Algebraic Geometrical Methods Research Articles

Local optimality of the critical lattice sphere-packing of regular tetrahedra

The critical lattice of the euclidean 3-dimensional space generated by the vertices of a regular tetrahedron T 1 of side length 1 provides a 3-solid sphere-packing of any regular tetrahedron T m derived by dilatation in the ratio m:1 from the centroid of T 1. By means of a (permutation group) orbit method it is proved that that packing is locally optimal for differentiable perturbations. By means of an algebraic geometric method it is proved that the packing even is locally optimal for any small perturbation.

Algebraic geometrical methods in Hamiltonian mechanics

During the nineteenth century one of the main concerns in mechanics was to solve Hamiltonian systems by quadrature in terms of elliptic and hyperelliptic functions. The vein of research, abandoned for nearly a century, was entirely revived by the recent findings about the Korteweg—de Vries equation. They shed new light and perspective onto problems of finite-dimensional mechanics, which has led to effective and systematic methods for deciding about the complete integrability of Hamiltonian systems. T he problems that can be captured by these methods have the common virtue that, when run with complex time, most (complex) trajectories are dense on complex algebraic tori; such a system is called algebraically completely integrable, which is stronger than the customary notion of analytic integrability. Many old and new systems enjoy this property and, in particular, a wide variety of systems that occur in the context of orbits in Kac—Moody Lie algebras. This paper explains how the behaviour at the ‘ blow-up ’ time of the solutions to the differential equations enables one to decide about their integrability and also how to derive precise information about the invariant surfaces of the system.

On the Zeta-Functions of Some Simple Shimura Varieties

In an earlier paper [14] I have adumbrated a method for establishing that the zeta-function of a Shimura variety associated to a quaternion algebra over a totally real field can be expressed as a product of L-functions associated to automorphic forms. Now I want to add some body to that sketch. The representation-theoretic and combinatorial aspects of the proof will be given in detail, but it will simply be assumed that the set of geometric points has the structure suggested in [13]. This is so at least when the algebra is totally indefinite, but it is proved by algebraic-geometric methods that are somewhat provisional in the context of Shimura varieties. However, contrary to the suggestion in [13] the general moduli problem has yet to be treated fully. There are unresolved difficulties, but they do not arise for the problem attached to a totally indefinite quaternion algebra, which is discussed in detail in [17].

A GEOMETRIC SPLITTING AND THE C∞ STABILITY OF MAPS FROM SEMIANALYTIC SETS

Abstract John Mather has proved that infinitesimal stability implies stability for proper maps in the category of smooth manifolds. This result gives a computable algebraic criterion for stability. In this paper it is shown that there is an extension of Mather's result when the range is only assumed to be a compact semianalytic set of some real Euclidean space—this class of spaces is an obvious maximal candidate for which computations can be carried out using only classical polynomial algebra. The proof depends on a splitting theorem for the restriction map from the smooth functions on a Euclidean space to those on a closed subset and is proved by an algebraic-geometric method derived from the work of B. Malgrange. No knowledge of functional analysis is assumed although an alternative analytic method for proving the main result is also indicated. Only simple applications are given (mostly to functions defined locally in the neighbourhood of an isolated hypersurface singularity of the type studied by J. Milnor and others) since the author intends to publish a fairly comprehensive study of stability (smooth and C°) of smooth maps on closed semianalytic sets.

On Solving Nonlinear Equations with a One-Parameter Operator Imbedding

One parameter operator imbedding to modify Newton method for solution of nonlinear equations

On the regularity of surfaces, III

1. In two previous papers the regularity of non-singular surfaces in higher space was investigated by geometrical methods based on theorems which are themselves deducible by algebraic-geometrical methods. Among the results obtained, one of the most important was a consequence of the following theorem, due to Severi: