In 1946, Erdős proposed the distinct distances problem: what is the least number of distances a set of N points in R^d spans? He computed that integer grids give an upper bound of O(N/(log N)^(1/2)) for d=2 and

N^(2/d) for d at least 3. For d=2, a nearly matching lower bound cN/log N was obtained much later in 2010 by Guth and Katz, who introduced influential tools such as polynomial partitioning. In this talk, I will outline their proof and the extra twists needed to prove a lower bound of

N^(2/3-o(1)) for d=3. Time permitting, I will briefly talk about the recent resolution of the distinct distances problem for d>=3 by OpenAI. 

This is based on joint work with Jonathan Tidor and Dmitrii Zakharov.