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East African Summer School on Commutative Algebra and Algebraic Geometry

August 9-20, 2004

University of Nairobi, Kenya

Part I : An introduction to Algebraic Geometry and Combinatorial Methods in representation theory.

Main topics:

- The Grassmann variety and its Schubert Cells
- Plücker coordinates and their quadratic equations
- The standard monomial basis of Hodge associated to Plücker coordinates
- The theory of standard bitableaux
- Representations of the linear and symmetric group

Part II : Ideals, varieties and algorithms

Main topics:

(i) Groebner basis basics orderings on monomials

- Division algorithm in the polynomial ring

- Monomials Ideals and Dickson’s lemma
- Hilbert Basis Theorem and Groebner Basis
- Buchberger’s algorithm
- Ideal membership problem, implication problem and elimination theory
- The geometry of elimination

- Symmetric polynomials
- Finite matrix groups and rings of invariants
- Generators for the ring of invariants

- Most computer work will be done by using the computer algebra package CoCoA.
- Besides the main minicourses, several other topics in computer
algebra and also some experiments of computer assisted research will be
offered.