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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.