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Li, Y. (2026)

Spectral localisers and aperiodic topological phases in noncommutative geometry

Doctoral Thesis
The dissertation offers new tools and insights to computation and modelling of topological phases, with an emphasis on the aperiodic setting.

The first part develops an abstract mathematical framework for computing topological invariants of quantum systems using finite spectral data, and provides a conceptual foundation for the odd spectral localiser introduced by Loring and Schulz-Baldes. To this end, we employ E-theory, a bivariant K-theory introduced by Connes and Higson. This framework also allows for a generalisation of the spectral localiser to the Hilbert module setting, thereby enabling finite-rank computations of the topological invariants associated with families of operators.

The second part of the dissertation further develops C*-algebraic models of aperiodic topological materials, and explicitly describes the comparison maps between them. This yields a characterisation of the robustness of topological phases and their invariants, determined by their...Show moreThe dissertation offers new tools and insights to computation and modelling of topological phases, with an emphasis on the aperiodic setting.

The first part develops an abstract mathematical framework for computing topological invariants of quantum systems using finite spectral data, and provides a conceptual foundation for the odd spectral localiser introduced by Loring and Schulz-Baldes. To this end, we employ E-theory, a bivariant K-theory introduced by Connes and Higson. This framework also allows for a generalisation of the spectral localiser to the Hilbert module setting, thereby enabling finite-rank computations of the topological invariants associated with families of operators.

The second part of the dissertation further develops C*-algebraic models of aperiodic topological materials, and explicitly describes the comparison maps between them. This yields a characterisation of the robustness of topological phases and their invariants, determined by their stability under perturbations.  We also provide alternative dynamical descriptions of one-dimensional aperiodic topological materials, and interpret the Cuntz-Pimsner model of Bourne and Mesland as a topological graph C*-algebra.

We introduce an inductive limit C*-algebra, called the symmetry-breaking Roe C*-algebra, which encodes the symmetry-breaking processes of lattices.  Its K-theory accommodates both strong and weak topological phases, and distinguishes them by divisibility. This allows us to interpret strong topological phases as those that are invariant under symmetry-breaking processes, corresponding to the independence from the specific microscopic partitioning of the material into unit cells.
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Noncommutative geometry
Bivariant K-theory
Spectral localisers
Topological insulators
Coarse geometry
Groupoid C*-algebras
Delone sets and aperiodic tilings
All authors
Li, Y.
Supervisor
Doelman, A.
Co-supervisor
Mesland, B.
Committee
Derks, G.L.A.; Spieksma, F.M.; Suijlekom, W.D. van; Willett, R.; Bourne, C.
Qualification
Doctor (dr.)
Awarding Institution
Mathematical Institute (MI), Faculty of Science, Leiden University
Date
2026-02-26

Funding

Sponsorship
NWO
Grant number
613.009.142