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Research

My mathematical world lies at the interface of topology, analysis and dynamical systems. I use topological methods to understand the spectra of linear differential operators. The main tool in my arsenal is the Maslov index, an intersection index for a path of Lagrangian planes. During my PhD, I have developed index theorems for one-dimensional Hamiltonian differential operators on both bounded and unbounded domains, with applications to the stability analysis of nonlinear waves.

Here's a fun fact about the Maslov index. The classical Sturm oscillation theorem, which relates the number of zeros of an eigenfunction to where in the sequence of eigenvalues the corresponding eigenvalue sits, describes the oscillations of a straight line through the origin in the plane. Its proof follows from the topological invariance of the Maslov index, which captures this winding. From this perspective one can generalise the classical Sturmian theorem to the Morse index theorem from variational calculus, and to spectral theorems for Hamiltonian ordinary differential operators.

Check out my slides from my award-winning talk.

Papers and preprints

  1. 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). (pdf)
  2. Detecting eigenvalues in a fourth-order NLS equation with a non-regular Maslov box. In preparation. (pdf)

    3. PhD Thesis

  3. Hamiltonian spectral theory and the Maslov index. (pdf)