2021 NORTH AMERICAN ANNUAL MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC

Abstract

of the invited 32nd Annual Gödel Lecture MATTHEW FOREMAN, Gödel diffeomorphisms. Department of Mathematics, University of California, Irvine, CA, USA. E-mail: mforeman@math.uci.edu Motivated by problems in physics, solutions to differential equations were studied in the late 19th and early 20th centuries by people like Birkhoff, Poincaré and von Neumann. Poincaré’s work was described by Smale in the 1960s as the qualitative study and von Neumann’s own description was the study of the statistical aspects of differential equations. The explicit goal was to classify this behavior. A contemporaneous problem was whether time forwards could be distinguished from time backwards. The modern formulation of these problems is to classify diffeomorphisms of smooth manifolds up to topological conjugacy and measure isomorphism and to ask, for a given diffeomorphism, whether T ∼= T –1. Very significant progress was made on both classes of problems, in the first case by people like Birkhoff, Morse and Smale and in the second case by Birkhoff, Poincare, von Neumann, Halmos, Kolmogorov, Sinai, Ornstein and Furstenberg. This talk applies techniques developed by Kechris, Louveau and Hjorth to these problems to show that the relevant equivalence relations are complete analytic. Moreover the collection of T that are measure theoretically isomorphic to their inverses is also complete analytic. Finally, the whole story can be miniaturized to show that the collection of diffeomorphisms of the two-torus that are measure theoretically isomorphic to their inverses is Π1-hard. 30