Based on our method for determining endomorphism rings, we describe an algorithm to compute decompositions of modules of finite-dimensional algebras over finite fields. The algorithm is implemented in M. Ringe, “The C-Meat-Axe."

We describe a computer-based database of polarized K3 surfaces and explain the meaning of the information it contains. In a precise sense, the database includes all K3 surfaces.

The semiperiodic behavior of the zeta function $\zeta(s)$ and its partial sums $\zeta_N(s)$ as a function of the imaginary coordinate has been long established. In fact, the zeros of a $\zeta_N(s)$, when reduced into imaginary periods derived from primes less than or equal to $N$, establish regular patterns. We show that these zeros can be embedded as a dense set in the period of a surface in $\mathbb{R}^{k+1}$, where $k$ is the number of primes in the expansion. This enables us, for example, to establish the lower bound for the real parts of zeros of $\zeta_N(s)$ for prime $N$ and justifies the use of methods of calculus to find expressions for the bounding curves for sets of reduced zeros in $\mathbb{C}$.

This paper presents some algorithmic techniques for computing explicitly the Noetherian operators associated with a class of ideals and modules over a polynomial ring. The procedures we include in this work can be easily encoded in computer algebra packages such as CoCoA.

The Fischer decomposition on $\mR^n$ gives the decomposition of arbitrary homogeneous polynomials in $n$ variables $(x_1,\dotsc,x_n)$ in terms of harmonic homogeneous polynomials. In classical Clifford analysis a refinement was obtained, giving a decomposition in terms of monogenic polynomials, i.e., homogeneous null solutions for the Dirac operator (a vector-valued differential operator factorizing the Laplacian $\Delta_n$ on $\mR^n$). In this paper the building blocks for the Fischer decomposition in the Hermitian Clifford setting are determined, yielding a new refinement of harmonic analysis on $\mR^{2n}$ involving complex Dirac operators commuting with the action of the unitary group.

Using geodesic currents, we provide a theoretical justification for some of the experimental results obtained by Haralick, Miasnikov, and Myasnikov via pattern-recognition methods regarding the behavior of Whitehead's algorithm on nonminimal inputs. In particular, we prove that the images of "random'' elements of a free group $F$ under the automorphisms of $F$ form "clusters'' that share similar normalized Whitehead graphs and similar behavior with respect to Whitehead's algorithm.

We aim to give a classification of Euclidian bijective polygonal piecewise isometries with a finite number of compact polygonal atoms. We rely on a specific type of triangulation process that enables us to describe a notion of combinatorial type similar to its one-dimensional counterpart for interval exchange maps. Moreover, it is possible to handle all the possible piecewise isometries, given two combinatorial types. We show that most of the examples treated in the literature of piecewise isometries can be retrieved by systematic computations. The computations yield new examples with apparently interesting behavior, but they still have to be studied in more detail. We also exhibit a new class of maps, the piecewise similarities, which fit nicely in this framework and whose behavior is shown to be highly nontrivial.

We consider the problem of obtaining good upper and lower bounds on the number of balanced Boolean functions in $n$ variables with degree less than or equal to $k$. This is the same as the problem of finding bounds on the number of codewords of weight $2^{n-1}$ in the Reed--Muller code of length $2^n$ and order $k$. We state several conjectures and use them to obtain good bounds. We believe that the conjectures will be highly useful for further research

A closed hyperbolic 3-manifold is exceptional if its shortest geodesic does not have an embedded tube of radius $\ln(3)/2$. D. Gabai, R. Meyerhoff, and N. Thurston identified seven families of exceptional manifolds in their proof of the homotopy rigidity theorem. They identified the hyperbolic manifold known as Vol3 in the literature as the exceptional manifold associated with one of the families. It is conjectured that there are exactly six exceptional manifolds. We find hyperbolic 3-manifolds, some from the SnapPea census of closed hyperbolic 3-manifolds, associated with five other families. Along with the hyperbolic 3-manifold found by Lipyanskiy associated with the seventh family, we show that any exceptional manifold is covered by one of these manifolds. We also find their group coefficient fields and invariant trace fields.

Serre's conjecture relates two-dimensional odd irreducible Galois representations over $\bar\F_p$ to modular forms. We discuss a generalization of this conjecture to higher-dimensional Galois representations. In particular, for $n$-dimensional Galois representations that are irreducible when restricted to the decomposition group at $p$, we strengthen a conjecture of Ash, Doud, and Pollack. We then give computational evidence for this conjecture in the case of three-dimensional representations.