Curriculum Vitae
CV
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
2020 - 2024
University of Wollongong
Doctor of Philosophy - Topological Levinson's theorem via index pairings and spectral flow
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.
2015- 2019
University of Wollongong
Bachelor of Mathematics (Advanced) - First Class Honours & Bachelor of Science (Physics)
Teaching
Lecturing
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.
C*-algebras (2024)
This was an honours level course introducing the theory of C*-algebras. I was the lecturer for the first several weeks.
Tutoring/Teaching Assistant
Multivariate and Vector Calculus (2020, 2021, 2022, 2023) - MATH201 at University of Wollongong
Real Analysis (2022) - MATH222 at University of Wollongong
Foundations of Engineering Mathematics (2018 Autumn and Spring, 2019 Autumn and Spring) - MATH141 at University of Wollongong
Essentials of Engineering Mathematics (2018, 2020, 2021) - MATH142 at University of Wollongong
Discrete Mathematics and Difference Equations (2023) - MATH123 at University of Wollongong
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)