Olav Arnfinn Laudal

The Legacy of Niels Henrik Abel

The collected works of Niels Henrik Abel, edited by Sophus Lie and Ludwig Sylow in 1881, contain 29 papers. The first was written in 1823 when Abel was 21 years old, and the last was jotted down on his deathbed in January and February 1829.

Noncommutative algebraic geometry

The need for a noncommutative algebraic geometry is apparent in classical invariant and moduli theory. It is, in general, impossible to find commuting parameters parametrizing all orbits of a Lie group acting on a scheme. When one orbit is contained in the closure of another, the orbit space cannot, in a natural way, be given a scheme structure. In this paper we shall show that one may overcome these difficulties by introducing a noncommutative algebraic geometry, where affine “schemes” are modeled on associative algebras. The points of such an affine scheme are the simple modules of the algebra, and the local structure of the scheme at a finite family of points, is expressed in terms of a noncommutative deformation theory proposed by the author in [10]. More generally, the geometry in the theory is represented by a swarm, i.e. a diagram (finite or infinite) of objects (and if one wants, arrows) in a given k-linear Abelian category (k a field), satisfying some reasonable conditions. The noncommutative deformation theory refered to above, permits the construction of a presheaf of associative k-algebras, locally parametrizing the diagram. It is shown that this theory, in a natural way, generalizes the classical scheme theory. Moreover it provides a promising framework for treating problems of invariant theory and moduli problems. In particular it is shown that many moduli spaces in classical algebraic geometry are commutativizations of noncommutative schemes containing additional information. 2000 Mathematics Subject Classification: 14A22, 16E, 16D90, 16G, 13D.

Local moduli and singularities

The prorepresenting substratum of the formal moduli.- Automorphisms of the formal moduli.- The kodaira-spencer map and its kernel.- Applications to isolated hypersurface singularities.- Plane curve singularities with k*-action.- The generic component of the local moduli suite.- The moduli suite of x 1 5 +x 2 11 .

Geometry of Time-Spaces: Non-commutative Algebraic Geometry, Applied to Quantum Theory

Introduction Phase Spaces and the Dirac Derivation Non-Commutative Deformations and the Structure of the Moduli Space of Simple Representations Geometry of Time-Spaces and the General Dynamical Law Interaction, Decoherence and Decay.

Cosmos and Its Furniture

In this note I shall continue the study of the geometry of the moduli space of pairs of points in three dimensions. I shall show that this space, \(\underline{\tilde{H}}\), is the base space of a canonical family of associative k-algebras in dimension 4. The study of the corresponding family of derivations leads to a natural way of introducing a spin structure in \(\underline{\tilde{H}}\), and it also furnishes a possible mathematical model for super symmetry and a Big Bang scenario in cosmology. These subjects are all treated within the setup of Laudal (Geometry of Time Spaces. World Scientific, Singapore, 2011).