## Research Interests

I am a mathematician interested in differential geometry, mathematical physics, and symplectic topology. I research (pseudo)-holomorphic curves in symplectic manifolds, and the related Gromov-Witten invariants.

I am working on a large project using a new type of space called an exploded manifold. On the small scale, these exploded manifolds look like ordinary manifolds, but on the large scale, they are piecewise-linear. I use exploded manifolds to study holomorphic curves — which arise in string theory as the world-sheets traced out over time by strings. Using exploded manifolds, these world-sheets act as usual on the small scale, but on the large scale, they look like piecewise-linear graphs reminiscent of interacting one-dimensional particle paths. For Gromov-Witten invariants, this can reduce the difficult problem of finding and counting holomorphic curves to the combinatorial problem of counting these piecewise-linear graphs — called tropical curves.

You can see slides to
an introductory talk on exploded manifolds by clicking
** here**. (There
are lots of pictures and some entertaining zooming around.)
Experts in
Gromov-Witten invariants should instead click
**here** — this more
advanced talk has just as many entertaining pictures, but skips background
to focus on the resulting `tropical' gluing
formula for Gromov-Witten invariants.

I invented exploded manifolds for my work, however algebraic geometers have related spaces called log schemes.
Experts in log geometry can
look at the paper ** here**, where I explain the relationship
between exploded manifolds and log schemes. Log-unaquainted
mathematicians can instead read ** these introductory notes**.

## Contact Information

*Office*: JD 2145

*Phone*: +61 2 6125 3720