Gröbner bases and their applications, Autumn 2011

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This module (for students aiming to obtain a Master's degree in Mathematics) is intended as an introduction to an important computational method of algebra, that of Gröbner bases [of defining relations of some algebraic object]. Defining something by generators and relations is a very common situation in mathematics: in geometry, it is often useful to define a geometric object as the set of points whose coordinates satisfy some equations, in algebra, one can present an algebraic structure (group, algebra etc.) by generators and defining relations etc. Gröbner bases give an algorithmic way to determine which relations follow from a given system, — in a sense, they generalise the notion of long division to the case of defining relations. The module also gives an introduction to basic homological algebra for associative algebras.

Syllabus

• Gröbner bases and elimination in the commutative case: long division, Gauss-Jordan elimination, general case of solving systems of arbitrary polynomial equations. Finiteness, universal Gröbner bases.
• Noncommutative associative algebras. Graded and filtered algebras. Growth. Hilbert series. Examples. Rationality of Hilbert series (Hilbert-Serre theorem, Govorov's theorem).
• Gröbner bases for associative algebras. Diamond Lemma. Examples. Application: Poincare-Birkhoff-Witt theorem via Gröbner bases. Gröbner bases for left/right modules.
• Basic homological algebra. (Co)bar homology, Ext and Tor groups. Anick's resolution for an augmented algebra with a Gröbner basis of relations. Koszul property vs PBW property. Examples.

Assessment

There will be several home assignments of varying lengths, depending on the topics covered. However, the primary method of assessment is via a written examination. The final mark is computed as MAX(F,0.8F+0.2C), where F is the percentage earned in the final exam, C is the percentage earned by means of continuous assessment (home assignments). Thus, if a student does well in the final exam paper, their possible poor performance during the semester would not affect the final mark. However, since the final exam paper questions are modelled on questions from home assignments, students are strongly recommended to attempt as many problems from home assignments as they can.

Literature

No specific textbooks are recommended; some handouts and xerox copies of relevant literature will be disseminated in class.