Cornell Math - MATH 732, Fall 2005

MATH 732: Computer Algebra Systems and Algorithmic Algebra (Fall 2005)

Instructor: Gerhard Michler

Meeting Time & Room

Almost all areas of algebra and related areas in mathematics have specialized computer algebra systems. It is the purpose of this course to give a survey of these systems and their use in computational algebra.

Most mathematicians have experience with the multiple purpose systems MAPLE/Matlab and MATHEMATICA. In this course we concentrate on these systems which are helpful in modern research in commutative and noncommutative algebra and their applications in discrete mathematics.

A very efficient system is MAGMA. It has powerful implementations of many useful algorithms in the areas:

a) Rings and fields,
b) Modules and algebras,
c) Groups and representation theory,
d) Elliptic curves and algebraic number fields,
e) Enumerative combinatorics,
f) Error correcting codes.

It does not provide access to its source code. Therefore the systems GAP (group theory, representation theory, character tables) and Chevie (symbolic calculations with generic character tables of groups of Lie type) will be surveyed as well.

As textbooks for this section I suggest:

J. J. Cannon & C. Playoust, An introduction to algebraic programming with MAGMA, Springer-Verlag, Berlin, 2001.

D. F. Holt, Handbook of computational group theory, Chapman and Hall/CRC, Boca Raton, 2005.

I will choose special topics from these books. In particular, I will address some applications of MAGMA and MAPLE in group theory in theoretical research projects. Furthermore, there will be demonstrations in class.

This course will also survey the following systems used in commutative algebra and number theory:

Macaulay 2,
see: Grayson & Stillman, A computer software system designed to support research in commutative algebra and algebraic geometry.

SINGULAR,
see: G. M. Greuel & G. Pfister, A SINGULAR introduction to commutative algebra, Springer-Verlag, Heidelberg, 2002.