Universality in computable dynamical systems: Old and new

TL;DR

This survey explores the deep connections between computability and dynamical systems, highlighting recent advances in Turing universality in fluid dynamics and topological theories, and discusses open problems in the field.

Contribution

It introduces modern perspectives on representing computability through dynamical systems, including Turing universality in fluid dynamics and topological field theories.

Findings

Review of recent work on dynamical universality

Introduction of Turing universality in fluid dynamics

Discussion of open problems in computable dynamical systems

Abstract

The relationship between computational models and dynamics has captivated mathematicians and computer scientists since the earliest conceptualizations of computation. Recently, this connection has gained renewed attention, fueled by T. Tao's programme aiming to discover blowing-up solutions of the Navier-Stokes equations using an embedded computational model. In this survey paper, we review some of the recent works that introduce novel and exciting perspectives on the representation of computability through dynamical systems. Starting from dynamical universality in a classical sense, we shall explore the modern notions of Turing universality in fluid dynamics and Topological Kleene Field Theories as a systematic way of representing computable functions by means of dynamical bordisms. Finally, we will discuss some important open problems in the area.