My research interests are, broadly speaking, in partial differential equations and analysis on manifolds. More specifically, here is a list of topics that have been of interest for me in the past:
- Heat equation on Riemannian manifolds. Heat equations in a geometric context are important, as exemplified by the famous Ricci flow (used by G. Perelman to solve the longstanding Poincare conjecture in 2004), which may be seen as a non-linear heat equation. A particular question I have worked on, is to understand the long-time behaviour of solutions of the heat equation on a non-compact Riemannian manifold, in terms of the geometry at infinity.
- Riesz transform on manifolds. The Riesz transform, traditionnally studied in the Euclidean space, is a singular integral operator, which is connected to regularity theory for pseudo-differential operators. I have studied this operator on on a Riemannian manifold, and more precisely studied how the curvature affects its behaviour.
- Hardy inequalities. These are inequalities that are connected with the spectral theory of second-order differential operators. Together with Y. Pinchover and M. Fraas, I have proposed a definition of optimal Hardy inequalities.
- Index of minimal surfaces. Roughly speaking, the index of a minimal surface is the maximal dimension of a space of variations of the surface that decrease the area. The index is related to the spectrum of a geometric Schrodinger operator (the Jacobi operator). Recently, I have studied the index of minimal surfaces in the unit ball with “free boundary”.