There exists a weakly mixing billiard in a polygon

This main result of this talk is that there exists a billiard flow in a polygon that is weakly mixing with respect to Lebesgue measure on the unit tangent bundle to the billiard. This strengthens Kerckhoff, Masur and Smillie's result that there exists ergodic billiard flows in polygons. The existence of a weakly mixing billiard follows, via a Baire category argument, from showing that for any translation surface the product of the flows in almost every pair of directions is ergodic with respect to Lebesgue measure. This in turn is proven by showing that for every translation surface the flows in almost every pair of directions do not share non-trivial common eigenvalues. This talk will explain the problem, related results, and approach. The talk will not assume familiarity with translation surfaces. This is joint work with Giovanni Forni.