Math 683 SPRING 2008, List of lectures
On this page I will post content of all lectures. All handouts also will be posted here.
Monday, March 31: Examples of actions of algebraic groups.
Wednesday, April 2: More examples.
Friday, April 4: Basic properties of orbits. Problems of linear algebra.
Monday, April 7: Quivers and problems of linear algebra. Finite, tame
and wild quivers.
Wednesday, April 9: Finite quivers are positive definite. Classification
of positive (semi) definite quivers.
Friday, April 11: Representations of quivers as a category. Direct sum
decompositions. Indecopmosable representations.
Monday, April 21: Krull-Remak-Schmidt theorem.
Wednesday, April 23: Reflection functors.
Thursday, April 24: Positive roots, Weyl group, Coxeter element.
Friday, April 25: Gabriel's theorem.
Monday, April 28: Examples for Gabriel's theorem: quiver D_4.
Wednesday, April 30: Geometric quotients.
Thursday, May 1: Examples for geometric quotients.
Friday, May 2: Rosenlicht's theorem.
Monday, May 5: Proof of Rosenlicht's theorem: G-invariant rational functions.
Wednesday, May 7: Proof of Rosenlicht's theorem: improvement of the open set.
Friday, May 9: Kac's theorem: statement and reduction to Lemmas A,B,C.
Monday, May 12: Lemma A: existense of indecomposable representations.
Wednesday, May 14: Lemma A: dimension estimates.
Friday, May 16: Lemma A: indecomposable modules with nontrivial endomorphisms ring.
Monday, May 19: End of proof of Lemma A. Plan of proof of Lemma B.
Wednesday, May 21 Reduction to positive characteristic. Number of points of
a variety over a finite field.
Friday, May 23: Lang's theorem.
Wednesday, May 28: Consequences of Lang's theorem. Minimal field of definition.
Thursday, May 29: Counting of absolutely irreducible representations over
a finite field.
Friday, May 30: Brauer's Lemma and independence of orientation. Here is the
Monday, June 2: Representations of affine quivers.
Wednesday, June 4: Hall algebra and Ringel's theorem.
Friday, June 6: Canonical decomposition.