What is Computer Algebra?

Algebra as a mathematical discipline arose from attempts to find solutions for equations. In computer algebra the electronic computer is used as a tool to solve equations. The methods developed in algebra are employed and adapted to constructive needs in order to design, implement and apply algorithms. What distinguishes computer algebra from numerical analysis is the aim to find exact rather than approximate solutions. This leads to the need to manipulate mathematical objects represented as symbols in software in a very precise way. Symbolic computation is therefore often used as a synonym for computer algebra.

Mathematical Aspects

There are many aspects to computer algebra; not only because the field of algebra itself is wide and there are many different types of equations to be studied, but also because the interaction with computers raises many issues relating to computer science. The focus in the Master Class will be on mathematical aspects of computer algebra. This involves the study and development of algorithms to perform algebraic tasks in an exact and efficient way. Although the practical performance of the algorithms is of great concern, precise complexity analysis of the algorithms will be of secondary importance. On the other hand, it will be instructive to observe the behaviour of various implementations of algorithms, for example in existing computer algebra systems.

Program 2002/2003

Coverage

Algebra comprises subjects like linear algebra, group and ring theory, commutative algebra, but there are also strong algebraic aspects to parts of number theory, geometry, and the theory of differential equations. The Master Class will cover important algorithms in these areas, stressing the common principles, and showing the main applications. We will show how to compute efficiently with groups, number fields, algebraic curves, and how to solve systems of polynomial equations and certain differential equations. Becoming aware of what can be computed with modern computer algebra systems in these areas, and how it is done is the main goal of the Master Class.

Systems

Hands-on experience with, and exercises to programme algorithms in some computer algebra systems will be an integral part of all courses. Besides a variety of smaller packages and systems for specific purposes, Magma and Maple will be used as general tools for instruction with the courses listed below.

Courses

We offer two semesters of four courses at the universities of Groningen, Nijmegen and Utrecht. The courses are accompanied by practice sessions, using software running on the UNIX systems of the participating universities.

What do we want?

If you are a talented and highly motivated student who is not afraid to be confronted with abstract mathematical concepts as well as with hands-on programming of algorithms, you may be eligible for participation in this Master Class. You should hold an undergraduate degree in Mathematics (or a related field with a strong mathematical background, such as Computer Science). Your background includes:

• knowledge of basic linear algebra, groups and rings;
  
• some experience with programming in a modern computer algebra system;
  
• the ability to communicate in English (very important!) and collaborate with your fellow students and master class teachers.
  

You will be expected to take actively part in the courses, as well as in the working sessions. Both concrete examples and abstract theory will be studied. Your knowledge will be further enhanced by working at exercisies and a research problem, and by presenting at least one lecture at the seminar.

Participants

Here is a list of the foreign participants of the Master Class.