3:00–4:00 pm Eckhart Hall, Room 202
Title: Squaring the circle with few and simple pieces
Abstract: In 1925 Tarski posed the problem of whether a solid two-dimensional square can be partitioned into finitely many pieces which can be rearranged by isometries to form a disk of the same area. In 1990 Laczkovich positively answered Tarski's question using the axiom of choice. Joint with Spencer Unger, we show that there is an explicitly definable Borel solution to this problem. Our solution uses recent progress in two research programs in mathematical logic and descriptive set theory. First, the theory of descriptive graph combinatorics, which studies definable/measurable solutions to combinatorial problems on infinite graphs. At their combinatorial core, equidecomposition problems like Tarski's circle squaring problem are perfect matching problems, and progress on the theory of definable matchings is central to our solution. Second, the study of the descriptive-set-theoretic complexity of actions of countable groups. In particular, our proof uses a recent result of Gao-Jackson-Krohne-Seward on the hyperfiniteness of Borel actions of abelian groups.
At the end of the talk, we will discuss some recent progress giving better upper bounds for the number of pieces needed to square the circle, using results from diophantine approximation. Diophantine approximation studies how well irrational numbers can be approximated by rationals. A landmark result in this area is Roth's 1955 theorem on diophantine approximation of algebraic irrationals. It is an open problem if the constants in Roth's theorem can be computed effectively, but partial results on this problem can be used to improve the upper bound on the number of pieces used to square the circle, via the quantitative bounds they give on the ergodic theorem for translation actions on the torus.