# Distinct distances on the plane (Frontiers in Mathematics)

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? Falconer conjectured in 1985 that delta(E) should have positive Lebesgue measure. In 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. In this talk, we will discuss these results, methods and the connection between the discrete and continuous version. (This is the first talk in the Frontiers in Mathematics Distinguished Lecture Series. See also the second talk in the series, on Friday, September 11 at noon.)