Bulletin

Issue: September 2005

Research in Action

Duality theory - Getting a fix on the doppelgangers

If you're a mathematician or a logician you'll know your duality theory.

Chances are you'll also know that the algebraists of La Trobe University's Department of Mathematics are internationally cutting-edge duality theorists.

They do after all have as an Associate Professor of Mathematics, Dr Brian Davey - one of the founding fathers of duality theory - to ensure that theoretical skills get a regular workout.

They now also have two ARC-funded post-doctoral research scientists working with Dr Davey to take natural duality theory into even higher realms, and to solve some of the most fundamental problems practitioners face in applying it.

Dr Jane Pitkethly, with a PhD from La Trobe University, and Dr Marcel Jackson, who did his PhD at the University of Tasmania but now has a mixed lecturing and research position in the Department of Mathematics at La Trobe, are both working on different aspects of natural duality theory.

What Dr Pitkethly seeks to resolve are big issues in the world of duality theory - fundamental questions dating back to the birth of the theory in 1980 - and Dr Jackson wants to know how its techniques can be applied in Semigroup Theory (an area of algebra related to automata and formal languages).

Dr Pitkethly, in effect, wants to establish the top and bottom lines of duality theory - including whether in some circumstances it doesn't work at all - so that algebraists, logicians, computer scientists and others who use it will not waste time in blind alleys looking for dualities that may not exist.

For the uninitiated, here's how Dr Jackson explains it: 'A duality means there are two sides to something, as with a mirror. When you look into a mirror, you see an opposite version of yourself, but "the person in the mirror" also sees their mirror image, which is you. These two acts cancel each other out.

'With natural duality theory, the notion of duality arises where you have an object, Object A, and you aim to find some kind of "doppelganger" of it - a fundamentally new object that lives in the mirror world. You want to have the doppleganger of the doppleganger of Object A to be essentially what you started with - Object A. Actually, you want a rule for finding these dopplegangers, for whole classes of objects. These notions of duality are everywhere in mathematics, but natural duality is a particularly powerful way in which they can arise.'

Get it? It is a mathematical tool for studying algebra by finding what looks like a double or parallel version of an original problem, and working 'in the mirror' to solve the original problem from a new and perhaps radically different perspective.

It can't be assumed however that dualities always exist, or that when they do they are an 'exact' duality: there are many possibilities from non-existent to full dualities to strong dualities and near-perfect to perfect dualities.

'What we call strong dualities are what we really want,' Dr Jackson explains, 'whereas a duality in general is slightly one-sided, like the mirror example; we exist on one side and when we look in the mirror there's a mirror image, but there's an asymmetry because you can't really start off in the mirror. This can be useful, but it's a whole lot better if everything works starting from the mirror side as well.'

Dr Jackson and his fellow algebraists of La Trobe's General Algebra Group will be just as happy to find that - while natural dualities are everywhere in algebra - they do not exist to the extent hoped for in semigroup theory.

'I'd be happy enough to prove that there is no algorithmic way to decide whether or not a duality exists. You're flying blind when you're trying to prove something, and you don't necessarily know it is possible.'

Dr Pitkethly hopes when her ARC-funded post-doctoral work is completed to give all practitioners of this elusive science more systematic techniques for recognising its limits and its potential - and cheerfully anticipates for every problem she solves she will discover a few more.