# Dr Anthony Henderson: First Christopher Heyde medallist

Dr Anthony Henderson, from the School of Mathematics and Statistics, won the 2011 Christopher Heyde Medal from the Australian Academy of Science - the first time the medal and $10 000 prize has been awarded.

The new award recognises distinguished research in the mathematical sciences by a researcher under 40 years of age, in one of three fields on a rotating basis: pure mathematics; applied, computational and financial mathematics; and probability theory, statistical methodology and their applications. Dr Henderson won his medal for research in pure mathematics.

"This is a great honour and I'm very pleased that the Australian Academy of Science has chosen to reward my research. I was quite surprised, since I'm only 34 and I thought it would go to someone closer to the age limit," said Dr Henderson.

The latest honorific medal to be added to the Australian Academy of Science's annual awards honours the contributions to mathematics made by Professor Christopher Heyde, who was the Foundation Dean of the School of Mathematical Sciences at the Australian National University, and Professor Emeritus of Statistics at Columbia University, New York.

"Awards for early career researchers are always welcome. I think they can play an important role in encouraging young people to stay in Australia or return here," said Dr Henderson.

"I hope that the Heyde Medal will become a major indicator of the work being done by young Australian mathematicians and statisticians, like the Pawsey Medal in physics," said Dr Henderson.

Dr Henderson works on geometric and combinatorial aspects of representation theory.

"When we talk about a 'group' in mathematics, we mean a collection of operations which preserve something. For example, in chemistry, if you have a molecule which has some symmetry, then there are rotations and reflections and so on which leave the molecule unchanged, and those form a group. In algebra, if you have a function involving several variables, then there may be some permutations of the variables which leave the function unchanged, and those form a group. These groups come with a kind of multiplication, because you can do one operation and then a second one, and that whole process is a third operation in the group," said Dr Henderson.

"My research is in representation theory, which studies how to represent one of these abstract groups of operations by a concrete collection of matrices of numbers, so that the multiplication is just the standard multiplication of matrices. These representations are very useful for calculations relating to the original molecule, or function, or whatever it may be; they're also needed to describe the states of various particles in quantum physics. But it can be hard to find representations, and there are some basic classes of groups where we still don't know them all after more than a hundred years of searching," explained Dr Henderson.

"My main contribution has been describing some new representations of old groups. The idea is that even if you start with the first kind of group - the symmetries of a molecule or other geometric object in ordinary space - there may be some group of the second kind, defined from a function or collection of functions, which has the same multiplication law. Once you know that, you can look at a different geometric object, the set of solutions when you equate all the functions to zero; the functions may involve many variables, so this solution set may live in some very high-dimensional space.

"Then there's a trick you can use, called homology, to produce matrices from this solution set and get a representation of your original group. This idea goes back decades, but what I've done is to find various new ways to make it work," said Dr Henderson.