19982021

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Research interests

The questions I am interested in mix geometry with partial differential equations.

Geometric evolution equations: here we deform geometric objects in a smooth way to help us understand their shape. This is a very powerful recent technique in geometric analysis, and has been crucial in solving many open geometric questions, including the Poincare conjecture (proved by Perelman).

Eigenvalue estimates: Every geometric object has a set of numbers attached to it, called the spectrum. These are like the resonant frequencies of a drum. I am interested in how the shape of the object affects the spectrum.

Capillary surfaces and the calculus of variations: Consider a meniscus--- the interface between the water in a glass and the air above it. The special shape of the meniscus is due to an interplay between the energy used in the water/glass interface and the air/water interface, and the gravitational energy--- the shape we see will minimise the sum of these energies. It will also dependent on the shape of the glass and the volume of the water. This is a typical problem in the calculus of variations, where minimizing a physical quantity can give rise to interesting geometric shapes.

Monash teaching commitment

Second semester 2017:

M41022 Partial Differential Equations

MTH3160 Functional Analysis 

Second semester 2018

MTH3160 Metric spaces, Banach spaces, Hilbert spaces 

 

 

Research area keywords

  • eigenvalues
  • parabolic equations
  • heat flow
  • geometric analysis
  • calculus of variations
  • geometric evolution equations
  • mean curvature flow
  • spectral theory

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