Supervisor: Dr. Glen Wheeler (Homepage, School of Mathe-matics & Applied Statistics, University of Wollongong)

Start: 2026 (flexible)

Stipend: AUD $35,522 p.a. (indexed; includes $3,000 top-up)

Dura-tion: 3.5–4 years

Location: Wollongong, NSW, Australia

Suggested PhD Topics

(P1) The family of H−k(ds) gradient flows.

Develop existence/uniqueness theory, weak/variational formulations, regularity, and long-time behaviour for curvature energies under negative-index Sobolev metrics along curves or surfaces. Questions include metric completeness, well-posedness across topological changes, energy dissipation, and convergence to canonical equilibria.

(P2) The H2(dµ)-gradient flow of the ideal energy (a second-order flow).

Study the flow generated by the H2 Riemannian metric on the space of immersions with area measure dµ. The completeness of the H2-metric is expected to aid stability analysis of constant-mean-curvature (CMC) surfaces. Targets include linear and nonlinear stability, spectral-gap estimates, Łojasiewicz–Simon type inequalities, and sharp convergence criteria.

Each project equips the candidate with cutting-edge knowledge in geometric flows with intersec-tions across PDE, functional analysis, and differential geometry.

What You Will Do

• Build rigorous PDE/variational theory for geometric gradient flows (well-posedness, regularity, asymptotics).
• Prove stability and convergence results (e.g. around CMC equilibria) using analytic and geometric tools.
• (Optional) Design numerical experiments (e.g. MATLAB/Python/Julia) to explore conjectures and guide analysis.
• Disseminate results via publications and international conferences.

Training & Environment

You will join an active group in analysis and geometry at the University of Wollongong, Training covers: high-order elliptic/parabolic PDE, calculus of variations, geometric analysis of curvature energies, Sobolev Riemannian metrics on shape spaces, and gradient-flow techniques (including metric-space and Łojasiewicz–Simon frameworks).