Differential geometry: I am interested in many areas of differential geometry, including

• geometry of curves, surfaces and hypersurfaces;
• Riemannian geometry; effects of local curvature conditions on global geometry and topology;
• geodesics, minimal surfaces, and related problems;
• isoperimetric inequalities;
• differential geometry associated with conformal transformations, projective transformations, affine transformations and other groups;
• general relativity and semi-Riemannian geometry;
• geometry of convex bodies;
• Finsler geometry.

Partial differential equations:

• Partial differential equations applied to differential geometry;
• heat flows applied to prove results in global differential geometry;
• moving curves, surfaces and hypersurfaces;
• deforming Riemannian metrics or connections;
• smoothing effects of heat flows;
• minimal surfaces, surfaces of prescribed curvature, and related problems;
• harmonic functions and harmonic maps, related variational problems;
• fully nonlinear elliptic and parabolic equations regularity theory;
• qualitative and asymptotic behaviour nonlinear wave equations,
• general relativity;
• eigenvalue problems arising in geometry;
• integrable systems arising in geometry.

Applications:

• tumbling stones and their eventual shapes;
• image processing;
• moving interfaces, phase changes, and related problems;
• reaction-diffusion systems in mathematical biology.