Dr Julie Clutterbuck
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.
 Clutterbuck, Julie Faye ; Schulze, Felix ; Schnuerer, Oliver. "Stability of translating solutions to mean curvature flow". In: Calculus of Variations and Partial Differential Equations. 2007 ; Vol. 29, No. 3. pp. 281-293. https://doi.org/10.1007/s00526-006-0033-1
 Andrews, Ben ; Clutterbuck, Julie Faye. "Proof of the fundamental gap conjecture". In: Journal of the American Mathematical Society. 2011 ; Vol. 24, No. 3. pp. 899 - 916. https://doi.org/10.1090/S0894-0347-2011-00699-1
 Athanassenas, Maria ; Clutterbuck, Julie. "A capillarity problem for compressible liquids". In: Pacific Journal of Mathematics. 2009 ; Vol. 243, No. 2. pp. 213 - 232.
 Bourni, Theodora ; Clutterbuck, Julie ; Nguyen, Xuan Hien ; Stancu, Alina ; Wei, Guofang ; Wheeler, Valentina-Mira. "The vanishing of the fundamental gap of convex domains". Pre-print. 2020. https://arxiv.org/abs/2005.11784