Research topics:

Algebra

There are two key areas of algebra catered for in the Department: group theory and ring theory.

The group theorists, John Curran and Dennis McCaughan, have recently been interested in finite p-groups, where p is a prime, and their automorphism groups. This has involved use of a recently developed computer package which is designed to carry out algebraic calculations and is particularly helpful for research in finite group theory. Those who are familiar with the definition of a group can see an example of their research in their recent article:
Central automorphisms that are almost inner, Communications in Algebra, vol. 29, pp. 2081-2087 (2001).

The ring theorist, John Clark, is mainly interested in non-commutative rings and how modules over these rings can be used to characterise them. Some examples of his recent research can be obtained from his personal web page.

Prospective postgraduate students nterested in doing research in algebra at Otago should contact one of these algebraists to get further details.

Analysis

Peter Fenton
I am interested in entire functions (i.e. functions that are analytic in the entire complex plane), subharmonic functions (the prototypical example being the log of the modulus of an analytic function, which is harmonic except at the zeros of the analytic function, where it is minus infinity), and to a lesser extent meromorphic functions (analytic except for poles).

For example: what can be said about the behaviour of the terms of the Taylor series of an entire function? How large is the smallest harmonic function that lies above a given subharmonic function? What is the relationship between the largest and smallest values of a subharmonic function on large circles centred on the origin?

A selection of papers can be seen here.

Discrete Mathematics

Robert Aldred
My interests and expertise lie in the broad area of Discrete Mathematics.

Primarily I am a Graph Theorist and work on problems which include paths and cycles in graphs, connectivity, colouring and factors in graphs. One example of a factor in a graph is a 1-factor or perfect matching. This is a set of independent edges which together cover all of the vertices in a graph. The partition of the vertex set into pairs induced by a perfect matching lends itself naturally to many real problems of assignment and there is much interest in which graphs admit perfect matchings and what conditions may be placed on the perfect matchings that exist (e.g. can we always find a perfect matching that includes some given set of independent edges or which avoids some given set of edges?). An interesting result along these lines was obtained by Professor Mike Plummer of Vanderbilt University and myself and it states that in any graph that can be embedded in the plane, there is a set of three independent edges e, f and g such that any perfect matching containing both e and f must also contain g.

While my work continues in the area of graph factors I am also involved in projects of a more combinatorial nature. I currently work with an active theory group in Computer Science investigating the classification and enumeration of permutations avoiding certain subpermutations.