We prove the Li-Yau gradient estimate for the heat kernel on graphs. The only assumption is a variant of the curvature-dimension inequality, which is purely local, and can be considered as a new notion of curvature for graphs. We compute this curvature for lattices and trees and conclude that it behaves more naturally than the already existing notions of curvature. Moreover, we show that if a graph has non-negative curvature then it has polynomial volume growth.

We also derive Harnack inequalities and heat kernel bounds from the gradient estimate, and show how it can be used to derive a Buser-type inequality relating the spectral gap and the Cheeger constant of a graph.

Necessary and sufficient conditions are given in order for a Borel measure on the Euclidean sphere to have an affine image that is isotropic. A sharp reverse affine isoperimetric inequality for Borel measures on the sphere is presented. This leads to sharp reverse affine isoperimetric inequalities for convex bodies.

Given a proper, cocompact action of a Lie groupoid, we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this pairing by applying the van Est map and algebraic index theory. Finally we discuss in examples the meaning of the index pairing and our index formula.

In this paper, we prove that for every bumpy Finsler $2k$-sphere $(S^{2k}, F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying the pinching condition $(\frac{\lambda}{\lambda + 1}) \lt K \leq 1$ either there exist infinitely many closed geodesics or there exist at least $2k$ non-hyperbolic closed geodesics. Due to the example of A. B. Katok, this estimate is sharp.

For any $n$-dimensional compact Riemannian manifold $M$ with smooth metric $g$, we construct a canonical $t$-family of isometric embeddings $I_t : M \to \mathbb{R}^{q(t)}$, with $t \gt 0$ sufficiently small and $q(t) \gg t^{-\frac{n}{2}}$. This is done by intrinsically perturbing the heat kernel embedding introduced in [BBG]. As $t \to 0_{+}$, asymptotic geometry of the embedded images is discussed.