AIS Invariant Theory (2011)

Speakers

• V. Balaji
• Pramath Sastry
• D. S. Nagaraj
• K. N. Raghavan
• A. J. Parameswaran
• Jugal Verma

 Syllabus:

• Revision of algebraic geometric methods, affine algebraic groups and elementary representation theory local algebra, Lie algebras and their representation theory, essential for the rest of the course.
•  Hilbert's fourteenth problem, Reynold's operator, Mumford's theorem on existence of quotients for linearly reductive groups, Weitzenbock's theorem, Fischer's theorem, classical invariant theory (first and second fundamental theorems), geometric reductivity, Mumford's conjecture (various proofs), quotients by reductive algebraic groups, basic GIT, Hilbert- Mumford criterion, and examples of moduli problems as applications of GIT such as endomorphisms of vector spaces, ordered points on the line, elliptic curves. Geometry of quotients, Cohen-Macaulay properties, Hochster-Roberts theorem, Boutot's theorem on rational singularites of good quotients in char 0, Kempf- Rousseau theory of instability, Luna's etale slice theorem, theorems of Luna- Richardson.

Suggested Texts:
1. Invariant Theory, by J. Fogarty.
2. Introduction to moduli problems and orbits spaces, by P. Newstead.
3. Geometric Invariant Theory, by Mumford, Fogarty and Kirwan,
4. Invariant Theory, by Springer.