A summarised version of my CV can be found in a single document here. The remainder of this page covers some of the important details of this document.
Education
- Doctor of Philosophy (Mathematics) , University of Wollongong, 2020 - 2024. Thesis title: Topological Levinson's theorem via index pairings and spectral flow.
- Bachelor of Mathematics (Advanced) - First Class Honours & Bachelor of Science (Physics), University of Wollongong, 2015 - 2019.
A copy of my thesis is available here. The abstract is as follows.
Using techniques from noncommutative geometry, we explore how Levinson's theorem from scattering theory can be interpreted in a topological manner. We use low-energy resolvent expansions to deduce that the wave operator for short range scattering has a particular universal form. The wave operator does not have this form when certain obstructions occur in the resolvent expansions in even dimensions. Using the form of the wave operator, we apply index theoretic techniques to interpret Levinson's theorem as an index pairing between the $K$-theory class of the unitary scattering operator and the $K$-homology class of the generator of dilations on the half-line. A careful analysis of the trace class properties of the scattering operator allows us to provide new proofs of Levinson's theorem in all dimensions. We also compute the spectral flow for Euclidean Schrödinger operators, giving another new proof of Levinson's theorem in all dimensions.
Teaching
All of the classes below were taught at the University of Wollongong.Lecturing
- MATH203 Linear Algebra (2026)
- MATH206 Complex Calculus and Analysis (2026)
- MATH123 Discrete Mathematics and Difference Equations (2025)
- MATH142 Essentials of Engineering Mathematics (2025)
- MTST399 Mathematics Capstone (2025)
- $C^*$-algebras (2024)
- This was an honours level course introducing the theory of $C^*$-algebras. I was the lecturer for the first several weeks.
- Spectral theory and transform methods (2022)
- This was an honours level course beginning with spectral theory for linear operators on Hilbert space and ending with an introduction to some concepts in scattering theory, with a focus on diagonalisations and integral transforms. I was the lecturer for this course, creating lecture notes, assessments and performing all marking.
Tutoring/Teaching Assistant
- MATH201 Multivariate and Vector Calulus (2020, 2021, 2022, 2023)
- MATH222 Real Analysis (2022)
- MATH141 Foundations of Engineering Mathematics (2018 Autumn and Spring, 2019 Autumn and Spring)
- MATH142 Essentials of Engineering Mathematics (2018, 2020, 2021)
- MATH123 Discrete Mathematics and Difference Equations (2023)
Selected Talks
- Topological Levinson's theorem and scattering theory
- Leibniz University Hannover Analysis Seminar (November 2024)
- Noncommutivaty Behind the Dunes at Delft University (October 2024)
- Australia-China-Japan-Singapore-U.S.- Index Theory Conference at University of Adelaide (June 2024)
- PDE and Analysis seminar at Australian National University (April 2024)
- ANZAMP Annual Meeting (February 2024)
- Threshold behaviour of the spectral shift function for Schrödinger operators
- University of Wollongong OANCG Seminar (November 2023)
- Spectral flow for Schrödinger operators
- University of Wollongong OANCG Seminar (August 2023)
- An index pairing in scattering theory
- AMSI Annual Conference, University of Queensland (December 2023)
- NSeaG2023 Workshop: Noncommutative Geometry and Operator Algebras, Hausdorff Institute, Bonn (May 2023)
- Southern Denmark University (February 2023)
- Levinson's theorem as an index pairing
- University of Wollongong OANCG Seminar (November 2022)
- AUSTMS Annual Conference, University of Newcastle (December 2021)
- What is a resonance?
- University of Wollongong OANCG Seminar (August 2020)