric viewpoint, this research project aims to enhance our understanding of these difficult and fundamental problems by substantially extending the known results, techniques, and tools. In addition to research activities, the PI will organize and initiate student-faculty research projects for a select group of undergraduate students at Michigan State University. The research projects will be enhanced by an undergraduate colloquium series and funded conference travel for the undergraduate students. The PI will continue his close involvement with graduate education and mentor graduate students and postdocs, suggesting problems related to the proposed research strands. The PI will study several interrelated problems revolving around the study of integral points on varieties. First, the PI will examine the problem of generalizing Siegel's theorem for integral points on curves to higher-dimensional varieties, along the lines of the PI's conjectures. In a second project, the PI will study integral points of bounded degree on curves. Then, in a different direction, aspects of the problem of effectively computing integral points on varieties will be studied, both for curves and higher-dimensional varieties. Fundamental tools for studying integral (or rational) points on varieties come from the subject of Diophantine approximation. One such key tool is Schmidt's Subspace Theorem. The PI will continue his recent work on proving generalizations of Schmidt's theorem, with a Schmidt-Wirsing type conjecture for algebraic points as an ultimate goal. The PI will study several "uniform boundedness" conjectures in arithmetic dynamics, a subject well-suited for applications of the above techniques and results. Lastly, by work of Vojta and others, there exists a surprising correspondence between statements and theorems in Diophantine approximation and statements and theorems in Nevanlinna theory. The PI plans to study the analogues of the above problems in Nevanlinna theory, with the hope of enriching both subjects.