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Research

I am broadly interested in studying the evolution and structure of topological dynamical systems. Most of my work centres on the connection between dynamical systems and associated operator algebras.

Publications and preprints

Sturmian subshifts and their C*-algebras

Journal of Operator Theory, to appear.

This paper investigates the structure of C*-algebras built from one-sided Sturmian subshifts. They are parametrised by irrationals in the unit interval and built from a local homeomorphism associated to the subshift. We provide an explicit construction and description of this local homeomorphism. The C*-algebras are *-isomorphic exactly when the systems are conjugate, and they are Morita equivalent exactly when the defining irrationals are equivalent (this happens precisely when the systems are flow equivalent). Using only elementary dynamical tools, we compute the dynamic asymptotic dimension of the (groupoid of the) local homeomorphism to be one, and by a result of Guentner, Willett, and Yu, it follows that the nuclear dimension of the C*-algebras is one.


Conjugacy of local homeomorphisms via groupoids and C*-algebras

with Becky Armstrong, Toke Meier Carlsen, and Søren Eilers, Ergodic Theory and Dynamical Systems, to appear.

We investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterise topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and \(C^*\)-algebras. This significantly generalises recent work of Matsumoto and of the second- and third-named authors.


Balanced strong shift equivalence, balanced in-splits, and eventual conjugacy

Ergodic Theory and Dynamical Systems, 2022.

We introduce the notion of balanced strong shift equivalence between square nonnegative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a sequence of out-splits and balanced in-splits; the latter move being a variation of the classical in-split move introduced by Williams in his study of shifts of finite type. We also relate one-sided eventual conjugacies to certain block maps on the finite paths of the graphs. These characterizations emphasize that eventual conjugacy is the one-sided analog of two-sided conjugacy.


\(C^*\)-algebras, groupoids and covers of shift spaces

with Toke Meier Carlsen, Transactions of the American Mathematical Society, 2020.

To every one-sided shift space \(\mathsf{X}\) we associate a cover \(\tilde{\mathsf{X}}\), a groupoid \(\mathcal{G}_\mathsf{X}\) and a \(C^*\)-algebra \(\mathcal{O}_\mathsf{X}\). We characterize one-sided conjugacy, eventual conjugacy and (stabilizer-preserving) continuous orbit equivalence between \(\mathsf{X}\) and \(\mathsf{Y}\) in terms of isomorphism of \(\mathcal{G}_\mathsf{X}\) and \(\mathcal{G}_\mathsf{Y}\), and diagonal-preserving \(^*\)-isomorphism of \(\mathcal{O}_\mathsf{X}\) and \(\mathcal{O}_\mathsf{Y}\). We also characterize two-sided conjugacy and flow equivalence of the associated two-sided shift spaces \(\Lambda_\mathsf{X}\) and \(\Lambda_\mathsf{Y}\) in terms of isomorphism of the stabilized groupoids \(\mathcal{G}_\mathsf{X} \times \mathcal{R}\) and \(\mathcal{G}_\mathsf{Y} \times \mathcal{R}\), and diagonal-preserving \(^*\)-isomorphism of the stabilized \(C^*\)-algebras \(\mathcal{O}_\mathsf{X} \otimes \mathbb{K}\) and \(\mathcal{O}_\mathsf{Y} \otimes \mathbb{K}\). Our strategy is to lift relations on the shift spaces to similar relations on the covers.

Restricting to the class of sofic shifts whose groupoids are effective, we show that it is possible to recover the continuous orbit equivalence class of \(\mathsf{X}\) from the pair \( (\mathcal{O}_\mathsf{X}, C(\mathsf{X}))\), and the flow equivalence class of \(\Lambda_\mathsf{X}\) from the pair \((\mathcal{O}_\mathsf{X} \otimes \mathbb{K}, C(\mathsf{X}) \otimes c_0)\). In particular, continuous orbit equivalence implies flow equivalence for this class of shift spaces.


\(C^*\)-simplicity and representations of topological full groups of groupoids

with Eduardo Scarparo, Journal of Functional Analysis, 2019.

Given an ample groupoid \(G\) with compact unit space, we study the canonical representation of the topological full group \([[G]]\) in the full groupoid \(C^*\)-algebra \(C^*(G)\). In particular, we show that the image of this representation generates \(C^*(G)\) if and only if \(C^*(G)\) admits no tracial state. The techniques that we use include the notion of groups covering groupoids. As an application, we provide sufficient conditions for \(C^*\)-simplicity of certain topological full groups, including those associated with topologically free and minimal actions of nonamenable and countable groups on the Cantor set.


Cuntz-Krieger algebras and one-sided conjugacy of shifts of finite type and their groupoids

with Toke Meier Carlsen, Journal of the Australian Mathematical Society, 2020.

A one-sided shift of finite type \((\mathsf{X}_A, \sigma_A)\) determines on the one hand a Cuntz–Krieger algebra \(\mathcal{O}_A\) with a distinguished abelian subalgebra \(\mathcal{D}_A\) and a certain completely positive map \(\tau_A\) on \(\mathcal{O}_A\). On the other hand, \((\mathsf{X}_A, \sigma_A)\) determines a groupoid \(\mathcal{G}_A\) together with a certain homomorphism \(\varepsilon_A\) on \(\mathcal{G}_A\). We show that each of these two sets of data completely characterizes the one-sided conjugacy class of \(\mathsf{X}_A\). This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.


Research talks

Invited talks

  • A meditation on flow, OANCG seminar, University of Wollongong (May 2022).
  • Ideal structure of C*-algebras, AMS Western Sectional Meeting (May 2022).
  • Algebraic invariants between symbolic dynamics and C*-algebras, Aberdeen Algebra Seminar (March 2022).
  • Flow equivalence and C*-algebras, joint KU/SDU operator algebra seminar, University of Southern Denmark, Denmark (Mar. 2022).
  • Orbit equivalence, flow equivalence, and C*-algebras, NYC Noncommutative Geometry seminar (online) (Sep. 2021). See slides.
  • Conjugacy of local homeomorphisms, Operator algebra and noncommutative geometry seminar (online), University of Wollongong, Australia (Apr. 2021).
  • The conjugacy problem for subshifts, UK Operator algebra Seminar (online), United Kingdom (Mar. 2021).
  • OA aspects of the Williams conjecture, Operator algebra Seminar (online), University of Southern Denmark/Copenhagen, Denmark (Feb. 2021).
  • Fine structure of C*-algebras associated to topological dynamics, Abend seminar (online), University of Western Sydney (Jun. 2020).
  • Fine structure of C*-algebras associated to topological dynamical systems, Seminar, Queen Mary University, England (Nov. 2019).
  • Fine structure of C*-algebras associated to topological dynamical systems, C*-algebra Oberseminar, WWU Münster, Germany (Nov. 2019).
  • Fine structure of C*-algebras associated to topological dynamical systems, Analysis seminar, KU Leuven, Belgium (Oct. 2019).
  • Structure-preserving *-isomorphisms of (groupoid) C*-algebras, Analysis seminar, University of Glasgow, Scotland (Jan. 2019).
  • Various *-isomorphisms of (groupoid) C*-algebras, Danish Operator Algebra Seminar, University of Southern Denmark, Denmark (Dec. 2018).
  • Symbolic dynamics and operator algebras, Operator algebra seminar, University of Copenhagen, Denmark (Sep. 2018).
  • C*-simplicity and representations of topological full groups, Workshop in operator algebras and dynamics, University of the Faroe Islands, the Faroe Islands (Jun. 2018).
  • Modeling symbolic dynamics using C*-algebras, Operator algebra seminar, University of Wollongong, Australia (Feb. 2018).
  • Modeling finite type symbolic dynamics using C*-algebras, Seminar, University of Western Sydney, Australia (Jan. 2018).
  • The C*-algebra associated to a symbolic dynamical system, Operator algebra research program, Universitát Áutonoma de Barcelona, Spain (May 2017).

Other presentations

  • Why construct C*-algebras from dynamics? A series of four lectures given at the University of Glasgow, United Kingdom (Jan--Feb 2022).
  • C*-algebras from symbolic dynamical systems, YMC*A, University of Münster, Germany (Aug 2021).
  • Conjugacy of local homeomorphisms, Summer school in Operator Algebras, Fields institute and University of Ottawa (June 2021).
  • Between symbolic dynamics and C*-algebras, 2 talks, Glasgow Working Seminar, University of Glasgow, United Kingdom (May 2020).
  • The conjugacy problem of symbolic dynamics, an operator algebraic approach OANCG Seminar, University of Wollongong, Australia (Nov 2020).
  • What's your problem? Virtual OANCG Seminar, University of Wollongong, Australia (May 2020).
  • Modeling topological dynamics in groupoids and C*-algebras, YMC*A, KU Leuven, Belgium (Aug. 2018).
  • Investigating symbolic dynamics using C*-algebras, AustMS, Macquarie University, Australia (Dec. 2017).
  • One-sided and two-sided dynamics and Cuntz-Krieger algebras, AMSSC 2017, University of Wollongong, Australia (Dec. 2017).
  • C*-algebras associated to symbolic dynamics, DUC*S, Aarhus University, Denmark (Jan. 2017).
  • A groupoid approach to Cuntz-Krieger algebras, PhD seminar, University of Copenhagen, Denmark (Sep. 2016).