Lecture Content：

Consider surfaces of genus at least 2. In the study of various properties of such surfaces, closed curves play an important role. When attempting to characterize closed curves on surfaces, we typically consider intersection numbers between closed curves. For closed curves without self-intersections, work by Dehn and Thurston shows that any such closed curve is determined, up to homotopy, by its intersection numbers with a finite number of preselected closed curves. However, this approach cannot be directly applied to curves with self-intersections. When trying to describe all closed curves on a surface, for any natural number k, there exist non-homotopic closed curves that share the same intersection numbers with any closed curve having k self-intersections. This phenomenon gives rise to the k-equivalence relation among curves on surfaces. In this talk, I will introduce and discuss the k-equivalence relation for closed curves on surfaces. This presentation is based on collaborative work with Hugo Parlier.

Speaker Introduction:

Binbin Xu is an Associate Professor at Nankai University, with research interests in geometric topology. His relevant papers have been published in prestigious journals including Ergodic Theory and Dynamical Systems and Transactions of the American Mathematical Society.