Special Colloquium
Spencer Unger
Tel Aviv University
A constructive solution to Tarski's circle squaring problem
Abstract: In 1925, Tarski asked whether a disk in R^2 can be partitioned into finitely many pieces which can be rearranged by isometries to form a square of the same area. The restriction of having a disk and a square with the same area is necessary. In 1990, Laczkovich gave a positive answer to the problem using the axiom of choice. We give a completely explicit (Borel) way to break the circle and the square into congruent pieces. This answers a question of Wagon. Our proof has three main components. The first is work of Laczkovich in Diophantine approximation. The second is recent progress in a program of descriptive set theory to understand the complexity of actions of amenable groups. The third is the study of flows in networks.
Monday December 4, 2017 at 3:00 PM in SEO 636