Several authors have considered the infinite parametric family of simplest quartic fields $K_t=\mathbb Q(\xi)$. In this paper, we explicitly give all generators of power integral bases in the ring of integers $\mathbb Z_K$ of$K_t$ assuming that $t^2+16$is not divisible by an odd square. We use a well known general algorithm for calculating power integral bases in quartic fields.

The Goresky-Kottwitz-MacPherson (GKM) graph is a combinatorial analogue of a compact connected symplectic manifold with a Hamiltonian action of a compact torus. This graph has been intensively studied by Guillemin and Zara, who discovered analogues in graph theory of classical results such as: symplectic reduction and "quantization and reduction commute.'' In this paper, we describe the implementation of algorithms illustrating their results.

An alternative to the traditional curve-straightening flow on periodic curves in surfaces is introduced. The implementation of this flow produces periodic geodesics in minutes rather than hours. The flow is also simpler to initiate since its use of a penalty method permits initial curves that are not necessarily in the surface. Compact and noncompact examples are provided as well as examples with trivial and nontrivial free homotopy classes. The explicit curve-straightening flow on circles in Euclidian space is derived to help check the consistency of the implementations.

We adapt the $p$-group generation algorithm to classify small-dimensional nilpotent Lie algebras over small fields. Using an implementation of this algorithm, we list the nilpotent Lie algebras of dimension up to 9 over $\F_2$ and those of dimension up to 7 over $\F_3$ and $\F_5$.

It is an open question whether tight closure commutes with localization in quotients of a polynomial ring in finitely many variables over a field. Katzman showed that tight closure of ideals in these rings commutes with localization at one element, if for all ideals {\small $I$} and {\small $J$} in a polynomial ring there is a linear upper bound in {\small $q$} on the degree in the least variable of reduced Gröbner bases in reverse lexicographic ordering of the ideals of the form {\small $J + \Fq I$}. Katzman conjectured that this property would always be satisfied. In this paper we prove several cases of Katzman's conjecture. We also provide an experimental analysis (with proofs) of asymptotic properties of Gröbner bases connected with Katzman's conjectures.

It is well known that the ring of polynomial invariants of a finite matrix group {\small $G$} becomes more and more messy the further one moves away from groups generated by pseudoreflections. Also, the number of generators has a tendency to become large. In this experimental study, we try to create evidence for the observation that these unpleasant properties might improve if one enlarges the ring of invariants in one way or another. For instance, {\small $G$} might fix a symplectic form. It can be used to turn the ring of invariants into a Lie algebra by introducing a {\small $G$}-invariant Poisson bracket. In the absence of an invariant symplectic form, one might still consider the Lie algebra of invariant polynomial vector fields simultaneously with the ring of polynomial invariants. If one is prepared to leave the realm of commutative rings and Lie algebras, one can also take {\small $G$}-invariant differential operators into account. Under these additional operations the number of generators necessary to create all invariants is often drastically decreased. A particularly nice situation arises if the group fixes a quadratic form and a symplectic form at the same time, because together they give rise to an endomorphism on the space of homogeneous invariants of any given degree that, for instance, can be used to single out more effective generators in the new sense. In the classical situation of fields of characteristic zero, to which this paper is restricted, the averaging operator is both theoretically and algorithmically an important tool, whose computational feasibility, however, decreases with increasing degrees. The methods presented here might also help to avoid this difficulty.

We present empirical results that suggest that there are real quadratic fields with properties similar to imaginary quadratic fields in terms of size and structure of the class group. Therefore, these class groups can also be used for encryption schemes such as the ElGamal scheme, where up to now, only class groups of imaginary quadratic fields have been considered. Some security aspects are also addressed.

There are many examples of self-similar tiles that are connected, but whose interior is disconnected. For such tiles we show that the boundary of a component of the interior may be decomposed into a finite union of pieces, each similar to a subset of the outer boundary of the tile. This is significant because the outer boundary typically has lower dimension than the full boundary. We describe a method to realize the outer boundary as the invariant set of a graph-directed iterated function system. The method works under a certain "finiteness'' assumption. While it is not clear that this assumption always holds, and it is problematic to give a rigorous proof that it holds even in cases where it is "visually clear'' that it holds, we give some examples where the method yields clear and nontrivial results. Details concerning the algorithms may be found at the website www.math.cornell.edu/~$sld32/Tiles.html.

We use graphs to define sets of Salem and Pisot numbers and prove that the union of these sets is closed, supporting a conjecture of Boyd that the set of all Salem and Pisot numbers is closed. We find all trees that define Salem numbers. We show that for all integers $n$ the smallest known element of the {\small$n$}th derived set of the set of Pisot numbers comes from a graph. We define the Mahler measure of a graph and find all graphs of Mahler measure less than {\small $\frac12(1+\sqrt{5})$}. Finally, we list all small Salem numbers known to be definable using a graph.

Continuing the work of Dunfield, we determine the fibred status of all the unknown hyperbolic 3-manifolds in the cusped census. We then find all the fibred hyperbolic 3-manifolds in the closed census and use this to find over 100 examples each of closed and cusped nonfibred virtually fibred census 3-manifolds, including the Weeks manifold. We also show that the corank of the fundamental group of every 3-manifold in the cusped and in the closed census is 0 or 1.