The PIMS Postdoctoral Fellow Seminar: Gregory Knapp

In 1909, Thue proved that when $F(x,y) \in \mathbb{Z}[x,y]$ is irreducible, homogeneous, and has degree at least 3, the inequality $|F(x,y)| \leq h$ has finitely many integer-pair solutions for any positive $h$. Because of this result, the inequality $|F(x,y)| \leq h$ is known as Thue’s Inequality and much work has been done to find sharp bounds on the number of integer-pair solutions to Thue’s Inequality. In this talk, I will describe different techniques used by Akhtari and Bengoechea; Baker; Bennett; Mueller and Schmidt; Saradha and Sharma; and Thomas to make progress on this general problem. After that, I will discuss some improvements that can be made to a counting technique used in association with "the gap principle’’ and how those improvements lead to better bounds on the number of solutions to Thue’s Inequality.

Speaker biography: Greg grew up in Centerville, Ohio in the US and graduated with his bachelor’s and master’s degrees from Case Western Reserve University. His master’s thesis was completed under the supervision of Colin McLarty and covered Minkowski’s Linear Forms Theorem in Elementary Function Arithmetic, a subsystem of first-order arithmetic. Greg then went to the University of Oregon for his PhD, and he completed his dissertation on the topic of polynomial root distribution and Thue equations under the supervision of Shabnam Akhtari. Greg is now a PIMS postdoctoral fellow at the University of Calgary where he is working with Khoa Nguyen. Outside of math, he enjoys rock climbing, disc golf, learning new hobbies, and playing all types of games.

This event is part of the Emergent Research: The PIMS Postdoctoral Fellow Colloquium Series.