Invariant theory has a history that is almost one-and-a-half centuries long. It owes its existence to problems from number theory, algebra and geometry, which appeared in the work of Gauss, Jacobi, Eisenstein and Hermite. Invariants came into existence as a tool to distinguish (and, ultimately, to classify) non-equivalent objects in algebraic problems where the equivalence relation is usually given by the action of a group on a set. Although the connection between invariants and orbits was essentially discovered in the work of Aronhold and Boole, a clear understanding of this connection has not been achieved until recently when invariant theory was in fact subsumed by a general theory of algebraic transformation groups. The aim of these lectures is to give an introduction to this exciting field of mathematics and to discuss some fundamental problems, e.g., Hilbert's 14th Problem, the rationality problem, the problems of constructive (computational) Invariant theory. Along the way we will describe many examples such as invariants of vectors and quadratic forms, invariants of binary forms, invariants of linear operators and of more general tensors. Some modern basic geometric tools and theories from algebraic transformation groups (quotients, Hilbert-Mumford theory of null-forms) will be developed, and it will be shown how to apply them to the problems discussed before. Finally, the discussion will go back to the so-called classical invariant theory which is still an important source for many unsolved questions and problems.

Zeit:       Mittwoch 10 - 12
Ort:        HG G 43 (Hermann-Weyl-Zimmer)
Beginn:  30.10.02
 

M. Struwe