A quantum approach to symbolic dynamics

ended

This project in pure mathematics aims to solve problems in symbolic dynamics (a mature branch of dynamics with important connections to coding theory, information theory, and operator algebras) using tools from the modern theories of quantum information and quantum groups. The central challenge dating back to at least the 1970s is to classify symbolic dynamical systems. My proposal pioneers a highly interdisciplinary and novel approach using the uniquely quantum phenomena of ‘entanglement’ and ‘noncommutativity’ as they appear in quantum information/quantum groups. The main objective is therefore to establish a fruitful connection between symbolic dynamics, quantum information theory and quantum groups to the benefit of all three communities. The proposal is based around two central open problems: how do we classify symbolic systems, and what are the symmetries of symbolic systems? A novel approach to these problems builds on the recent advances made in part by the supervisor that connect graph theory with quantum information/quantum groups. I introduce a new nonlocal game (the Conjugacy Game) that will define ‘quantum conjugacies’, and a notion of quantum automorphism that will define ‘quantum symmetries’ of symbolic systems. This marks the first interaction between quantum information/quantum groups and symbolic dynamics. The ideas are therefore very timely, will have high impact, and set a precedent for future uses of quantum mathematics in other domains and specifically other classes of dynamics (e.g. multidimensional subshifts and algebraic actions). The fellowship will strengthen my academic career via the opportunity to learn new tools in quantum mathematics and apply them to solve important problems in dynamics. It will also allow me to establish new collaborations with leading experts in dynamics and quantum information/quantum groups that will play a key role in my goal of obtaining a permanent academic job and an investigator grant to build a research g