Seminar| Institute of Mathematical Sciences
Time: Wednesday, January 17th , 2024 , 15:00-16:00
Speaker: Zhipeng Lu, Shenzhen MSU-BIT University
Abstract: In this series of lectures, I will try to cover as much materials as I personally know of the current state of the art of the distinct distances problem and relatives in incidence geometry.
First, we will investigate the historical background since Erd\H{o}s, who conjectured a lower bound for the case of Euclidean spaces based on results from number theoretic results on Euclidean lattices. This should include the work of Landau, Ramanujan, Conway-Sloane, Moree-Osburn etc. in the aspect of lattices, and the work of Szemer\'{e}di-Trotter, Sz\'{e}kely, Pach-Sharir, Solymosi-Vu etc. in the aspect of incidence geometry.
Second, I will introduce the SOTA work of Guth-Katz based on the framework of Elekes-Sharir. The focus should be on their polynomial partitioning method and how they breakthrough the classical incidence bound of Szemer\'{e}di-Trotter in the case of $\mathbb{R}^3$.
At the end, if time permits, I will show how the polynomial partitioning method revises derivation of classical results of incidence geometry and new ones. Moreover, I would like to introduce our own work and perspective on the problem on more general surfaces and lattices.