We use the Valentiner action of $\mathcal A_6$ on $\mathbb C \mathbb P^2$ to develop an iterative algorithm for the solution of the general sextic equation over $\mathbb C$, analogous to Doyle and McMullen's algorithm for the quintic.

The 3n+1-problem is the following iterative procedure on the positive integers: the integer n maps to n/2 or 3n+1, depending on whether n is even or odd. It is conjectured that every positive integer will be eventually periodic, and the cycle it falls onto is $1\mapsto 4\mapsto 2\mapsto 1$. We construct entire holomorphic functions that realize the same dynamics on the integers and for which all the integers are in the Fatou set. We show that no integer is in a Baker domain (domain at infinity). We conclude that any integer that is not eventually periodic must be in a wandering domain.

We present a method for computing generic homoclinic tangencies in the complex Hénon map, based on analytic parametrizations of the stable and unstable manifold, and we discuss applications and consequences of the existence of such tangencies.

Affine spheres with definite and indefinite Blaschke metric are discretized in a purely geometric manner. The technique is based on simple relations between affine spheres and their duals which possess natural discrete analogues. The geometry of these duality relations is discussed in detail. Cauchy problems are posed and shown to admit unique solutions. Particular discrete definite affine spheres are shown to include regular polyhedra and some of their generalizations. Connections with integrable partial difference equations and symmetric mappings are recorded.

This article is the result of experiments performed using computer programs written in the GAP language. We describe an algorithm which computes a set of rational functions attached to a finite Coxeter group W. Conjecturally, these rational functions should be polynomials, and in the case where W is the Weyl group of a Chevalley group G defined over $\mathbb F_q$, the values of our polynomials at q should give the number of $\mathbb F_q$-rational points of Lusztig's special pieces in the unipotent variety of G. The algorithm even works for complex reflection groups. We give a number of examples which show, in particular, that our conjecture is true for all types except possibly $B_n$ and $D_n$.

We study the calculation of the volume of the polytope $B_n$ of n x n doubly stochastic matrices (real nonnegative matrices with row and column sums equal to one). We describe two methods. The first involves a decomposition of the polytope into simplices. The second involves the enumeration of "magic squares", that is, n x n nonnegative integer matrices whose rows and columns all sum to the same integer.

We have used the first method to confirm the previously known values through n=7. This method can also be used to compute the volumes of faces of $B_n$. For example, we have observed that the volume of a particular face of $B_n$ appears to be a product of Catalan numbers. We have used the second method to find the volume for n=8, which we believe was not previously known.

We consider a bounded Rooms and Passages region $\Omega$ on which the negative Neumann laplacian (restricted to the orthogonal complement of the constant functions) does not have a compact inverse and hence has an essential spectrum. We try to understand how such spectra may be approximated by results from a sequence of finite-dimensional problems. Approximations to this laplacian on finite-dimensional structures have only eigenvalues for spectra. Our strategy is to attempt to discern how results on increasingly better approximating structures point to spectral results in the limiting case.