The Hamiltonian of the quantum nonlinear Schrodinger equation is a selfadjoint operator on Fock space whose eigenstates are given by the Bethe Ansatz. The quantum inverse scattering method of the physics literature introduced two families of operators which have been claimed to satisfy important properties. Operators of one family are diagonal on the Bethe Ansatz eigenstates, while the other family creates the Bethe Ansatz eigenstates. In the present work we examine these operators. VVe establish some of the claims and show the inconsistency of others.

On a tree, the Martin boundary for positive eigenfunctions of the "Laplacian" or other suitable difference operators is known to coincide with the natural boundary of the tree. In this survey, operators on a finite product of trees are considered. Old and new results are described. In the case when all the trees are homogeneous, we let the operator be a positive linear combination of the Laplacians in the factor trees. If at least one of the trees is not $\mathbb{Z}$, the corresponding Martin boundary is nontrivial for all sufficiently large eigenvalues, and is given as the product of the natural boundaries of the trees times a hyper surface which depends on the eigemalue. The situation is similar to that of a polydisc. There is a pointwise convergence theorem at the boundary. For $\mathbb{Z}^n$ however, the boundary is a one-point set. To get a nontrivial boundary here, one can consider instead an operator with drift.

Relationships are described between the supports of functions on the d-dimensional torus and the convergence of, their multiple Fourier series both at 0 and in compact neighbourhoods of O.

I shall try in this paper to give a brief survey of a few recent and very exciting developments in the application of analysis on Lipschitz manifolds to geometric topology. As will eventually become apparent, this work involves both operator algebras (especially the connection between C*-algebras and K-theory) and harmonic analysis (in the literal sense of analysis of harmonics, i.e., of the spectrum of the Laplacian) in the proofs, though not in the statements of most of the theorems. Some of these results could only be obtained with great difficulty (if at all) by more traditional topological methods. I will give references to the literature but no proofs. The parts of this work that are my own are joint work with Shmuel Weinberger [10].

The objective of this paper is to provide an overview of results obtained recently by the author and M. Takesaki on the classification, up to cocycle conjugacy, of actions of discrete amenable groups on injective factor von Neumann algebras of type $III_{\lambda}, \lambda \neq 1$. Details will appear elsewhere, [14]. We also describe the conclusions of preceding work of Oceanu, [10] extending previous work of Connes [1], [2] and Jones [7], and of Sutherland and Takesaki, [13], on actions of discrete amenable groups and groupoids on injective semi finite von Neumann algebras.