We consider germs of mappings of a line to contact space and classify the first simple singularities up to the action of contactomorphisms in the target space and diffeomorphisms of the line. Even in these first cases there arises a new interesting interaction of local commutative algebra with contact structure.

The trace formula contains terms on the spectral side that are constructed from unramified automorphic $L$ -functions. We shall establish an identify that relates these terms with corresponding terms attached to endoscopic groups of $G$ . In the process, we shall show that the $L$ -functions of $G$ that come from automorphic representations of endoscopic groups have meromorphic continuation.

A reflection is an invertible linear transformation of a vector space fixing a given hyperplane, its axis, vectorwise and a given complement to this hyperplane, its center, setwise. A reflection torus is a one-dimensional group generated by all reflections with fixed axis and center.

In this paper we classify subgroups of general linear groups (in arbitrary dimension and defined over arbitrary fields) generated by reflection tori.

Let $G$ be a finite group generated by (pseudo-) reflections in a complex vector space and let $g$ be any linear transformation which normalises $G$ . In an earlier paper, the authors showed how to associate with any maximal eigenspace of an element of the coset $gG$ , a subquotient of $G$ which acts as a reflection group on the eigenspace. In this work, we address the questions of irreducibility and the coexponents of this subquotient, as well as centralisers in $G$ of certain elements of the coset. A criterion is also given in terms of the invariant degrees of $G$ for an integer to be regular for $G$ . A key tool is the investigation of extensions of invariant vector fields on the eigenspace, which leads to some results and questions concerning the geometry of intersections of invariant hypersurfaces.

In this paper we describe a canonical basis for the equivariant $K$ -theory (with respect to a ${{\mathbf{C}}^{*}}$ -action) of the variety of Borel subalgebras containing a subregular nilpotent element of a simple complex Lie algebra of type $D$ or $E$ .

The semi-affine Coxeter-Dynkin graph is introduced, generalizing both the affine and the finite types.

It is shown here that, for $n\ge 2$ , the $n$ -torus is the only $n$ -dimensional compact euclidean space-form which can admit a regular or chiral tessellation. Further, such a tessellation can only be chiral if $n\,=\,2$ .

We determine and completely describe all pure realizations of the finite regular toroidal polyhedra of types {3, 6} and {6, 3}.

Lattices and $\mathbb{Z}$ -modules in Euclidean space possess an infinitude of subsets that are images of the original set under similarity transformation. We classify such self-similar images according to their indices for certain $4D$ examples that are related to $4D$ root systems, both crystallographic and non-crystallographic. We encapsulate their statistics in terms of Dirichlet series generating functions and derive some of their asymptotic properties.

An isomorphism invariant is an expression, defined for a configuration in the projective plane, which takes the same value for all isomorphic configurations. Examples are given as well as a general method (Nehring sequences) for constructing such invariants.

Partial answers are given to two questions. When does a lattice $\Lambda $ contain a sublattice ${\Lambda }'$ of index $N$ that is geometrically similar to $\Lambda $ ? When is the sublattice “clean”, in the sense that the boundaries of the Voronoi cells for ${\Lambda }'$ do not intersect $\Lambda $ ?

Matrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive $n$ -space or hyperbolic $(n+1)-\text{space}\,{{\text{H}}^{n+1}}$ . For small $n$ , these may be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of ${{\text{H}}^{n+1}}$ . We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z})$ , the Gaussian modular (Picard) group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z}[i])$ , and the Eisenstein modular group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z}\left[ \omega\right])$ .