Research_

Manifolds and cell complexes

“Manifolds are around us in many guises.” —Bill Thurston

Manifolds are central objects in the sciences: they arise as solution sets to equations, as graphs of functions, as parameter spaces. How can one tell two manifolds apart? What is the most natural geometry for a manifold?

Manifolds are higher-dimensional generalisations of shapes such as piece of string or the surface of a sphere. A small part of a manifold looks like the familiar Euclidean space, but the overall shape may be very different.

Fundamental research questions in this area are often not only interesting in their own right, but their resolution for special classes of manifolds is important for applications in mathematics and the sciences.

For example, methods to distinguish two given manifolds can help detecting mutation in DNA or determining the shape of our universe.

Determining natural geometric structures on a given manifold, and being able to make exact measurements, has applications to optimisation problems and magnetic resonance imaging.

Research problems of the group at the University of Sydney are often motivated by classification problems of manifolds and their distinguishing algebraic, combinatorial or geometric properties, and are related to geometric group theory, topological quantum field theories, gauge theory and algebraic geometry.

- Dalyell students - please enquire about possible projects.
- Summer vacation scholarships - a great way to see what research is all about.
- Honours students - several projects are available each year.

For domestic and international students, you can find details of possible scholarships here.