Problems in the Geometry of Numbers and Diophantine Analysis

A fundamental and central question in mathematics, going back to antiquity, concerns understanding integer solutions to polynomial equations. This question has been a source of inspiration in modern mathematics, as the techniques used to study this question have helped shape the foundation of several areas such as number theory, algebra, and analysis. This work will bring new perspectives to the field by studying various families of polynomial equations and by finding innovative applications in the rich area of number theory. The project will also enhance the training of graduate students and support mentoring activities and research opportunities for undergraduate students. The principal investigator aims to utilize tools from classical Diophantine analysis and modern applications of analytic number theory and arithmetic geometry to study basic arithmetic structures. This work will combine ideas from the geometry of numbers with both algebraic and analytic tools to study problems such as representation of integers by binary forms and orders in number fields, as well as various problems about heights in number fields. 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.