Leírás
We go through the history of measurable perfect matchings from the Banach-Tarski paradox via circle-squaring and the report of recent progress. We show that the Hall condition is essentially sufficient in the hyperfinite, one-ended, bipartite case. This allows us to characterize bipartite Cayley graphs with factor of iid perfect matchings extended the Lyons-Nazarov theorem. We apply these to Gardner's conjecture for uniformly distributed sets, to balanced orientations, and to get new, simple proofs of the measurable circle-squaring. We prove the analogous theorems in the context of rounding flows, too. On the other hand, we construct $d$-regular treeing (for every $d>2$) without measurable perfect matching. Partially joint work with Matt Bowen and Marcin Sabok.
Zoom link: https://zoom.us/j/93746696898?pwd=b1J2MnEwMVdDVElPUFRkYWdtVXdWdz09
Meeting ID: 937 4669 6898
Passcode: 280561