Pure Math Colloquium

Thomas Parker, Michigan State University

“Paths of holomorphic curves and the GV conjecture”

This is a talk on counting solutions of non-linear elliptic PDEs. After presenting the basic idea, I will explain, from two completely different perspectives, how the search for simple examples leads – rather surprisingly – to problems considered by string theorists. Following this path, I will explain a 1998 conjecture of physicists R. Gopakumar and C. Vafa about spaces called Calabi-Yau 3-folds, and give a geometric interpretation. At the end, I will outline a geometric proof, with pictures, of the GV conjecture (joint work with E. Ionel) based on the idea of using deformations to count the contributions of “clusters” of curves.