## Verksamhet under hösten 1999

Sällskapet håller tre möten under hösten: tisdagen den 26 oktober, tisdagen 16 november samt ordinarie årsmöte tisdagen den 14 december.
Mötena börjar klockan 18.30 i sal C i Matematikhuset. Alla, även icke-medlemmar, är varmt välkomna! Före mötena, från klockan 18.15, serveras gratis förfriskningar. Efter varje möte inbjuds alla till en eftersits med mat och dryck till självkostnadspris.

Den 26 oktober håller Amiran Ambroladze, LTH, föredraget

### General orthogonal polynomials

The lecture is concerned with the asymptotic behaviour and zero distribution of general orthogonal polynomials (all definitions will be given). We will give a tutorial survey of the main results in this area and discuss some open problems. As an example, we give here one of the open problems: It is easy to prove that if the support of a measure belongs to the interval [-1,1], then the zeros of the corresponding orthogonal polynomials also belong to this interval. Now let us assume that the interval [-1/2, 1/2] has no intersection with the support of the measure. Is it true that there exists an infinite subsequence of the orthogonal polynomials which have no zeros in the interval [-1/2, 1/2]? (For the full sequence of the orthogonal polynomials the assertion is not true: If we take a measure symmetric about the origin, then the zeros of the corresponding orthogonal polynomials are also symmetric, which, in particular, means that the odd polynomials have a zero at the origin.)

Den 16 november håller Carsten Thomassen, DTU Lyngby, föredraget

### Patterns and map colourings on surfaces

Two of the best known results on maps or graphs on the sphere are the characterization of the regular polyhedra (the tetrahedron, the cube, the octahedron, the icosahedron, and the dodecahedron) and the 4-colour theorem. The former is easy, the latter is complicated. Both results have analogues for other surfaces. For example, every map on the torus can be coloured in 7 colours and not necessarily in 6 colours. Inspired by Kempe's incomplete proof of the 4-colour conjecture, P.J.Heawood gave in 1890 a formula for the smallest number of colours needed to colour any map on a fixed surface (other than the sphere). Although Heawood's claim was correct it took about 80 years before a proof was completed (By Ringel and Youngs). In this talk we discuss the regular tilings of higer surfaces. Only the torus and the Klein bottle have infinitely many, and we describe them completely with only finitely many exceptions. Also, there is a 5-colour theorem for each fixed surface S in the following sense: There is a finite set of (forbidden) maps on S such that any map on S can be 5-coloured if and only if it does not contain any of the forbidden maps. The list of forbidden maps is known only for the sphere (where it is empty), the projective plane (where there is only one forbidden map) and for the torus (where there are four forbidden maps). There is no 4-colour theorem of this type, except for one surface: the sphere.

Den 14 december så följer vi upp succén från våren, då Lars Gårdings prisbelönta dialog Matematiken, Livet och Döden framfördes, genom att presentera akt 2

### Hos Djävulen

I denna episod träffar John von Neumann några intressanta personer, t.ex. så diskuteras teologi med Augustinus. Bland skådespelarna finner vi bl.a. Magnus Fontes som Djävulen och Per-Anders Ivert som John von Neumann.