## Terence Harris

**Biography**

Terence Harris is a student studying mathematics at the University of New South Wales. Some of his interests include point set topology, and the relationship between game theory and differential equations. He will be commence honours study in 2015.

**Noncommutative calculus**

There is a rich theory of multivariable calculus that can be used to study mappings between euclidean spaces, and more generally between spaces that are locally euclidean (e.g, Riemannian manifolds). Currently there is interest in Carnot-Caratheodory spaces and “metric spaces”; these are locally euclidean spaces in which it is possible to move only in certain directions at any point, but it is still possible to get from any point to any other point. These spaces are naturally modelled on nilpotent Lie groups in the same way that riemannian manifolds are modelled on euclidean spaces. There are various projects in this context:

The derivative is well understood, but the integral is not, because integrals involve addition, and implicitly assume commutativity in their denition. How can we get around this?

In multivariable calculus, it is known which dierential forms are dierentials of other dierential forms. In the Carnot{Caratheodory context, the so-called Rumin complex explains this. But we only know how this works for a few simple examples. Looking at a few more examples is the rst step towards gaining insight into this problem.

Differentiable Carnot-Caratheodory mappings are often automatically smooth. The reason for this is only understood for a few examples. Looking at some more examples should enable us to distill out the essence of why this happens.