Let M be a smooth connected non-compact geodesically complete Riemannian manifold, Δ denote the Laplace operator associated with the Riemannian metric, n ≥ 2 be the dimension of M. Consider the heat equation on the manifold
u_{t} - Δu = 0,
where u = u(x,t), x ∈ M, t > 0. The heat kernel p(x,y,t) is by definition the smallest positive fundamental solution to the heat equation which exists on any manifold (see [Ch], [D]). The...

We introduce a new method for obtaining heat kernel on-diagonal lower bounds on non-
compact Lie groups and on infinite discrete groups. By using this method, we are able to
recover the previously known results for unimodular amenable Lie groups as well as for
certain classes of discrete groups including the polycyclic groups, and to give them a
geometric interpretation. We also obtain new results for some discrete groups which admit
the structure of a semi-direct product or of a wreath product....

We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities
on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack
inequalities under certain non-uniform changes of the weight. We also prove necessary and
sufficient conditions for the Harnack inequalities to hold on complete non-compact
manifolds having non-negative Ricci curvature outside a compact set and a finite first
Betti number or just having asymptotically...

We prove two-sided estimates of heat kernels on non-parabolic Riemannian manifolds with ends, assuming that the heat kernel on each end separately satisfies the Li-Yau estimate.

We introduce the notion of fundamental groupoid of a digraph and prove its basic properties. In particular, we obtain a product theorem and an analogue of the Van Kampen theorem. Considering the category of (undirected) graphs as the full subcategory of digraphs, we transfer the results to the category of graphs. As a corollary we obtain the corresponding results for the fundamental groups of digraphs and graphs. We give an application to graph coloring.

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