Distinct distances on the plane

Abstract: Given N distinct points on the plane, what is the minimal number of distinct distances between them? This problem was posed by Paul Erdos in 1946 and essentially solved by Guth and Katz in 2010.

We are going to consider a continuous analogue of this problem, the Falconer distance problem. Given a set $E$ of dimension $s>1$, what can we say about its distance set $\Delta(E)=\{ |x-y|: x,y\in E\}$? Falconer conjectured in 1985 that $\Delta(E)$ should have positive Lebesgue measure. In the recent years, people have attacked this problem in different ways (including geometric measure theory, Fourier analysis, and combinatorics) and made some progress for various examples and for some range of $s$.

analysis of PDEsclassical analysis and ODEsfunctional analysismetric geometry

Audience: researchers in the topic

HA-GMT-PDE Seminar

Series comments: Description: A senior graduate student/postdoc series in harmonic analysis, geometric measure theory, and partial differential equations.

Meetings will be weekly, and usually will occur on Monday, with some exceptions. For each talk, the Zoom link is made available in the website too. Contact Bruno Poggi at poggi008@umn.edu if you would like to subscribe to the seminar mailing list.