Andy Miller, Ph.D., of the University of Oklahoma, will present at the Mathematics Colloquium at 4:10 p.m. on Friday, Feb. 9, in Room 202, Ritter Hall.
Miller will present, “The Arithmetic of Right-Angled Coxeter Groups.” Each Coxeter group G with n generators has an n-dimensional representation which is orthogonal with respect to a canonical bilinear form B(x,y). The associated quadratic form B(x,x) will be Diophantine when G is right-angled. By construction, for each integer D, the Coxeter group G acts on the set Sol(D) of primitive integer solutions to B(x,x)=D and this set admits a partial ordering whose components coincide with the orbits of the G action.
All of these constructs can be directly defined simply in terms of the Coxeter graph associated with G. The talk will focus on describing some low dimensional cases of this construction. Examples tend to have interesting geometric interpretations involving such concepts as Apollonian circle packings, Pythagorean triples, and Conway's pictorial realizations of binary forms and their rivers.
Refreshments at 3:30pm in the lobby of Ritter Hall. The talk to follow.