# Reductions for Distinct Distances Problems in R^d

### Wednesday, December 21st, 2016, 16:10

### Schreiber 309

### Reductions for Distinct Distances Problems in R^d

### Adam Sheffer, CalTech

### Abstract:

ErdÅ‘s' distinct distances problem asks for the minimum number of distances that can be spanned by a set of n points in R^d. A few years ago Guth and Katz almost completely settled the planar case of this problem. However, the distinct distances problem remains open for any dimension d>2.

The Guth-Katz proof was based on reducing the distinct distances problem into an incidence problem (adapting previous ideas of Elekes and Sharir). Simple attempts to extend this approach to higher dimensions lead to complicated incidence problems that are hard to study. In this talk we will present a more involved reduction that leads to simpler incidence problems. This reduction is based on the Lie group Spin(n).

Joint work with Sam Bardwell-Evans.