Some problems in algebraic group theory

For smooth actions of compact Lie groups on differentiable manifolds, the existence of a smooth slice transversal to each orbit gives a clear description of the local structure. In 1973, D. Luna proved the existence of a slice in the etale topology at a closed orbit, for reductive algebraic groups acting on an affine variety, over an algebraically closed field of characteristic zero. This thesis explores the extent to which Luna's methods work over an arbitrary field. Conditions for the quotient of a morphism to be etale are given, necessary and sufficient conditions are given for the existence of a slice on a smooth affine scheme, and a new proof is given of the isomorphism of the unipotent variety of a split connected, simple, semisimple algebraic group with the nilpotent variety of its Lie algebra

Some problems in number theory that arise from group theory

In this expository paper, we present several open problems in number theory that have arisen while doing research in group theory. These problems are on arithmetical functions or partitions. Solving some of these problems would allow to solve some open problem in group theory

Quadratures of Pontryagin extremals for optimal control problems

We obtain a method to compute effective first integrals by combining Noether's principle with the Kozlov-Kolesnikov integrability theorem. A sufficient condition for the integrability by quadratures of optimal control problems with controls taking values on open sets is obtained. We illustrate our approach on some problems taken from the literature. An alternative proof of the integrability of the sub-Riemannian nilpotent Lie group of type (2,3,5) is also given.control theory group (cotg)CEOCFCTPOCI/MAT/55524/200

Renormalization group in Statistical Mechanics and Mechanics: gauge symmetries and vanishing beta functions

Two very different problems that can be studied by renormalization group methods are discussed with the aim of showing the conceptual unity that renormalization group has introduced in some areas of theoretical Physics. The two problems are: the ground state theory of a one dimensional quantum Fermi liquid and the existence of quasi periodic motions in classical mechanical systems close to integrable ones. I summarize here the main ideas and show that the two treatments, although completely independent of each other, are strikingly similar.Comment: Plain TeX, 20 pages 7 figures (included in ps format

Twining characters and orbit Lie algebras

We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched.Comment: 6 pages, LaTeX, Talk given by C. Schweigert at the XXI international colloquium on group theoretical methods in physics, July 1996, Goslar, German