We discuss a recently discovered connection between the discrete spectrum and the essential spectrum of Schr"odinger operators in one or two space dimensions. The situation is particularly interesting on the half-line since new phenomena occur in this case due to boundary effects. For example, we show that the existence of singular spectrum embedded in the essential spectrum implies that the discrete spectrum is infinite. The proof starts out by relating the problem at hand to the theory of orthogonal polynomials on the unit circle via the Szeg"o and Geronimus transformation. This transformation yields estimates on the potential, which can then be fed into an analysis of the non-linear Fourier transform arising in the Pr"ufer reformulation of the time-independent Schr"odinger equation.

Ideal plane flows are incompressible inviscid two dimensional fluids, described mathematically by the Euler equations. Infinitely many steady states exist. The stability of these steady states is a very classical problem initiated by Rayleigh in 1880. It is also physically very important since instability is believed to cause the onset of turbulence of a fluid. Nevertheless, progress in its understanding has been very slow. I will discuss several concepts of stability and some linear stability and instability criteria. In some cases nonlinear stability and instability can be showed to follow from linear results. I will also describe some methods and techniques developed recently for stability problems, one of which is to use the geometrical properties of the dynamical system for the particle trajectories.

In classical probability, the Gaussian, Chi-square, and Beta are three of the most studied distributions, with wide applicability. In the last century, matrix equivalents to these three distributions have emerged from nuclear physics (Gaussian ensembles) and multivariate statistics (Wishart and MANOVA ensembles). Their eigenvalue statistics have been studied in depth for three values of a parameter (\beta = 1,2,4) which defines the "threefold way" and can be thought of as a counting tool for their real, complex, or quaternion entries.

The re-examination of the Selberg integral formula, in the late '80s, has brought the advent of general \beta-ensembles, which subsume the classical cases, and for which the Boltzmann parameter \beta acts as an inverse temperature. Their eigenvalue statistics interpolate between the isolated instances 1,2, and 4, offering a "behind the scenes" perspective.

With the discovery of matrix models for the general \beta-ensembles in the early 00's, we have entered a new stage in the understanding of the complex phenomena that lie beneath the threefold way. While the \beta = 1,2,4 cases are and will always be special, we believe that the future of the classical ensembles is written in terms of a continuous \beta>0 parameter.

The importance of the conserved quantities were well recoganized in the physical sciences. In this lecture, through several examples, we shall illustrate the fundamental roles played by such conserved quatities in the multiscale analysis. One therefore has to put such quatities into serious considerations also in both modelings and computations.

Network structures, such as polycrystals, foams, and biological tissues consist of space-filling polyhedral grains, bubbles, or cells, respectively. The irregularity of such physical polyhedra derives from the fact that their faces consist of mixed shapes, their edges are of varying length, and the positions of their vertices are arranged non-symmetrically. The geometric complexity of irregular polyhedra forming 3-d networks makes their analysis difficult. To help circumvent this difficulty, average N-hedra (ANHs) are proposed as a set of regular polyhedra consisting of N identical faces. ANHsonly a few of which are constructibleact as abstract proxies for each corresponding topological class of irregular network polyhedra with the same number of faces. This study provides a comparison of the intrinsic areas and volumes of ANHs with data estimated numerically using Brakkes Surface Evolver simulations for a range of constructible polyhedral cells. Evolver data show that for every topological class, ANHs always provide a sharp upper bound for the isoperimetric quotient, which is a measure of the inverse energy cost of the cell. Of special interest also is demonstrating that the critical ANH, which has both zero mean and Gaussian curvature, satisfies Kusners bound for the average number of faces in a minimally partitioned network. In Euclidean 3-space the requisite number of faces is satisfied exactly by the critical ANH. The critical ANH, therefore, statistically represents the abstract unit cell that solves the so-called Kelvin Problem for a space-filling 3-d network exhibiting the minimum partition energy or surface area. This limit remains of particular interest in the case of annealed polycrystals and dry foams, as it establishes the lower bound of the energy cost of cellular random structures with a given metric gauge.