Finite geometry, graph theory and design theory
Research has been done together with Ball and Mazzocca on the
existence of 
-arcs in projective planes of orde q.
A famuous 40 year old problem has been solved by showing that such arcs
do not exist in planes of odd order.
In joint work with Ball, Storme and Szönyi the question of the number of directions determined by the graph of a function on a finite field is almost completely answered. This gives a solution to a 25 year old problem of Rédei.
In joint work with Storme and Szönyi the lower bounds for the size of multiple blocking sets in Desarguesian projective planes have been improved considerably. In most cases the exact structure of such a blocking set meeting the lower bound has been determined.
Together with Szönyi the nonexistence of certain blocking sets of almost Rédei type has been proven.
MDS codes have been studied in terms of arcs in projective spaces.
Togeteher with Cooperstein Lie incidence systems have been investigated.
Consider an embedding of a geometry 
into a projective space.
Let 
be the geometry of points and lines of the projective
space that (as point sets) are in the image of the point set of 
.
The general question is: describe 
.
We have given an answer to this question
in case 
is a Lie incidence geometry.
In joint work with R. Blok it is determined which Lie incidence systems are spanned by an appartment.
Point line geometries with big subspaces have been studied. Thereby the connections with Zara graphs, polar spaces and Fischer spaces were investigated.
Euclidean lattices generated by vectors with minimal norm 3
have been studied. This work is a continuation of work by Neumaier.
Thereby we obtained characterizations of some lattices
related to the sporadic simple group 
.
All distance regular graphs without 3-claws have been classified.
The spectra of several association schemes have been determined.
A chapter on strongly regular graphs has been written for the Handbook of Combinatorial Designs.
In joint work with Colbourn and Dinitz a chapter on orthogonal Latin squares for the Handbook of Combinatorial Designs has been written.
Theory of groups and Lie algebras
It has been studied to what extend a finite subgroup H of an algebraic group G leaves invariant a characteristic lattice.
First it appeared, that under some weak conditions for H
there is a unique splitting field for H.
This, however, does not imply that it is easy to find this unique field.
Together with Tiep, this field has been determined in the special case
of a Jordan subgroup, which was a good step in the right direction.
For, if R is de ring of integers of the minimal splitting field,
then one might wonder whether there is a definition for a subgroup 
such that H is the group H is the subgroup of points defined over R,
i.e., 
.
This is possible for the case 
and H one of the 4 Lie primitive
subgroups contained in G, as was shown in joint work with Nebe and Plesken.
The general case is still under consideration.
Together with David Wales the 
-module of cubic forms modulo
the third powers has been investigated. Here k is an algebraic field of characteristic 3.
It appeared that this module admits finitely many 
-orbits.
The article on this research, which will appear in the Journal of Algebra
contains an algorithm in a nutshell, how to rewrite a general
form to a representative of one of the orbits.
This algorithm has been implemented in the case of 3 dimensions
as an interactive program that can be tested via the Proceedings of the Organics Workshop in Vancouver.
With Gábor Ivanyos and David Wales work has been done on an algorithm to determine the radical of a finite dimensional algebra. Thereby we studied and simplified Rónyai's polynomial algorithm. In particular we were able to avoid a lift to the integers which was necessary in Rónyai's approach. The result can be used to check whether two algebra representations are equivalent or not.
With G. Ivanyos, A. Kuronya and L. Rónyai we studied the Levi decomposition of a Lie algebra in characteristic zero.
It has been proven yet that the existing algorithm for this is polynomial. By making some adjustments to that algorithm we can prove it to be polynomial. Furthermore a new algorithm for computing the nil radical of a Lie algebra has been developed. Moreover, structure computations of a Lie algebra have been studied.
In joint work with S. Elashvili we have studied centralisors
of classes of nilpotent elements in 
.
It appeared that
all of these centralizors are of index 8.
With T. Breuer some of the above algorithms have been implemented in GAP.
Work has been done on a conjecture of Deligne regarding the existence of a tensor category in which the representation categories of the exceptional groups and some other simple algebraic groups are specialisations.
The conjecture is based on a surprising patern in the decomposition of the tensor powers of the adjoint representation of these groups. An article will appear in Comptes Rendues with the results of the decomposition of the fourth tensor power. We are working on the fifth power. We are still looking for some combinatorial model for the tensor category and have made some steps in the right direction.
In joint work with J. van Bon diagram geometries of type
and 
have been studied. The 7-dimensional
representation of the groups of type 
has been characterized
by geometric means as well as the spin representation of the groups of type 
.
The connection between cotriangular geometries and a class of Lie algebras introduced by Kaplanski has been studied. A characterisation of these Lie algebras has been given. As a consequence of this characterisation the automorphism group and the algebra of derivations of these Lie algebras could be determined. Thereby we proved an improved version of a conjecture of Rodman and Weichsel.
Computer algebra, computers and mathematics
Together with Heck an article has been written on the experience with Cam design.
An article on the use of Gröbner bases in the theory of error correcting codes has been written.
An introductory article on the fundamental algorithms of permutation groups has been written. Partially these algorithms have been implemented in Maple and Mathematica.
The path model of Littelmann is studied and some algorithms have been implemented in GAP.
Some work has been done on the COCO (`coherent configurations') computer algebra system.
Much time has been spent on projects on the interaction with a computer, in particular on the integration of computer algebra systems with proof verification systems and interactive books.