Matt Foreman : Hilbert's 10th problem for measure preserving transformations.

Posted by Filippo Calderoni , part of the Logic Seminar.

At

Feb. 25, 2020, 3 p.m.

In

427 SEO

Abstract

In his seminal 1932 paper von Neumann asked whether it is possible to tell whether time is going forwards or time is going backward from the statistics of a measure preserving system. It was not until 1941 that Anzai produced the first example of a measure preserving system where $T$ is not isomorphic to $T^{-1}$. In this talk I show that there is a one-to-one, primitive recursive map $F$ that maps Gödel numbers of $\Pi^0_1$-sentences to recursive, measure preserving, invertible, ergodic diffeomorphisms of the 2-torus such that: \[\phi \mbox{ is true if and only if } F(\phi) \mbox{ is isomorphic to }F(\phi)^{-1}.\] As corollaries there are non-isomorphic ergodic measure preserving diffeomorphism of the torus: - $T_{RH}$ such that $T_{RH}\cong T_{RH}^{-1}$ if and only if the Riemann Hypothesis is true. - $T_{GC}$ such that $T_{GC}\cong T_{GC}^{-1}$ if and only if Goldbach's Conjecture is true. - $T_{ZFC}$ such that $T_{ZFC}\cong T_{ZFC}^{-1}$ if and only if ZFC is consistent. - $T$ such that $T\cong T^{-1}$ but this is independent of ''ZFC + there is a supercompact cardinal."