We study rational maps of the real or complex projective plane of degree two or more, concentrating on those that map a genus-one curve onto itself, necessarily by an expanding map. We describe relatively simple examples with a rich variety of interesting dynamical behaviors that are perhaps familiar to the applied dynamics community but not to specialists in several complex variables. For example, we describe smooth attractors with riddled or intermingled attracting basins, and we observe ``blowout'' bifurcations when the transverse Lyapunov exponent for the invariant curve changes sign. In the complex case, we prove that the genus-one curve (a topological torus) can never have a trapping neighborhood, yet it can have an attracting basin of large measure (perhaps even of full measure). We also describe examples in which there appear to be attracting Herman rings (topological cylinders mapped to themselves with irrational rotation number) with open attracting basin. Section 8 provides a more general discussion of Herman rings and Siegel disks for arbitrary holomorphic maps of $\bP^2(\C)$, and the last section outlines open problems

The arithmetic--geometric mean algorithm for calculating elliptic integrals of the first type was introduced by Gauss. The analogous algorithm for abelian integrals of genus $2$ was introduced by Richelot (1837) and Humbert (1901). We present the analogous algorithm for abelian integrals of genus 3.

We present a constructive method to convert the (two-dimensional) Jacobi--Perron evolution of $(x_1,x_2) \in [0,1]^2$ into the corresponding evolution of Podsypanin, and vice versa. A similar approach allows us to extend the result to Brun's algorithm. The method, based on the techniques of singularization and insertion, is built up in steps. From experiments, we assume that each step terminates after finitely many states almost everywhere.

In this paper we compute $\gamma_{K,2$ for $K=\mathbb{Q}(\rho)$, where $\rho$ is the real root of the polynomial $x^3 -x^2 +1 =0$. We refine some techniques introduced in Baeza, et al. to construct all possible sets of minimal vectors for perfect forms. These refinements include a relation between minimal vectors and the Lenstra constant. This construction gives rise to results that can be applied in several other cases.

We demonstrate how to construct three-dimensional compact hyperbolic polyhedra using Newton's method. Under the restriction that the dihedral angles must be nonobtuse, Andreev's theorem provides as necessary and sufficient conditions five classes of linear inequalities for the dihedral angles of a compact hyperbolic polyhedron realizing a given combinatorial structure $C$. Andreev's theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Our construction uses Newton's method and a homotopy to explicitly follow the existence proof presented by Andreev, providing both a very clear illustration of a proof of Andreev's theorem and a convenient way to construct three-dimensional compact hyperbolic polyhedra having nonobtuse dihedral angles.

As an application, we construct compact hyperbolic polyhedra having dihedral angles that are (proper) integer submultiples of $\pi$, so that the group $\Gamma$ generated by reflections in the faces is a discrete group of isometries of hyperbolic space. The quotient $\mathbb{H}^3/\Gamma$ is hence a compact hyperbolic 3-orbifold, of which we study the hyperbolic volume and spectrum of closed geodesic lengths using SnapPea. One consequence is a volume estimate for a ``hyperelliptic'' manifold considered in Mednykh and Vesnin.

Currently, all major implementations of cyclotomic fields as well as number fields are based on a dense model in which elements are represented either as dense polynomials in the generator of the field or as coefficient vectors with respect to a fixed basis. While this representation allows for the asymptotically fastest arithmetic for general elements, it is unsuitable for fields of degree greater than $10^4$ that arise in certain applications such as character theory for finite groups. We propose instead a sparse representation for cyclotomic fields that is particularly tailored to representation theory. We implemented our ideas in MAGMA and used it for fields of degree greater than $10^6$ over $\Q$

We study the simplest fragment of topological theory: those statements that can be expressed using one set variable, interior and closure operators, and inclusion. We introduce a formal system that is simple enough to be implemented on a computer and exhaustively studied and yet rich enough to be sound and complete for the fragment of theory under consideration. This fragment is rich enough to capture concepts such as regular open sets, extremal disconnectedness, partition topologies, and the nodec property.