Lectures

Schedule of Lectures |  List of Lectures

Projective planes and loops by Franz Kalhoff (Dortmund University, Germany)

This lecture covers some topics regarding the connections between projective planes and loops. The loops arising from a projective plane P actually stem from the webs of P, i.e. from certain geometric substructures of P. However, our concern is not the one-to-one correspondence between webs and loops. We are mainly interested in the impact the surrounding projective plane has on the arising loops, and therefore the emphasis of this lecture lies on double loops and ternary fields associated to projective planes.

After an introduction to projective planes, their coordinatizations and their global and local automorphisms, we restrict our attention to two main points, namely to the Lenz-Barlotti classification of projective planes and its consequences on their double loops, and secondly to the notion of the extended radical (a certain normal subloop of the multiplicative loop of a ternary field) and its geometric aspects.

GAP and loops by Gabor Nagy (Bolyai Institute, Szeged, Hungary) & Petr Vojtechovsky (University of Denver, USA)

[Presentation 1]   

Even small computational problems in nonassociative algebra are difficult to tackle by hand. We have therefore developed a package for GAP (Groups, Algorithms, and Programming) called LOOPS, that is tailored for calculations with small quasigroups and (mainly) loops. The goal of these lectures is to explain what the capabilities of the package are, and how to use it. As an illustration, we will present several recently solved problems in which LOOPS played an important role.

If possible, the participants should bring their own laptop.

Connected transversals and multiplication groups of loops by Markku Niemenmaa (University of Oulu, Finland)

The left and right translations of a loop generate a group called the multiplication group of the loop. The multiplication group can be characterized in purely group theoretic terms and the notion of connected transversals is central to this characterization. We start by considering the basic properties of connected transversals in a group and we continue by proving group theoretic results which have direct loop theoretic interpretations. Among other things we show that the inner mapping group of a loop can never be a finite nontrivial cyclic group and we also show that finite loops with abelian inner mapping groups are centrally nilpotent.

Quasigroup representations by Jonathan D. H. Smith (Iowa State University, Ames, USA)

The course is designed to give a brief introduction to the three main aspects of quasigroup representation theory: character theory, permutation representations, and module theory.

There will be three lectures. The first is introductory in nature, establishing notation, basic concepts, typical examples, and some slightly more advanced concepts, such as relative multiplication groups and central isotopy, which are needed for the representation theory.

The remaining lectures will cover the three branches of representation theory in turn. The approach will be to give a fairly detailed and careful coverage of the rudiments, at a level accessible to beginning graduate students. The aim is to prepare participants for their own exploration of the more advanced techniques and research problems.

Transversals in latin squares by Ian Wanless (Monash University, Australia)

[Presentation]

A latin square of order n is an n by n array of n symbols in which each symbol occurs exactly once in each row and column. A transversal of such a square is a set of n entries such that no two entries share the same row, column or symbol. Transversals are closely related to the notions of complete mappings and orthomorphisms in quasigroups, and are fundamental to the concept of mutually orthogonal latin squares.

This course provides a brief survey of the literature on transversals. We cover (1) existence and enumeration results, (2) generalisations of transversals including partial transversals and plexes, (3) the special case when the latin square is a group table, (4) a connection with covering radii of sets of permutations.

The subject is littered with tantalising conjectures and open problems, many of which will be discussed.