Let F(x, y) be a binary form of degree at least 3 which is irreducible over the rationals. For any fixed non-zero integer m, Thue showed that the equation F(x, y) =m has at most finitely many solutions in integers x, y. I will discuss some fundamental work on bounding the number of integral solutions of such equations, which I learned during my graduate study at UBC. Then I will share some of my related results, including some current work in progress which aims to improve upper bounds on the number of solutions of index (and discriminant) form equations in quartic number fields.
Shabnam Akhtari is a number theorist at Pennsylvania State University. She studied mathematics at Sharif University of Technology in Tehran, Simon Fraser University, and University of British Columbia. She received her Ph.D at UBC under the supervision of Professor Mike Bennett. After graduation in 2008, she was a postdoctoral fellow at Max Planck Institute in Bonn, and CRM, Montreal. She is now a professor of Mathematics at Pennsylvania State University. Before joining Penn State, she was a faculty member at University of Oregon.