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 Final Exam.
• Monday, June 2: Representations of affine quivers.
• Wednesday, June 4: Hall algebra and Ringel's theorem.
• Friday, June 6: Canonical decomposition.
• THE END