# Mathematics

Mathematics research in the School builds around three broad and overlapping research groupings.

## Discrete Mathematics, Theoretical Computer Science and Logic

Research in this group builds on ideas from combinatorics, mathematical logic and general algebra.

Areas that are actively being developed include recursion theory, complexity of computation, reverse mathematics, algorithmic randomness, algebraic logic, modal logic, model theory, matroid theory, graph theory, set theory, universal algebra, Hopf algebras, lattices and more.

We have strong links with the Philosophy programme and the School of Engineering and Computer Science and joint seminars involving staff and postgraduate students from all three Schools are held frequently.

The School also holds a regular informal seminar series on matroid theory. Anyone interested in participating should contact Geoff Whittle.

## Analysis, Geometry and Number Theory

These research areas belong largely to the domains of continuous mathematics and number fields, having strong connections with calculus but also incorporating many ideas from algebra.

Research currently undertaken by mathematicians in this group includes Banach algebras, elliptic curves, functional and global analysis, invariant theory and singularity theory with applications to robotics and non-standard analysis.

## Applied Mathematics, Theoretical Physics and Modelling

Applied mathematics ranges from theoretical physics to industrial mathematics and from geophysics to fractal geometry. The mathematical ideas involved are equally varied, from structures on manifolds, through differential equations to numerical methods and techniques of modelling.

Research interests of this group include black holes, cosmology, quantum field theory, general relativity, industrial, geophysical, medical and biophysical applied mathematics, mathematical physics and robot kinematics.

The group collaborates with physicists and geophysicists at Victoria University and maintains lively interactions with scientists and mathematicians in government research organisations and universities in New Zealand around the world.

The Gravity Group also contributes to research in mathematics.

## More Marsden Magic

Two mathematicians are among the 25 Victoria academics to be awarded Marsden Fund research grants in the 2014 round.

Dimitrios Mitsotakis has been awarded a Fast Start grant (for early career researchers). His project is entitled "Numerical solution of time-dependent multi-dimensional nonlinear dispersive wave equations with applications to coastal hydrodynamics”.He explains the goal of the project as follows.The mathematical modeling of water waves continues to attract great interest in the scientific community, including mathematics, physics, and engineering. Models that describe water waves are systems of nonlinear partial differential equations. Computing the solutions of these systems requires efficient and accurate numerical methods. Our objective is to develop a theoretical and computational research framework through which we will systematically address wave systems. The new tools willimprove the speed and the accuracy of tsunami early-warning systems, which are critical for mitigating the devastating effects of tsunamis.

Matt Visser is a world-renowned expert on the theory of gravitation. This will be his fourth Marsden project and it concerns "The final stages of the Hawking evaporation of black holes”.Stephen Hawking predicted in 1974 that black holes are not entirely black - they emit radiation and slowly evaporate due to subtle quantum effects. The final stages of this process, when the black hole has become relatively small, continue to generate heated debate and confusion among experts, even after 40 years of intensive research. Matt plans a renewed attack on this problem, based on a three-fold approach: investigating the types of horizon that can occur and their implications for the internal structure of black holes, investigating the quantum energy conditions and the properties of any resulting spacetime singularities, and finally a careful analysis of both the differences and similarities between black-hole and ordinary thermodynamics.

These projects bring to 34 the number of Marsden grants on which Victoria mathematicians and statisticians have been principal investigators, in the 20 years the Fund has been running. The total value of those grants is in excess $12.5M.This remarkable ongoing record places the School of Mathematics and Statistics at the forefront of Victoria’s leading research standing in New Zealand.