Undergraduate Math Club Seminar

We return to the Erdős Distinct Distances Problem in R^2, as introduced in the earlier talk by Felix Weilacher: what is the least number of distinct distances a set of n points in R^2 can determine? We will begin by looking at an outline of the proof given by Guth and Katz of the best known lower bound, which relies heavily on turning the problem of counting repeated distances into one of counting incidences of lines in R^3. We will then look at work that is being done with Adam Sheffer to extend this proof to a general analogue of the original problem, namely the Erdős Distinct Distances Problem in the d-sphere. In particular, we will look in detail at the subsets T_{AB} of Spin(d) consisting of transformations bringing a given point A in S^d to a given point B, and we will look at how one can map these subsets onto d(d-1)/2-dimensional planes in R^(d(d+1)/2).