Redaktion: K. Sigmund, G. Greschonig (Univ. Wien, Strudlhofgasse ...

48 Hauptvorträge
Modern Developments in Invariant Theory
VLADIMIR POPOV
Department of Mathematics, Moscow State Technical University MGIEM
Bol’shoi Trekhsvyatitel’skii per, 3/12, 109028, Moscow, Russia
popov@ppc.msk.ru
1. Brief historical overview
2. Biregular Invariant Theory
Generators and relations. Hilbert’s 14th problem. Constructive Invariant Theory:
explicit bounds. Good properties in Invariant Theory, three sources: Commutative
Algebra, Algebraic Geometry, Representation Theory. Finiteness theorems.
Explicit classifications of actions with good properties.
3. Birational Invariant Theory
Birational classification of actions. Essential dimension of algebraic groups and
Hilbert’s 13th problem.
[1] V. L. Popov, E. B. Vinberg, Invariant Theory, Encycl. of Math. Sci., Algebraic
Geometry. IV, Springer Verlag, Vol. 55, 1994, 123–284.
[2] Z. Reichstein, On the notion of essential dimension for algebraic groups,
Transformation Groups 5 (2000), No. 3, 265–304.
[3] H. Derksen, G. Kemper, Computational Invariant Theory, forthcoming:
Springer Verlag, 2002.
Singular reduction and Hamiltonian dynamics
TUDOR RATIU
Département de mathématiques, Ecole polytechnique fédérale de Lausanne
CH-1015 Lausanne, Schweiz
Tudor.Ratiu@epfl.ch
http://dmawww.epfl.ch/ratiu/index.html
This talk will present generalizations of the Weinstein-Moser theorem about the
existence of periodic orbits around stable equilibria of Hamiltonian systems. The
results discussed deal with symmetric Hamiltonian systems and address the existence
and the lower estimates on the number of bifurcating relative equlibria and
relative periodic orbits emanating from equlibria and relative equlibria. The case
of formally unstable critical elements will also be discussed. Techniques of singular
reduction are used to obtain some of these results. Special emphasis will be put
on the so called “reconstruction equations” which mirror the equations of motion
in the singularly reduced spaces but expressed on the original smooth manifold.
The results are obtained by combining techniques of singular reduction, normal
forms, and topological estimates. If time permits, the symmetric Hamiltonian