Research and Publications

Research Interests

Most of my work involves using techniques from operator algebras, index theory and Kasparov theory to better understand topological phases of matter and quantum mechanical systems. This also raises questions about real K-theory (and KK-theory) which have largely been absent in the mathematics literature. I'm also interested in applications of index theory to dynamical systems.

Preprints/Miscellaneous

C. Bourne. The K-theoretic classification of topological insulators and superconductors. Submitted chapter for the updated Encyclopedia of Mathematical Physics, 16 pages. pdf.

Published articles

C. Bourne. Index theory of chiral unitaries and split-step quantum walks. SIGMA, 19, Paper No. 053 (39 pages). 2023. DOI (open access), arXiv.

C. Bourne. Locally equivalent quasifree states and index theory. J. Phys A: Math. Theor., 55, no. 10, 104004 (38 pages), 2022. DOI, arXiv.

C. Bourne, A. L. Carey, M. Lesch and A. Rennie. The KO-valued spectral flow for skew-adjoint Fredholm operators. J. Topol. Anal., 14, no. 2, 505-556, 2022. DOI, arXiv.

C. Bourne and B. Mesland. Localised module frames and Wannier bases from groupoid Morita equivalences. J. Fourier Anal. Appl., 27, no. 4, Paper No. 69 (39 pages), 2021. DOI, arXiv.

C. Bourne and Y. Ogata. The classification of symmetry protected topological phases of one-dimensional fermion systems. Forum Math. Sigma, 9, Paper No. e25 (45 pages), 2021. DOI (open access), arXiv.

C. Bourne, J. Kellendonk and A. Rennie. The Cayley transform in complex, real and graded K-theory. Internat. J. Math., 31, no. 9, 2050074 (50 pages), 2020. DOI, arXiv.

C. Bourne and H. Schulz-Baldes. On Z_2-indices for ground states of fermionic chains. Rev. Math. Phys., 32, no. 9, 2050028 (69 pages), 2020. DOI, arXiv.

C. Bourne and B. Mesland. Index theory and topological phases of aperiodic lattices. Ann. Henri Poincaré, 20, no. 6, 1969-2038, 2019. DOI, arXiv.

C. Bourne and A. Rennie. Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases. Math. Phys. Anal. Geom., 21, no. 3, article 16 (62 pages), 2018. DOI, arXiv.

C. Bourne and E. Prodan. Non-commutative Chern numbers for generic aperiodic discrete systems. J. Phys A: Math. Theor., 51, no. 23, 235202 (38 pages), 2018. DOI, arXiv. Best Paper Prize 2020 for J Phys A (Mathematical and Theoretical).

C. Bourne, J. Kellendonk and A. Rennie. The K-theoretic bulk-edge correspondence for topological insulators. Ann. Henri Poincaré, 18, no. 5, 1833-1866, 2017. DOI, arXiv.

C. Bourne, A. L. Carey and A. Rennie. A noncommutative approach to topological insulators. Rev. Math. Phys., 28, no. 2, 1650004 (51 pages), 2016. DOI, arXiv. Paper featured as part of the 'Best of 2016' for Reviews in Mathematical Physics.

C. Bourne, A. L. Carey and A. Rennie. The bulk-edge correspondence for the quantum Hall effect in Kasparov theory. Lett. Math. Phys., 105, no. 9, 1253–1273, 2015. DOI, arXiv.

Refereed Conference Proceedings

C. Bourne and H. Schulz-Baldes. Applications of semifinite index theory to weak topological phases. In D. R. Wood et al. (editors), 2016 MATRIX Annals, 203-227, Springer International Publishing, 2018. Article, DOI, arXiv.

PhD Thesis

Topological states of matter and noncommutative geometry, Australian National University, 2015. Available for download from ANU Digital Repository. Direct link.