Speaker: Prof. Yumeng Ou

Abstract: Given a compact set E in the Euclidean space and consider different sets of point configurations generated by E (e.g. distinct distances, chains, trees, or triangles contained in E). It is natural that the larger the set E is, the larger all these sets of point configurations should be. There are many different questions one can ask concerning how exactly their sizes depend on that of E, and our main topic of the talk, Falconer type problems, is such an example. Thanks to the recent developments in the field, some of the point configuration problems got almost trivialized, but some others remain challenging. I’ll introduce some basic ideas (mostly from harmonic analysis and geometric measure theory) behind these problems. No prior knowledge required.