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. This is joint work with Andrew Marks.
2018-11-22 14:30 to 15:30
Manchester Building (Hall 2), Hebrew University Jerusalem