PhD in Pure Mathematics, 2002
I completed my PhD in pure mathematics in 2002 under the supervision of Klaus Ecker at Monash University. The title of my thesis was "The surface area preserving mean curvature flow". After completing post-doctoral positions at the Australian National University, in 2005 I took up an ARC Postdoctoral Fellowship as part of a joint ARC grant with Ben Andrews. In July that year I moved to the University of Wollongong, combining a lecturing position with the ARC research grant. In January 2018 I moved to the University of Newcastle where I am now Head of Mathematics Discipline.
I am a geometric analyst whose main interests are geometric evolution problems and associated elliptic equations. I work in particular with fully nonlinear second order elliptic and parabolic partial differential equations and with higher (even) order nonlinear equations with iterated divergence structure. I use techniques from the analysis of these equations together with differential geometry, topology and functional analysis. Particular questions of interest to me include short- and long-term behaviour of curvature flows of curves and of hypersurfaces and associated self-similar solutions, possibly with boundary conditions. Such work has practical applications in for crystal growth, annealing metals, image analysis and modelling fire-fronts and the shape of biomembranes.
- Associate Professor, Director of the Institute for Mathematics and its Applications, University of Wollongong, Wollongong, New South Wales, Australia, 2014 — 2018
- Associate Professor, Head of Mathematics Discipline, University of Newcastle, Newcastle, New South Wales, Australia, 2018 — present
- 2018—2020 Australian Research Council (ARC) Discovery Project “Parabolic methods for elliptic boundary value problems”, joint with Prof Neil Trudinger and Dr Valentina Wheeler
- 2015—2018 ARC Discovery Project “Higher order curvature flow of curves and hypersurfaces”, joint with Prof Ben Andrews and Dr Glen Wheeler
- 2012—2014 ARC Discovery Project “New directions in geometric evolution equations”, joint with Prof Ben Andrews
- 2005—2009 ARC Discovery Project “Singularities and surgery in curvature flows”, joint with Prof Ben Andrews
- Mixed volume preserving curvature flows, Calc. Var. PDE 24 (2005) no. 2, 131—154.
- The surface area preserving mean curvature flow, Asian J. Math. 7 (2003) no. 1, 7—30.
- (with B Andrews) Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature, Trans. Amer. Math. Soc. 364 (2012) no. 7, 3427—3447.
- (with B Andrews, Y Zheng) Contracting convex hypersurfaces by curvature, Calc. Var. PDE 47 (2013) no. 3-4, 611—665.
- (with B Andrews, M Langford) Non-collapsing in fully non-linear curvature flows, Ann. De l’Institut Henri Poincaré (C) Nonlinear Analysis 30 (2013), no. 1, 23—32.
- The mixed volume preserving mean curvature flow, Math. Z. 246 (2004), 155—166.
- (with N Thamwattana, J Hill) Energy density functions for protein structures, Q. Journal Mech. Appl. Math. 61 (2008) no. 3, 431—451.
- (with G. Wheeler) A classification theorem for Helfrich surfaces, Math. Ann. 357 (2013) no. 4, 1485—1508.
- (with G. Wheeler, G. Williams) Lifespan theorem for constrained surface diffusion flows, Math. Z. 269 (2011) no. 1-2, 147—178.
- (with B Andrews, M Langford) Convexity estimates for hypersurfaces moving by convex curvature functions, Analysis & PDE 7 (2014) no. 2, 407—433.