This lecture will focus on combinatorial geometry theorems concerning finite point sets. It will center on proving the existence of small subsets such that, when enclosed by a convex shape (e.g., a ball), a significant proportion of the entire point set is also guaranteed to be captured within that shape.

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.

This lecture focuses on promoting the spirit of "Two Bombs, One Satellite," a symbol of dedication and innovation in China's history, which has significantly influenced the development of the country's high-tech sectors. The session will explore how this pioneering spirit can continue to inspire current technological advancements and strategic achievements in China.