We develop an efficient technique for computing relative class numbers of imaginary abelian fields, efficient enough to enable us to easily compute relative class numbers of imaginary cyclic fields of degrees $32$ and conductors greater than $10^{13}$, or of degrees $4$ and conductors greater than $10^{15}$. Acccording to our extensive computation, all the $166204$ imaginary cyclic quartic fields of prime conductors $p$ less than $10^7$ have relative class numbers less than $p$/2. Our major innovation is a technique for computing numerically root numbers appearing in some functional equations.

There is a well-understood connection between polynomials and certain simple algebraic dynamical systems. In this connection, the Mahler measure corresponds to the topological entropy, Kronecker's Theorem relates ergodicity to positivity of entropy, approximants to the Mahler measure are related to growth rates of periodic points, and Lehmer's problem is related to the existence of algebraic models for Bernoulli shifts. There are similar relationships for higher-dimensional algebraic dynamical systems.

We review this connection, and indicate a possible analogous connection between the global canonical height attached to points on elliptic curves and a possible 'elliptic' dynamical system.

We report the results of some computer experiments on the orbit structure of the discrete maps on a finite set which arise when an expanding map of the circle is iterated "naively" on the computer. We also comment on what mathematical questions ought to be answered in order to account for the reliability in practice of orbit following on the computer as an indicator of the ergodic properties of the underlying map.

We consider the question of deciding whether or not certain subgroups of certain cocompact Kleinian groups are geometrically finite or infinite. This is reduced to a question about similarity interval exchange maps. The concept of trimming produces an practical algorithm to settle the latter question. This algorithm has been implemented on a computer and the results are perplexing.

We discuss the properties of $p$-adic analogues of the logistic and Smale horseshoe maps, and adapt them to form possible practical pseudo-random number generators. The properties of these practical modifications are studied in detail in the case $p=2$.

We find asymptotic formulas as $n\to\infty$ for the coefficients $a(r\hbox{,}\,n)$ defined by $$ \prod_{\nu=1}^\infty\,(1-x^\nu)^{-\nu^r} =\sum_{n=0}^\infty a(r\hbox{,}\,n)x^n\hbox{.} $$ (The case $r=1$ gives the number of plane partitions of $n$.) Generalized Dedekind sums occur naturally and are studied using the Finite Fourier Transform. The methods used are unorthodox; many of the computations are not justified but the result is in many cases very good numerically. The last section gives various formulas for Kinkelin's constant.

We compute the degree 3 homology of GL(3,\,\funnyZ) with coefficients in the module of homogeneous polynomials in three variables of degree $g$ over $\mathbb F_p$, for $g\leq 200$ and $p\leq 541$. The homology has a "boundary part'' and a "quasicuspidal'' part which we determine.

By conjecture a Hecke eigenclass in the homology has an attached Galois representation into $GL(3,\,\bar{\mathbb F}_p$). The conjecture is proved for the boundary part and explored experimentally for the quasicuspidal part.

We present an enumeration scheme for all blueprints of an $A_2(p)$ building (with $p$ prime). We then provide a computer based proof that the $\tilde A_n(\mathbb Q_2)$ buildings do not conform to a blueprint for $n\ge3$.