The curve shortening flow is a simple and beautiful example of a geometric heat flow, the family of equations which includes the Ricci flow used by Perelman to prove the Poincare conjecture as well as many other interesting examples. 

On a (compact) Riemannian manifold there is a natural differential operator on functions, the Laplacian. The eigenvalues of this operator are important invariants of the manifold, and there are many interesting results which relate the eigenvalues to other geometric quantities...

A two dimensional Riemannian manifold is an abstract surface sitting nowhere in particular, but which somehow has the structures imposed on it that a surface gets by sitting in Euclidean space, such as tangent spaces, a metric etc.

The LASSO and penalized likelihood methods have become an extremely hot topic in statistics over the past decade, as they offer a computationally efficient method of variable selection particularly in high dimensional situations.

Minimal surfaces are surfaces which are critical points of the area functional, are are characterised by the vanishing of their mean curvature.

The basic problem is to understand under what conditions it is possible to find a convex surface of prescribed Gauss curvature which also satisfies some boundary conditions.