Finite Quantum Chaos
We compare the statistics of the spectrum for various linear operators --- Schroedinger operators from quantum mechanics, Laplace operators of manifolds arising as quotients of arithmetic groups acting on the upper half plane, and finally adjacency operators of finite upper half plane graphs. These graphs may be viewed as a finite model for arithmetical quantum chaos. C.L. Chai and W.-C. W. Li have shown that the histograms of the spectra of the finite upper half plane graphs approach the Wigner semi-circle distribution as the number of vertices of the graph approaches infinity. That is, the spectral distribution approximates that of a random real symmetric matrix. The histograms for differences of adjacent eigenvalues for finite upper half plane graphs appear to approach the Poisson density, as do the level spacings for arithmetical quantum chaos. The eigenfunctions for finite upper half planes have contours that roughly show the same short of chaos as those for arithmetical quantum chaos, at least for Maass wave forms for the modular group of 2x2 integer matrices of determinant one.
Finding topology in a factory: configuration spaces
A. Abrams and R. Ghrist
As a Source of interesting examples of topological spaces, the modern automated warehouse is surprisingly productive. We discuss a class of configuration spaces that arises naturally in the solution of industrial problems associated with safe multiple-agent robot control. The configuration spaces are easy to define, yet quite difficult to visualize. One particularly pleasant example is the configuration space of two labeled robots on a complete graph of five vertices: this space is topologically equivalent to a genus six surface.
Constructions using a compass and twice-notched straightedge
It is not possible to trisect an arbitrary angle or double the cube using only a straightedge and compass. However, if we are clumsy enough to scratch our straightedge in two places, then those notches can be used to trisect an arbitrary angle or find the cube root of an arbitrary length. These results were known to the ancient Greeks. In this paper, we will place these results in a modern setting, demonstrate that the twice-notched straightedge can be used to construct the regular 7-gon, 9-gon, and 13-gon, and show that it can even be used to construct points whose coordinates are the roots of quintic polynomials that are not solvable by radicals. We will also demonstrate some of its limitations; for example, one cannot construct the regular 23-gon or 29-gon using only a compass and a twice-notched straightedge.
Hexagonal Economic Regions Solve the Location Problem
Frank Morgan and Roger Bolton, Williams College
The Location Problem in economics asks where to place centers of production in order to minimize the average distance from a consumer to the nearest center. We show that in a certain mathematical sense the ideal solution places them at the centers of regular hexagonal tiles.
Finite Groups of Matrices Whose Entries Are Integers
James Kuzmanovich and Audrey Pavlichenkov
Finite groups of matrices appear early as examples in a first course in abstract algebra, and most of these example are given with integral entries. While these groups provide a setting in which to illustrate new concepts and to pose problems, they also have surprising and beautiful properties. For example, Minkowski proved the unexpected result that GL(n,Z), the group of n x n matrices having inverses whose entries are also integers, has only finitely many isomorphism classes of finite subgroups. As a consequence, there are only finitely many possible orders for elements of GL(n,Z); fortunately, the possible orders can be determined using linear algebra. In general, the number of possible orders increases as n increases, but even here we have the surprising result that no new possible orders are obtained when going from GL(2k,Z) to GL(2k+1,Z). This paper is an exposition of these and other related results and questions.
Problems and Solutions