Mathematics
Research
Broadly speaking, I am interested in two of the main sub-branches of functional analysis - operator theory and operator algebras. If functional analysis can be considered an intersection between analysis and algebra, then operator theory corresponds to 'more analysis' and operator algebras corresponds to 'more algebra'.
On the operator theory side of things: I work in the asymptotics theory of operator semigroups, both strongly continuous and discrete, and its applications to partial differential equations and dynamical systems.
This means that I look at systems that evolve to an equilibrium through time (continuously or in steps) and use abstract methods to quantify the rate at which this change happens. This can be used, for example, to obtain the rate at which the energy of a system with damping decays to zero.
I am interested in both the development of new asymptotic theory and the application of theory (both new and old) to concrete models and systems.
On the operator algebra side of things: I am involved in various projects to do with C*-algebras. These are operator algebras that have deep connections and applications to quantum physics. In particular, I am working on C*-algebras that encode information from (higher-rank or topological) graphs and groupoids (group-like objects with a partially defined multiplication and multiple identities), the study of which involves operator K-theory.
Papers
2020
[1] 'Optimal energy decay in a one-dimensional wave-heat system with infinite heat part' with David Seifert. Journal of Mathematical Analysis and Applications 482 (2020), 123563; pre-print can be found on arXiv here.
[2] 'Optimal energy decay in a one-dimensional wave-heat-wave system'. Semigroups of operators: theory and applications, ed. by J. Banasiak, A. Bobrowski, M. Lachowicz, and Yu. Tomilov, Springer Proceedings in Mathematics and Statistics, Springer, Cham, 2020; pre-print can be found on arXiv here.
[3] 'Direct integrals of strongly continuous operator semigroups'. Journal of Mathematical Analysis and Applications 489 (2020), 124176; pre-print can be found on arXiv here.
[4] 'Optimal rates of decay in the Katznelson-Tzafriri theorem for operators on Hilbert spaces' with David Seifert. Journal of Functional Analysis 279(2020), 108799 ; pre-print can be found on arXiv here.
2021
[5] 'A categorical approach to operator semigroups'. Semigroup Forum 102 (2021) 495-516; pre-print can be found on arXiv here.
2023
[6] 'Stably finite extensions of rank-two graph C*-algebras' with Astrid an Huef and Aidan Sims. Journal of Operator Theory 90 (2023), 263–310; pre-print can be found on arXiv here.
[7] 'Reconstruction of topological graphs and their Hilbert bimodules' with Rodrigo Frausino and Aidan Sims. Proceedings of the Royal Society of Edinburgh Section A: Mathematics (2023; first view); pre-print can be found on arXiv here.
Pre-prints:
[8] 'A twist over a minimal étale groupoid that is topologically nontrivial over the interior of the isotropy' with Becky Armstrong, Aidan Sims and Yumiao Zhou; pre-print can be found on arXiv here.
Other:
[9] 'Microthesis: Asymptotics of Operator Semigroups and Applications'. London Mathematical Society Newsletter 507 (2023), 34-36.
Teaching:
Lead Tutor in Mathematics (2024) at St Andrew's College within the University of Sydney.
Co-lecturer for MATH305 - Partial Differential Equations (Autumn Semester 2022 and 2023) at the University of Wollongong.
Co-lecturer for MATH123 - Discrete Mathematics and Difference Equations (Spring Semester 2022 and 2023) at the University of Wollongong
Lecturer and Coordinator for Honours Course - Introduction to C*-algebras (Autumn Semester 2021) at the University of Wollongong.
Teaching Assistant for B4.1 - Banach Spaces (Michaelmas 2016), B4.2 - Hilbert Spaces (Hilary 2017), C4.2 - Linear Operators (Hilary 2018; unofficial class tutor) and Class Tutor for B4.1 - Functional Analysis I (Michaelmas 2018), B4.2 - Functional Analysis II (Hilary 2019) at the University of Oxford.
Tutor for MATH2320/3116/6110 - Advanced Analysis 1: Metric Spaces and Applications (Semester 1 2016) at the Australian National University.
Tutor for MATH1002 - Linear Algebra (Semester 1 2015), MATH1003 - Integral Calculus and Modelling (Semester 2 2015) at the University of Sydney.
Academic Assistant (2017/18) at St John's College, Oxford.
Private tutor for primary and secondary mathematics (2011-14).
Research Talks:
Extensions of C0-semigroups in Categorical Language at the Young Functional Analysts’ Workshop, Newcastle upon Tyne, March 2018.
Asymptotics of Operator Semigroups: Quantified Katznelson–Tzafriri Theorems at Newcastle University (upon Tyne) Analysis Seminar October, 2019.
Rates of Decay in the Classical Katznelson–Tzafriri Theorem Part II: The Story Six Years Later at the University of Oxford Functional Analysis Seminar October, 2019.
Asymptotics of Operator Semigroups: Strongly Continuous and Discrete at the University of Glasgow Analysis Seminar, February 2020.
Asymptotics of Operator Semigroups: Strongly Continuous and Discrete at the University of Reading Analysis Seminar, March 2020.
Asymptotics of Operator Semigroups at the University of Wollongong Operator Algebras and Noncommutative Geometry Seminar, December 2020.
Stably finite extensions of rank-2 graph C*-algebras at the Australian Mathematical Society Conference, Newcastle/Online, December 2021.
Stably finite extensions of rank-2 graph C*-algebras at the University of Wollongong Operator Algebras and Noncommutative Geometry Seminar February 2022.
Optimal rates of decay in the Katznelson–Tzafriri theorem for operators on Hilbert spaces at the New Challenges in Operator Semigroups Conference, Oxford, July 2022.
Stably finite extensions of rank-2 graph C*-algebras at the Young Mathematicians in C*-algebras Conference, Oslo, August 2022.
Asymptotics of Operator Semigroups at the University of New South Wales Pure Mathematics Seminar, September 2022.
Quantified Asymptotics of Operator Semigroups at the Victoria University of Wellington Operator Algebras Seminar, November 2022.
Optimal rates of decay in the Katznelson–Tzafriri theorem for operators on Hilbert spaces at the Australian Mathematical Society Conference, Sydney, December 2022.
Optimal energy decay in one-dimensional wave-heat systems at the Australian Mathematical Society Conference, Sydney, December 2022.
K-theory for algebras associated to commuting Hilbert bimodules at the University of Wollongong Operator Algebras and Noncommutative Geometry Seminar, June 2023.
K-theory for algebras associated to commuting Hilbert bimodules at the Operator Algebras and applications: Symmetry and Structure Conference, Toronto, September 2023.
K-theory for algebras associated to commuting Hilbert bimodules at the Victoria University of Wellington Operator Algebras Seminar, October 2023.
Grants
University of Wollongong AEGiS Connect Grant — Lambda-bundles and Groupoid Twists (AUD $13,101; 2023) held as Lead Chief Investigator with Aidan Sims as second Chief Investigator and Becky Armstrong as Associate Investigator.
Supervision
Rodrigo Frausino — PhD in Mathematics, Thesis: KMS-states and Reconstrution of Graphs from C*-algebras (2021 - Now; co-supervised with Aidan Sims)
Yumiao Zhou — Master of mathematics, Thesis: Twists of Deaconu–Renault Groupoids (2021; co-supervised with Aidan Sims).
Organisational Experience
Co-organiser of the University of Wollongong Noncommutative Geometry and Operator Alegbras Seminar (2022 - 2023; co-organised with Alex Mundey).
Co-organiser of the Noncommutative Geometry and Operator Algebras Special Session at the 2021 Australian Mathematical Society Conference.
Academic Appointments:
Associate Research Fellow at the University of Wollongong under ARC grant DP180100595 held by Aidan Sims (October 2020 - December 2023).
Research Affiliate at the University of Sydney at the University of Sydney (September 2023 - Now, appointment for 3 years).
Lead Tutor in Mathematics at St Andrew's College within the University of Sydney (February 2024 - Now).
Education:
Bachelor of Science (Advanced Mathematics) at the University of Sydney - Honours Class I and the University Medal graduated in 2015.
Doctor of Philosophy (Mathematics) at the University of Oxford graduated in 2020. My thesis titled 'Asymptotics of operator semigroups and applications' can be accessed on the Oxford Research Archive here and I have written a Microthesis for the Newsletter of the London Mathematical Society summarising my thesis for a general mathematical audience which can be accessed here .
You can trace my mathematical genealogy here through Sir Isaac Newton to Galileo Galilei.