Geometry & Topology
To understand a basic difference between topology and geometry, imagine a circle. Now ask yourself: Is it a perfect circle? If so, what radius is it? If you cannot immediately answer these questions, your concept of a circle is topological, to do with form rather than precise rigid/geometric shape. A topological circle can manifest geometrically in many different ways ... as an almost perfect circle (as drawn by someone with a steady hand), as ellipses of different shape and size, as pieces of string with ends tied together ... But another thing is very clear: a "circle" is different from a line or an interval.
In a similar way, the 2-dimensional surfaces of a ball or donut are intrinsically different, no matter how you try to stretch or distort. Each in its own way nonetheless has very nice possible geometric shapes: a sphere may manifest as a soap bubble wobbling in the air, but also tries to form a perfectly uniformly curved surface: a "perfect" sphere. On the other hand, if you slice an actual donut into two pieces with a knife, several possibilities arise for the general "shape" of the two pieces ... they may both be rings, may both be like bent balls, one bit may be a ball and the other like the original donut ... all depending on how the cut was made. There are only finitely many topological possibilities, but infinitely many geometric ones.
The research group at Melbourne studies the interplay of geometry and topology, as well as some applications to processes in industry, and models of biological systems. For example, DNA can form closed loops, and there are serious difficulties understanding or modelling enzyme actions which permit the unlinking of two copies after replication. Recent work in mathematical physics of elementary particles and statistical mechanics has provided some ways to try to understand this phenomena. Related mathematics also occurs when considering the "perfect" geometric forms of possible 3-dimensional topological spaces. The most important -- hyperbolic geometry -- is intimately connected with geometric transformations which occur in the theory of relativity.
When equations are used to describe things, such as the relationships of different lengths of components of a mechanical robot, the geometry and topology of possible shapes is again important to understand and describe: Fixing a length is akin to slicing a donut as described above. Problems related to combinatorial optimization often appear, which commonly are of great interest in industry.
Being able to describe, display and compute such objects is also an active research area in this group.