Dr Wheeler’s research lies at the intersection of differential geometry and partial differential equations. He works particularly on geometric evolution equations, especially higher-order curvature flows, and on variational and geometric PDE motivated by questions in elasticity, geometric inequalities, and moving-front problems. A central theme of his work is the development of new analytical tools for difficult flows that lie beyond standard theory, especially higher-order problems, problems with boundary conditions, and problems involving non-local effects.

His current research spans higher-order geometric flows, Sobolev gradient flows, elastica and Helfrich-type energies, inverse and invariant flows, and non-local PDE related to bushfire dynamics and moving fronts. Recent projects include work connected with the Cartan–Hadamard conjecture, Yau’s problem on the equivalence of plane curves, mathematically rigorous bushfire models, entropy and Sobolev gradient flows, and new variational problems in geometric analysis. The unifying aim is to produce mathematically deep results that also open new directions for modelling, simulation, and application.

Potential HDR supervision topics include geometric analysis; higher-order geometric flows; curvature-driven PDE; Sobolev gradient flows; elastica, Willmore, and Helfrich-type variational problems; inverse and invariant geometric flows; non-local PDE; moving-front models; bushfire dynamics; thin-film and interface problems; and mathematically rigorous modelling at the interface of pure and applied analysis.