NSF Award
2316946
Integer and rational solutions to polynomial equations have intrigued humans for thousands of years, from the triangle side lengths considered by the Pythagoreans to Fermat’s Last Theorem, which was only proven at the end of the twentieth century. There is a deep link between the rational solutions to a polynomial equation and geometry of the shape formed by the complex numbers that satisfy the equation, which for a widely-studied class of equations looks like a many-holed donut. This link is perhaps best illustrated by Faltings’ Theorem, a seminal modern result which shows that the number of holes in the shape defined by the complex number solutions imposes strong constraints on the number of rational solutions to the equation. This grant investigates another aspect of this relationship, exploring how the geometry relates to the rate at which new solutions to the equation are found by augmenting the potential solutions beyond the rational numbers. In addition, the PI will host career panels, workshops, and colloquia which will aid in recruitment and retention in the sciences of our students, the majority of whom are underrepresented. The PI will also run an art prize to foster inclusion of the arts in mathematics education, as well as visibility, representation, and inclusion of underrepresented math students. <br/><br/>The projects in this grant build on previous work on a geometric facet of the inverse Galois problem. In their program on Diophantine stability, Mazur and Rubin suggest studying a curve C by understanding the set of number fields generated over the rationals by a single point of C; in particular, they ask to what extent the set of such field extensions determines the curve. The projects proposed in this grant address an inverse direction of this, how the geometry of the curve influences the aforementioned set of field extensions. The proposed projects address the size of this set in terms of the degree and discriminant of the fields from an arithmetic statistics perspective, measuring the asymptotic growth of this set as a function of the discriminant bound. Additional directions which will be explored include restricting this count to field extensions with specified Galois groups, and field extensions which increase the rank of the Jacobian. The projects expand on ongoing work by the PI and her collaborators, using tools such as geometry of numbers, Hilbert irreducibility, Newton polygons, and linear optimization.<br/><br/>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.
