E0 231 Algorithmic Algebra (Jan-Apr 2010)

Lectures | Seminars

Lectures

• 6 JAN: Organizational Lecture: Division algorithm, Algorithmic and existential proof
• 13 JAN: Examples: System of linear equations, Division algorithm in one and more than one variables
• 15 JAN: Polynomial equations and affine varieties
• 20 JAN: Properties of affine varieties, Rings, ideals, structure and representation of ideals generated by sets, Vanishing ideal of a set
• 22 JAN: Algebra and Geometry dictionary
• 27 JAN: Monomial ordering, multivariate division algorithm, (Remarks on well orders), Notions on reduction, some properties on monomial orders
• 29 JAN: Monomial ideas, structure of polynomials in monomial ideas
• 3 FEB: Statement of Dickson's lemma and its consequences, Hilbert bases theorem and definition of Grobner Bases
• 5 FEB: Some characterizations of Grobner Bases
• 10 FEB: Noetherian condition (ACC) and HBT
• 24 FEB: S-Polynomials, Buchberger criterion, Buchberger algorithm and its correctness
• 1 MAR: Minimal, Reduced Grobner bases and its uniqueness
• 3 MAR: Elimination orders and ideals
• 5 MAR: Elimination theorem and its applications
• 10 MAR: Rings quotient rings, Isomorphism theorems
• 12 MAR: Quotient rings, maximal ideals, Zorns lemma, Axiom choice
• 15 MAR: More on maximal and prime ideals
• 17 MAR: Correspondence theorem for groups and rings, characterization of maximal and prime ideals
• 22 MAR: Adjunction Rings and Direct Products
• 24 APR: Dickson's Lemma
• 29 MAR: Dickson's Lemma
• 31 MAR: Dickson's Lemma
• 5 APR: Dicksons's Lemma
• 7 APR: Dickson's Lemma
• 16 APR: k-vector basis of residue class ring modulo an ideal
• 21 APR: Algebraically closed fields, Radical ideals, Hilbert Nullstellensatz (without proof)
• 26 APR: Grobner Bases characterization of Zero dim ideals, Universal Grobner Bases
• Written notes on Modules, Noetherian Modules, Syzygies, Proof of Buchberger Criterion

Seminars

• Border Bases by Prabhanjan
• Algebraic characteriation of uniquely vertex colorable graphs Jasine Babu
• Classifying polynomials and identity testing by Vivek
• Integer Programming by Anirudh
• Grobner Bases over rings by Prashant Puranik

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