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Place:\u00a0Room 614 in the Education & Sciences Building<\/p>\n
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Abstract<\/u><\/strong>:<\/u><\/strong><\/p>\n 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.\u00a0 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.<\/p>\n Tea will be served before the talk (at 13:50<\/strong>).<\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" Place:\u00a0Room 614 in the Education & Sciences Building 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…
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