Matthew Foreman
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View article: The isomorphism problem for group actions
The isomorphism problem for group actions Open
We discuss the isomorphism problem for ergodic actions of locally compact groups. In particular we show that the conjugacy relation is not Borel for ergodic measure preserving actions of indicable groups.
View article: Where do you put a telescope? How do you understand Covid concentrations?
Where do you put a telescope? How do you understand Covid concentrations? Open
This note discusses dynamical systems-systems that evolve through time. We start with two contemporary examples illustrating the qualitative and the quantitative behavior of dynamical systems. These are two broad categories, usually called…
View article: Open questions in descriptive set theory and dynamical systems
Open questions in descriptive set theory and dynamical systems Open
This file is composed of questions that emerged or were of interest during the workshop "Interactions between Descriptive Set Theory and Smooth Dynamics" that took place in Banff, Canada on 2022.
View article: A Preprinting Protocol for Designing a 3D-Printed Prosthesis for a Young Child With an Upper Extremity Difference
A Preprinting Protocol for Designing a 3D-Printed Prosthesis for a Young Child With an Upper Extremity Difference Open
Date Presented 04/01/2022 Upper extremity differences can produce challenges in childhood occupations. 3D printing is emerging in rehabilitation science for the creation of assistive technology because of its customizability and cost-effec…
View article: Anti-classification results for smooth dynamical systems. Preliminary Report
Anti-classification results for smooth dynamical systems. Preliminary Report Open
The paper considers the equivalence relation of conjugacy-by-homeomorphism on diffeomorphisms of smooth manifolds. In dimension 2 and above it is shown that there is no Borel method of attaching complete numerical invariants. In dimension …
View article: The complexity of the Structure and Classification of Dynamical Systems
The complexity of the Structure and Classification of Dynamical Systems Open
This is an expository paper about the Borel complexity of structure and classification theorems. It sorts several classical problems relative to known benchmarks of complexity. As a corollary various problems proposed by people such as von…
View article: Measure preserving diffeomorphisms of the torus are unclassifiable
Measure preserving diffeomorphisms of the torus are unclassifiable Open
In 1932 von Neumann proposed classifying the statistical behavior of differentiable systems. In modern language this is interpreted as classifying diffeomorphisms of compact manifolds up to measure isomorphism. This paper proves that this …
View article: Naive Descriptive Set Theory
Naive Descriptive Set Theory Open
The paper is a naive introduction to descriptive set theory. It is aimed mathematicians without a background in logic. The goal is to provide the basic facts used for applications of descriptive set theory to other areas of mathematics, pa…
View article: Odometer Based Systems
Odometer Based Systems Open
Construction sequences are a general method of building symbolic shifts that capture cut-and-stack constructions and are general enough to give symbolic representations of Anosov-Katok diffeomorphisms. We show here that any finite entropy …
View article: Godel Diffeomorphisms
Godel Diffeomorphisms Open
A basic problem in smooth dynamics is determining if a system can be distinguished from its inverse, i.e., whether a smooth diffeomorphism $T$ is isomorphic to $T^{-1}$. We show that this problem is sufficiently general that asking it for …
View article: Games with Filters
Games with Filters Open
This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call \emph{Welch games}. Player II having a winning strategy in the Welch game of length $\omega$ on $\kappa$ is …
View article: Games with Filters
Games with Filters Open
This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call \emph{Welch games}. Player II having a winning strategy in the Welch game of length $ω$ on $κ$ is equivalent…
View article: Rank-One Transformations, Odometers, and Finite Factors
Rank-One Transformations, Odometers, and Finite Factors Open
In this paper we give explicit characterizations, based on the cutting and spacer parameters, of (a) which rank-one transformations factor onto a given finite cyclic permutation, (b) which rank-one transformations factor onto a given odome…
View article: From odometers to circular systems: A global structure theorem
From odometers to circular systems: A global structure theorem Open
The main result of this paper is that two large collections of ergodic measure preserving systems, the Odometer Based and the Circular Systems have the same global structure with respect to joinings. The classes are canonically isomorphic …
View article: What is ... a Borel reduction?
What is ... a Borel reduction? Open
Borel reductions provide a method of proving that certain problems are impossible using countably infinitary techniques based on countable information and provide a hierarchy of difficulty for classification problems.This is illustrated wi…
View article: Measure Preserving Diffeomorphisms of the Torus are Unclassifiable
Measure Preserving Diffeomorphisms of the Torus are Unclassifiable Open
The isomorphism problem in ergodic theory was formulated by von Neumann in 1932 in his pioneering paper Zur Operatorenmethode in der klassischen Mechanik (Ann. of Math. (2), 33(3):587--642, 1932). The problem has been solved for some class…