Education, Master Class, Master Class 2002/2003, Detailed Course Content

Detailed Contents of the Courses
1st Semester


Applications of (elliptic) curves

The course concentrates on algorithms dealing with algebraic curves over finite fields. In the case of elliptic curves, we discuss applications to integer factorization and primality testing and to cryptography. More general curves are used in the theory of error correcting codes; the course will treat constructions of such codes and decoding methods.

Lecturer: Jaap Top (Groningen)

Course Material: this information will follow.

Computational methods for differential equations

The Risch algorithm is discussed, which determines whether a given function is the derivative of an `elementary' function. Next, some differential Galois theory is treated leading to the Kovacic algorithm, which determines this Galois group for second order linear differential equations. The case of equations with at most two singularities will be studied in detail. Finally, representation theory of finite groups will be used to construct equations with prescribed Galois group.

Lecturer: Marius van der Put (Groningen)

In this course we will look at various algorithms for solving problems in algebraic number theory. Taking the problem of finding the factorization of integers as a starting point, we will then discuss computational methods for determining properties of algebraic number fields and their subrings, leading to a description of the number field sieve. Along the way we encounter problems involving the factorization of polynomials, the solution of certain Diophantine equations, and the structure of unit and class groups of rings of integers. Some applications in cryptography will be pointed out. If time permits algorithms for the determination of Galois groups of number fields and the analogy with algebraic function fields may also be explored. Demonstration of the implementation in Magma of most of the algorithms discussed will be used to generate examples.

Lecturer: Wieb Bosma (Nijmegen)

The course will deal with various problems which are approached in a constructive way using representations of (mostly finite) groups. It is assumed that the fundamental facts about representation theory are known, e.g. definitions of representations, G-modules, characters, character table, irreducibility. Also a basic knowledge of general group theory should be present, e.g. on permutation groups and finitely presented groups. The following topics are planned: (1) Analysis of finitely presented groups Usually, finitely presented groups are approached by rewriting methods or coset enumeration. If these methods fail to reveal information about a group (typically arising from topology), one can try to construct representations of the group and thus obtain epimorphic images. It will be demonstrated how techniques as p-adic lifting or cohomology computations can be used to prove infinity of a group and to obtain infinitely many images. (2) Rational representations Over finite fields there is a powerful machinery to construct and decompose representations into irreducibles (Meat-Axe). In characteristic 0 there are fundamental problems with these methods, however, in certain cases a refined approach often succeeds. By an alternative approach it is feasible to compute endomorphism rings of rational representations up to degrees of a few hundred. This gives a further method to decompose representations. (3) Higher dimensional crystallography Classical crystallographic space groups are extensions of a 3-dimensional integral lattice by a finite integral matrix group of degree 3 (the point group) acting on it. This concept can be generalized to arbitrary dimensions and gives rise to a higher-dimensional (mathematical) crystallography. In an algorithmic approach to this area, methods will be presented that allow to construct, enumerate and classify crystallographic groups. A typical task would be to construct all space groups with a given type of point group.

Lecturer: Bernd Souvignier (Nijmegen)

In this series of lectures we look at the representation theory of semisimple Lie algebras, and the corresponding Chevalley groups. Various combinatorial objects play a major role, which makes this theory very suitable for doing computations. Indeed, since the early seventies people have been performing computations in this area. We start by looking at root systems, and their Weyl groups. These objects are of vital importance in the representation theory of semisimple Lie algebras. Algorithms for working with Weyl groups often lie at the heart of algorithms for computing with representations of semisimple Lie algebras. Subsequently we will consider semisimple Lie algebras and their irreducible representations. We will describe algorithms for working with these objects. For example, we will look at Littelmann's path model. This recent development, which uses only concepts like root systems and Weyl groups, allows one to compute very useful data concerning representations of semisimple Lie algebras. It is possible to give a version of the theory which entirely works over the integers. Hence we can reduce everything modulo a prime number. This leads to the Chevalley groups. Algorithms for working with these groups and their representations will also be treated. If time permits we will look at some more specialized subjects, such as quantum groups and canonical bases, and Hecke algebras and Kazhdan-Lusztig polynomials.

Lecturer: Willem de Graaf (Utrecht)