The book covers the latest research in the areas of mathematics that deal the properties of partial differential equations and stochastic processes on spaces in connection with the geometry of the underlying space. Written by experts in the field, this book is a valuable tool for the advanced mathematician.

• Contents
• Preface
• Part I: Fractals and graphs
• Stability of heat kernel estimates and parabolic Harnack inequalities for general symmetric pure jump processes
• The pointwise existence and properties of heat kernel
• Resistance estimates and critical exponents of Dirichlet forms on fractals
• A survey on unbounded Laplacians and intrinsic metrics on graphs
• Energy measures for diffusions on fractals: a survey
• Hyperbolic graphs induced by iterations and applications in fractals
• Geometric implications of fast volume growth and capacity estimates
• Parabolic index of an infinite graph and Ahlfors regular conformal dimension of a self-similar set
• Metrics and uniform Harnack inequality on the Strichartz hexacarpet
• Part II: Euclidean spaces and manifolds
• Analysis on manifolds and volume growth
• Geometric analysis on manifolds with ends
• A matrix Harnack estimate for a Kolmogorov type equation
• Entropy power concavity inequality on Riemannian manifolds and Ricci flow
• Fractional differential operators and divergence equations
• Interior gradient estimates for mean curvature type equations and related flows
• An alternate induction argument in Simons’ proof of holonomy theorem
• Higher integrability for nonlinear nonlocal equations with irregular kernel
• On nonexistence results of porous medium type equations and differential inequalities on Riemannian manifolds
• Index