Greatest Common Divisors, Integral Points, and Diophantine Approximation

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The project studies topics at the core of arithmetic and number theory. One of the most basic objects in mathematics is the greatest common divisor of two integers. The project will investigate generalizations and analogues of recently developed inequalities for greatest common divisors, and their connections with Vojta’s conjecture, a central and far-reaching conjecture. Another fundamental and important question, going back to antiquity, concerns understanding integer solutions to polynomial equations. The work will bring new ideas and perspectives to this important question, including research toward methods allowing one to algorithmically compute all integer solutions to large classes of equations. A fundamental tool for studying such equations comes from the subject of Diophantine approximation, which in its most basic form studies how well a real number can be approximated by rational numbers. The project will study various generalizations of one of the primary results in this subject, Schmidt’s subspace theorem. This research has close connections to and consequences for diverse areas of mathematics beyond number theory, including complex analysis and geometry. Additionally, the project will support a wide range of mentoring activities and research opportunities, involving the training of undergraduate students, graduate students, and postdoctoral researchers.The research supported by this award will involve the study of several questions revolving around greatest common divisors, integral points, Diophantine approximation, and their interrelations. The first set of projects center on the investigator's recent higher-dimensional generalization of results on greatest common divisors of polynomials evaluated at S-units. The project will study generalizations and extensions related to Vojta's conjecture, analogues in function fields, including applications, and a novel approach to analogous problems for abelian varieties. A second set of projects will focus on integral points on varieties. First, the project will continue work on aspects of Siegel’s theorem for integral points of bounded degree on curves. Second, the investigator will study an approach to effectively proving Siegel’s theorem for genus two curves. Fundamental tools for studying these problems come from the subject of Diophantine approximation, where Schmidt's subspace theorem is a central result.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.