This includes partial differential equations of elliptic and parabolic types on Riemannian manifolds

Geometric Analysis on Riemannian manifolds, graphs, and metric spaces

This includes partial differential equations of elliptic and parabolic types on Riemannian manifolds,
analysis and random walks on graphs, function theory and Markov processes on fractal spaces, etc.
The main direction of my research is investigation of the global properties of solutions to elliptic
and parabolic equations in connection with the geometry "in the large" of the underlying space.
Here are some examples of such properties: heat kernel estimates, Liouville properties,
recurrence and non-explosion of the heat semigroup, estimates of the eigenvalues of the Laplace
and Schrödinger operators on various spaces, long time behavior of random walks and diffusions, etc.