I am a geometric analyst. I am interested in partial differential equations and differential geometry, in particular geometric evolution equations, such as surface diffusion flow, Helfrich flow, Willmore flow, and mean curvature flow.
Subfields that I have work in include
- Calculus of variations and the study of functionals of submanifolds (such as the area functional, Willmore functional, Helfrich functional, ...)
- Local and global differential geometry
- Higher order elliptic and parabolic partial differential equations on manifolds
My research lies in the intersection of differential geometry and partial differential equations (PDE). I have a particular focus on nonlinear higher-order geometric PDE, in particular the class of equations known as curvature flow.
Curvature flow are evolution equations that describe how a geometric object (we call this a manifold) changes over time. The velocity of the flow is given by internal and external forces, usually arising from physical considerations. This makes curvature flow a natural fit for applications. In recent times, curvature flow have also proved to be powerful tools for the solution of long-standing problems in mathematics. For instance, the Poincare Conjecture was solved using Ricci flow, a flow of Riemannian metrics. There are many such applications, for instance of extrinsic curvature flow to geometric inequalities, such as the isoperimetric inequality. This is a relationship between area and length; in particular, it quantifies the idea that as you pack in more volume with fixed area, the shape of the region inside becomes round. We see this for instance when blowing up a balloon.
If my work is to have a single unifying characteristic, it is the specific study and consideration of difficult (but well-motivated) flows that require the development of new tools to treat. This is why much of my research concerns flows that are higher-order or have boundary conditions. Both of these cases prevent application of standard theory and techniques, but are absolutely critical from an applications perspective.
Here are some specific developments:
Year | Title |
---|---|
2015 | Awarded by: Funding Scheme: Discovery Projects |
Degree | Research Title | Advisee | |
---|---|---|---|
Doctor of Philosophy | Non-Compact Higher-Order Curvature Flow | ||
Doctor of Philosophy | Curvature Flows for Bushfire Modelling | ||
Doctor of Philosophy | Gradient Flow of the Dirichlet Energy for the Mean Curvature of Closed Surfaces | Fang, Yanqin | |
Doctor of Philosophy | To be confirmed. | Whale, Ben | |
Doctor of Philosophy | Physics-inspired deep neural networks for computer vision | Rice, Brad |
My research lies in the intersection of differential geometry and partial differential equations (PDE). I have a particular focus on nonlinear higher-order geometric PDE, in particular the class of equations known as curvature flow.
Curvature flow are evolution equations that describe how a geometric object (we call this a manifold) changes over time. The velocity of the flow is given by internal and external forces, usually arising from physical considerations. This makes curvature flow a natural fit for applications. In recent times, curvature flow have also proved to be powerful tools for the solution of long-standing problems in mathematics. For instance, the Poincare Conjecture was solved using Ricci flow, a flow of Riemannian metrics. There are many such applications, for instance of extrinsic curvature flow to geometric inequalities, such as the isoperimetric inequality. This is a relationship between area and length; in particular, it quantifies the idea that as you pack in more volume with fixed area, the shape of the region inside becomes round. We see this for instance when blowing up a balloon.
If my work is to have a single unifying characteristic, it is the specific study and consideration of difficult (but well-motivated) flows that require the development of new tools to treat. This is why much of my research concerns flows that are higher-order or have boundary conditions. Both of these cases prevent application of standard theory and techniques, but are absolutely critical from an applications perspective.
Here are some specific developments:
Year | Title |
---|---|
2015 | Awarded by: Funding Scheme: Discovery Projects |
Degree | Research Title | Advisee | |
---|---|---|---|
Doctor of Philosophy | Non-Compact Higher-Order Curvature Flow | ||
Doctor of Philosophy | Curvature Flows for Bushfire Modelling | ||
Doctor of Philosophy | Gradient Flow of the Dirichlet Energy for the Mean Curvature of Closed Surfaces | Fang, Yanqin | |
Doctor of Philosophy | To be confirmed. | Whale, Ben | |
Doctor of Philosophy | Physics-inspired deep neural networks for computer vision | Rice, Brad |