In this expository article we give an overview of recent developments in the study of optimal Hardy-type inequalityin the continuum and in the discrete setting. In particular, we present the technique of the {\emph supersolution construction} that yield “as large as possibleȍ Hardy weightswhich is made precise in terms ofthe notion of criticality. Instead of presenting the most general setting possible, we restrict ourselves to the case of the Laplacian on smooth manifolds and bounded combinatorial graphs. Although the results hold in far greater generality, the fundamental phenomena as well as the core ideas of the proofs become especially clear in these basic settings.

In an appendix to an earlier paper \cite{BDS-sewing} we showed how to construct tunnels of positive scalar curvature and of arbitrarily small length and volume connecting points in a three dimensional manifold of constant positive sectional curvature. Here we generalize the construction to arbitrary dimensions and require only positivity of the scalar curvature.

We consider a countable tree $T$, possibly having vertices with infinite degree, and an arbitrary stochastic nearest neighbour transition operator $P$. We provide a boundary integral representation for general eigenfunctions of $P$ with eigenvalue $\lambda \in \C$, under the condition that the oriented edges can be equipped with complex-valued weights satisfying three natural axioms. These axioms guarantee that one can construct a $\lambda$-Poisson kernel. The boundary integral is with respect to distributions, that is, elements in the dual of the space of locally constant functions. Distributions are interpreted as finitely additive complex measures. In general, they do not extend to $\sigma$-additive measures: for this extension, a summability condition over disjoint boundary arcs is required. Whenever $\lambda$ is in the resolvent of $P$ as a self-adjoint operator on a naturally associated $\ell^2$-space and the diagonal elements of the resolvent (“Green function”) do not vanish at $\lambda$, one can use the ordinary edge weights corresponding to the Green function and obtain the ordinary $\la$-Martin kernel.

We then consider the case when $P$ is invariant under a transitive group action. In this situation, we study the phenomenon that in addition to the $\lambda$-Martin kernel, there may be further choices for the edge weights which give rise to another $\lambda$-Poisson kernel with associated integral representations. In particular, we compare the resulting distributions on the boundary.

The material presented here is closely related to the contents of our “companion” paper\cite{PiWo}.

In this paper, we survey some results on infinite planar graphs with nonnegative combinatorial curvature, related to the total curvature, the number of vertices with positive curvature and the automorphism group.

A Laplacian eigenfunction on a manifold or a metric graph imposes a natural partition of the manifold or the graph. This partition is determined by the gradient vector field of the eigenfunction (on a manifold) or by the extremal points of the eigenfunction (on a graph). The submanifolds (or subgraphs) of this partition are called Neumann domains. Their counterparts are the well-known nodal domains. This paper reviews the subject of Neumann domains, as appears in recent publications and points out some open questions and conjectures. The paper concerns both manifolds and metric graphs and the exposition allows for a comparison between the results obtained for each of them.

Ramanujan graphs have fascinating properties and history. In this paper we explore a parallel notion of Ramanujan digraphs, collecting relevant results from old and recent papers, and proving some new ones. Almost-normal Ramanujan digraphs are shown to be of special interest, as they are extreme in the sense of an Alon-Boppana theorem, and they have remarkable combinatorial features, such as small diameter, Chernoff bound for sampling, optimal covering time and sharp cutoff. Other topics explored are the connection to Cayley graphs and digraphs, the spectral radius of universal covers, Alon's conjecture for random digraphs, and explicit constructions of almost-normal Ramanujan digraphs.

In this article we prove that antitrees with suitable growth properties are examples of infinite graphs exhibiting strictly positive curvature in various contexts: in the normalized and non-normalized Bakry-Émery setting as well in the Ollivier-Ricci curvature case. We also show that these graphs do not have global positive lower curvature bounds, which one would expect in view of discrete analogues of the Bonnet-Myers theorem. The proofs in the different settings require different techniques.

Internal diffusion-limited aggregation (IDLA) is a stochastic growth model on a graph G which describes the formation of a random set of vertices growing from the origin (some fixed vertex) of G. Particles start at the origin and perform simple random walks; each particle moves until it lands on a site which was not previously visited by other particles. This random set of occupied sites in G is called the IDLA cluster. In this paper we consider IDLA on Sierpinski gasket graphs, and show that the IDLA cluster fills balls (in the graphmetric) with probability 1.

We present a general framework for thermodynamic limits and its applications to a variety of models. In particular we will identify criteria such that the limits are uniform in a parameter.All results are illustrated with the example of eigenvalue counting functions converging to the integrated density of states.In this case, the convergence is uniform in the energy.

We discuss a construction which associates to a KdV equation the lamplighter group. In order to establish this relation one uses automata and random walks on ultra discrete limits. We present it in a more general context.

We discuss optimal lower boundsfor eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero eigenvalue in the finite volume case and the first eigenvalue of the Dirichlet Laplacian on subsets that satisfy natural geometric conditions.

We give an overview over recent results that deal with Riemannian manifolds whose Ricci curvature is mostly non-negative in an integral sense, in particular quantified in terms of a Kato condition.