Adam Rennie
Publications
For copyright reasons, the following publications are mostly pre-publication penultimate versions. Most are also available on the arXiv.Book chapters
- An analytic approach to spectral flow in von Neumann algebras, with M-T. Ben- emeur, A. Carey, J. Phillips, F. Sukochev, K. Wojciechowski, in ‘Analysis, Geometry and Topology of Elliptic Operators-Papers in Honour of K. P. Wojciechowski’, World Scientific, 2006, 297–352.
- Modular index invariants of Mumford curves, with A. Carey, M. Marcolli, in ‘Noncom- mutative Geometry, Arithmetic, and Related Topics’, eds Caterina Consani and Alain Connes, John Hopkins University Press, 2011, 31–73.
- Spectral triples: examples and index theory, with A. Carey, J. Phillips, in ‘Noncommu- tative Geometry and Physics: Renormalisation, Motives, Index Theory’, Ed. A. Carey, EMS, 2011, 175–265.
- The bulk-edge correspondence via Kasparov theory, to appear in the Encyclopedia of Math- ematical Physics, 2nd Ed., doi:10.1016/B978-0-323-95703-9.00042-2, 2024.
Journal Articles
- Commutative geometries are spin manifolds, Reviews in Mathematical Physics, 13, (2001) 409–464.
- Smoothness and locality for nonunital spectral triples, K-Theory, 28, (2003), 127–165.
- The Hochschild class of the Chern character of semifinite spectral triples, with A. Carey, J. Phillips, F. Sukochev, Journal of Functional Analysis, 213 (2004) 111–153.
- Spectral triples associated to degenerate metrics, Reviews in Mathematical Physics, 16 No. 1 (2004) 125–146.
- Spectral triples associated to degenerate metrics, Reviews in Mathematical Physics, 16 No. 1 (2004) 125–146.
- The local index formula in semifinite von Neumann algebras I: spectral flow, with A. Carey, J. Phillips, F. Sukochev, Advances in Mathematics, 202 No. 2 (2006) 451–516.
- The local index formula in semifinite von Neumann algebras II: the even case, with A. Carey, J. Phillips, F. Sukochev, Advances in Mathematics, 202 No. 2 (2006) 517–554.
- The noncommutative geometry of graph C*-algebras I: the index theorem, with D. Pask, Journal of Functional Analysis, 233 (2006) 92–134.
- Asymptotics of zeta functions and Dixmier traces, with A. Carey, A. Sedaaev, F. Sukochev, Journal of Functional Analysis, 249 (2007), 253–283.
- The Chern character of semifinite spectral triples, with A. Carey, J. Phillips, F. Sukochev, Journal of Noncommutative Geometry, 2, no. 2 (2008) 141–193.
- The noncommutative geometry of k-graph algebras, with D. Pask, A. Sims, Journal of K-Theory, 1, no. 2 (2008) 259–304.
- Orbifolds are not commutative geometries, with J. Varilly, Journal of the Australian Math- ematical Society, 84, no.1 (2008) 109–116.
- Spectral flow invariants and twisted cyclic theory for the Haar state on SUq(2), with A. Carey, K. Tong, Journal of Geometry and Physics, 59 (2009), 1431–1452.
- Noncommutative manifolds from graph and k-graph C∗-algebras, with D. Pask, A. Sims, Communications in Mathematical Physics, 292 (2009), 607–636.
- Noncommutative Atiyah-Patodi-Singer boundary conditions and index pairings in KK- theory, with A. Carey, J. Phillips, Journal fu ̈r die reine und angewandte Mathematik, 643 (2010), 59–109.
- Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras, with A. Carey, J. Phillips, Journal of K-Theory, 6 no. 2 (2010), 339–380.
- Twisted cyclic theory, equivariant KK-theory and KMS states, with A. Carey, S. Neshveyev, R. Nest, Journal fu ̈r die reine und angewandte Mathematik, 650 (2011), 161–191.
- Families of type III KMS states on a class of C∗-algebras containing On and QN, with A. Carey, J. Phillips, I. Putnam, Journal of Functional Analysis, 260 (6), (2011), 1637–1681.
- Integration on locally compact noncommutative spaces, with A. Carey, V. Gayral, F. Sukochev, Journal of Functional Analysis, 263 (no. 2) (2012), 383–414.
- KK-theory and spectral flow in von Neumann algebras, with J. Kaad, R. Nest, J. K- Theory, 10 (no. 2) (2012), 241–277.
- A residue formula for the fundamental Hochschild 3-cocycle for SUq(2), with U. Kr ̈ahmer, R. Senior, Journal of Lie Theory, 22 (no. 2) (2012), 557–585.
- Riemannian manifolds in noncommutative geometry, with S. Lord, J. V ́arilly, Journal of Geometry and Physics, 62 (2012), 1611–1638.
- Pseudo-Riemannian spectral triples and the harmonic oscillator, with K. van den Dungen, M. Paschke, J. Geom. Phys. 73 (2013), 37–55.
- The K-theory of Heegaard quantum lens spaces, with P. Hajac, B. Zielinski, Journal of Noncommutative Geometry, 7 (no. 4) (2013), 1185–1216.
- Twisted cyclic cohomology and modular Fredholm modules, with A. Sitarz, M. Yamashita, Sigma 9 (2013) 051, 15 pages.
- Dense domains, symmetric operators and spectral triples, with I. Forsyth, B. Mesland, New York Journal of Mathematics, 20 (2014), 1001–1020.
- The resolvent cocycle in twisted cyclic cohomology and a local index formula for the Podle ́s sphere with R. Senior, Journal of Noncommutative Geometry, 8 (no. 1) (2014), 1–43.
- The bulk-edge correspondence for the quantum Hall effect in Kasparov theory, with C. Bourne, A. Carey, Letters in Mathematical Physics, 105 (2015), 1253–1273.
- Spectral flow for nonunital spectral triples, with A. Carey, V. Gayral, J. Phillips, F. Sukochev, Canad. J. Math., 67 (4) (2015), 759–794.
- A noncommutative framework for topological insulators, with C. Bourne, A. Carey, Reviews in Mathematical Physics, Vol. 28, No. 2 (2016) 1650004.
- Universal measurability and the Hochschild class of the Chern character, with A. Carey, F. Sukochev, D. Zanin, J. Spectr. Theory, 6 (2016), 1–41.
- Indefinite Kasparov modules and pseudo-Riemannian manifolds, with K. van den Dungen, Annales Henri Poincar ́e 17 (2016), 3255–3286.
- Nonunital spectral triples and metric completeness in unbounded KK-theory, with B. Mes- land, Journal of Functional Analysis, 271(9) (2016), 2460–2538.
- Generalised time functions and finiteness of the Lorentzian distance, with Ben Whale, Journal of Geometry and Physics, 106 (2016), 108–121.
- The K-theoretic bulk-edge correspondence for topological insulators, with C. Bourne and J. Kellendonk, Annales Henri Poincar ́e, 18, No. 5 (2017), 1833–1866.
- The extension class and KMS states for Cuntz–Pimsner algebras of some bi-Hilbertian bimodules, with D. Robertson, A. Sims, Journal of Topology and Analysis, 09 No. 02, (2017) 297–327.
- Groupoid algebras as Cuntz-Pimsner algebras, with D. Robertson, A. Sims, Mathematica Scandinavica 120 No. 1 (2017), 115–123.
- Groupoid Fell bundles for product systems over quasi-lattice ordered groups, with D. Robertson, A. Sims, Mathematical Proceedings of the Cambridge Philosophical Society, 163, No. 3 (2017), 561–580.
- Non-commutative vector bundles for non-unital algebras, with A. Sims, SIGMA 13, (2017), 041, 12 pages, arXiv:1612.03559.
- Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases, with Chris Bourne, Mathematical Physics, Analysis and Geometry, 21 no. 3 (2018), 62 pages.
- Factorisation of equivariant spectral triples in unbounded KK-theory, with I. Forsyth, J. Aust. Math. Soc., 107 (No. 2) (2019), 145–180.
- Shift-tail equivalence and an unbounded representative of the Cuntz-Pimsner extension, with M. Goffeng and B. Mesland, Ergodic Theory and Dynamical Systems, 38 (no. 4) (2018), 1389–1421.
- The Cuntz-Pimsner extension and mapping cone exact sequences, with Francesca Arici, Math. Scand. 125 (No. 2) (2019), 291–319.
- Boundaries, spectral triples and K-homology, with Iain Forsyth, Magnus Goffeng and Bram Mesland, Journal of Noncommutative Geometry, 13 (No. 2) (2019), 407–472.
- Untwisting twisted spectral triples, with Magnus Goffeng, Bram Mesland, International Journal of Mathematics, 30 (No. 14) (2019), 1950076 (48 pages).
- Constructing KMS states from infinite-dimensional spectral triples, with Magnus Goffeng, Alexandr Usachev, Journal of Geometry and Physics, 143 (2019), 107–149.
- Poincar ́e duality for Cuntz-Pimsner algebras of bimodules, with Dave Robertson, Aidan Sims, Advances in Mathematics, 347 (2019), 1112–1172.
- The Cayley transform in real, complex and graded K-theory, with Chris Bourne, Jo- hannes Kellendonk, International Journal of Mathematics, 31 (no. 9) (2020), 2050074.
- The Godbillon-Vey invariant in equivariant KK-theory, with Lachlan MacDonald, Annals of K-Theory, 5 (no. 2) (2020), 249–294.
- A Cuntz-Pimsner Model for the C∗-algebra of a Graph of Groups, with Alex Mundey, Journal of Mathematical Analysis and Applications, 496 (2) (2021), 124838.
- The KO-valued spectral flow for skew-adjoint Fredholm operators, with Chis Bourne, Alan Carey, Matthias Lesch, Journal of Topology and Analysis, 14 (02) (2022), 505-556.
- Friedrichs angle and alternating projections in Hilbert C∗-modules, with Bram Mesland, Journal of Mathematical Analysis and Applications, 516 (2022), 126474, 19pp.
- Curvature of differentiable Hilbert modules and Kasparov modules, with Bram Mesland, Walter van Suijlekom, Advances in Mathematics, 402 (2022), 108128, 52pp.
- Addendum to: Friedrichs angle and alternating projections in Hilbert C∗-modules, with Koen van den Dungen, Bram Mesland, Journal of Mathematical Analysis and Applications, 522 (1) 2023, 127002.
- Levinson's theorem for two-dimensional scattering systems: it was a surprise, it is now topological!, with A. Alexander, D. T. Nguyen, S. Richard, Journal of Spectral Theory, 14 (3) (2024), 991--1031, arXiv:2403.17617.
- Topological Levinson’s theorem in the presence of embedded thresholds and discontinu- ities of the scattering matrix, with V. Austen, D. Parra, S. Richard, accepted to appear in Reviews in Mathematical Physics, arXiv:2403.17617.
- The structure of the wave operator in four dimensions in the presence of resonances, with A. Alexander, Letters in Mathematical Physics, 114 (5), 122 (2024), arXiv:2311.16438.
- Levinson’s theorem as an index pairing, with Angus Alexander, Journal of Functional Analysis, 286 (5) March 2024, 110287 (31 pages).
- Splittings for C*-correspondences and strong shift equivalence, with K. Brix and A. Mundey, Math. Scand., 130 (1) (2024), 101–148.
- Curvature and Weitzenbock formula for spectral triples, with Bram Mesland, arXiv:2404.07957, accepted to appear in Mathematische Nachtrichten.
- Morphisms of Cuntz-Pimsner algebras from sub-correspondences, with K. Brix and A. Mundey, arXiv:2311.16600, accepted July 2024 to appear in JAustMS, special Raeburn edition.
- Existence and uniqueness of the Levi-Civita connection on noncommutative differential forms, with B. Mesland, Advances in Mathematics, 468 (2025), 110207, arXiv:2403.13735.
- Spectral flow and Levinson's theorem for Schroedinger operators, with A. Alexander, Annals of Functional Analysis, 16, article 53, (2025), arXiv:2405.19571.
- Curvature and Weitzenboeck formula for the Podleś quantum sphere, with B. Mesland, to appear in Communications in Mathematical Physics, arXiv:2406.18483.
Conference Proceedings
- Drainage of Meringue Mixture Froth, with M.D. Mantle, P. Stevenson, 14th International Congress on Rheology, Korea (2004), 3pp.
- Semifinite spectral triples associated with graph C∗-algebras, with A. Carey, J. Phillips, in ‘Traces in Number Theory, Geometry and Quantum Fields’, Vieweg, editors S. Albeverio, M. Marcolli, S. Paycha, J. Plazas, (2008) 35–56.
- Type III KMS states on a class of C*-algebras containing On and QN and their modular index, with A. Carey, J. Phillips, I. Putnam, in ‘Perspectives on Noncommutative Geometry’, Fields Institute Communications no 61, Eds. M. Khalkhali, G. Yu, 2011, 29–40.
- The local index formula in noncommutative geometry revisited, with A. Carey, J. Phillips, F. Sukochev, in ‘Noncommutative Geometry and Physics 3’, eds G. Dito, M. Kotani, Y. Maeda, H. Moriyoshi, T. Natsume, S. Watamura, World Scientific, 2013, 3–36.
- Semi-finite noncommutative geometry and some applications, with A. Carey, J. Phillips, in ‘Noncommutative Geometry and Physics 3’, eds G. Dito, M. Kotani, Y. Maeda, H. Moriyoshi, T. Natsume, S. Watamura, World Scientific, 2013, 37–58.
Preprints
- Conformal transformations and equivariance in unbounded KK-theory, with A. Masters, arXiv:2412.17220
- Comparison of Levi-Civita connections in noncommutative geometry, with Alexander Flamant, Bram Mesland, arXiv:2505.18692
- Limit curve theorems for incomplete metric spaces and the null distance on Lorentzian manifolds, with Ben Whale, arXiv:2511.05847
Grad students
PhD students
- Ada Masters (current,UOW): Conformal dynamics and Kasparov theory.
- Tom Pedersen (2022,UOW): Equilibrium states for graph of group C*-algebras.
- Angus Alexander (2024,UOW): Topological Levinson’s theorem via index pairings and spectral flow.
- Alexander Mundey (2020,UOW): The noncommutative dynamics and topology of iterated function systems.
- Lachlan MacDonald (2019,UOW): Characteristic classes of foliated manifolds in noncommutative geometry.
- Andreas Andersson (2016,UOW): The noncommutative Gohberg-Krein theorem.
- James Fletcher (2016,UOW): Product systems as iterated Cuntz-Pimsner algebras.
- Christopher Bourne (2015,ANU): Topological states of matter and noncommutative geometry.
- Iain Forsyth (2016,ANU): Boundaries and equivariant products in unbounded Kasparov theory.
- Koenraad van den Dungen (2016,ANU): Lorentzian geometry and pohysics in Kasparov's theory.
- Roger Senior (2011,ANU): Modular spectral triples and KMS states in noncommutative geometry.
Masters students
Available for:
- Tutoring university level mathematics subjects
- Consulting on university mathematics degrees and postgraduate education
I am an Honorary Professor of mathematics at the University of Wollongong. My research interests are in noncommutative geometry and index theory and the applications to operator algebras, dynamical systems and mathematical physics.
Contact: rennie.on.the.roadATgmail.com.
Contact: rennie.on.the.roadATgmail.com.
