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Interests: spectral theory, stability of nonlinear waves, symplectic geometry, PDE's, dynamical systems.
My mathematical world lies at the interface of topology, analysis and dynamical systems. I use topological methods to understand the spectra of ordinary differential operators. The main tool in my arsenal is the Maslov index, an intersection index for a path of Lagrangian planes. So far I have studied index theorems for one-dimensional Hamiltonian differential operators on both bounded and unbounded domains, with applications to the stability analysis of nonlinear waves.
Papers and preprints:
Hamiltonian spectral flows, the Maslov index, and the stability of standing waves in the nonlinear Schrodinger equation. SIAM Journal on Mathematical Analysis (SIMA). 55 (5) pp. 4998-5050. DOI:10.1137/22M1533797. With Graham Cox, Robert Marangell and Yuri Latushkin (2023). DOI. Arxiv.
Detecting eigenvalues in a fourth-order NLS equation with a non-regular Maslov box. To appear in Journal of Differential Equations. 447 113649. DOI: 10.1016/j.jde.2025.113649. With Robert Marangell (2025). DOI. Arxiv.
PhD Thesis:
Hamiltonian spectral theory and the Maslov index. (pdf)
Talks:
You may have heard me yapping about the Maslov index here:
Seminars:
University of Sydney PDE seminar - The stability of nonlinear waves and the Maslov index - September 2025
Boston University Applied Math seminar - Detecting eigenvalues in a fourth-order nonlinear Schrödinger equation with a non-regular Maslov box - November 2024
Auburn University Applied and Computational Math seminar - Hamiltonian spectral theory via the Maslov index - August 2024
University of Sydney PhD "Defence" - Hamiltonian spectral theory via the Maslov index - March 2024
Special sessions:
Waves 2025 (Session: Recent advances in stability of nonlinear waves) - Fourth order NLS and the Maslov index - April 2025
AIMS 2023 (Session: Geometric methods in spectral theory of travelling waves and patterns) - Hamiltonian spectral flows, the Maslov index, and the stability of NLS standing waves - June 2023
AustMS 2022 (Session: Dynamical systems and ergodic theory) - Counting eigenvalues in Hamiltonian systems via the Maslov index - December 2022 - check out my slides from this award-winning talk
Spectrum of an s-dependent linearised NLS operator, restricted to a family of subintervals [0,sL], associated with a standing wave solution to NLS with cubic potential which satisfies Dirichlet boundary conditions on [0,L]. Red: purely imaginary eigenvalues; blue: purely real; pink: complex.
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