# Dr Brett Parker

## Research interests

Broadly, I am interested in mathematical physics, differential geometry and topology, symplectic topology, string theory, Gromov-Witten invariants, log geometry and tropical geometry.

Specifically, I am currently studying holomorphic curves in symplectic manifolds by applying a degeneration in which holomorhpic curves, which are two dimensional objects converge in some sense to tropical curves, which are piecewise linear one dimensional graphs. This allows me to express Gromov-Witten invariants as a sum of contributions from tropical curves, where the contribution from each tropical curve is calculated from relative invariants associated to the vertices of the curve.

The mathematical formulation of this uses exploded manifolds, which are related to log geometry and tropical geometry. A large amount of my work has gone towards defining the category of exploded manifolds and proving the properties required for Gromov-Witten invariants in the category of exploded manifolds.

## Publications

- Parker, B 2018, 'De Rham theory of exploded manifolds', Geometry and Topology, vol. 22, no. 1, pp. 1-54.
- Parker, B 2017, 'Three Dimensional Tropical Correspondence Formula', Communications in Mathematical Physics, vol. 353, no. 2, pp. 791-819.
- Parker, B 2015, 'Holomorphic curves in exploded manifolds: Compactness', Advances in Mathematics, vol. 283, pp. 377-457.
- Parker, B 2012, 'Log geometry and exploded manifolds', Abhandlungen aus dem Mathematischen Seminar der UniversitÃ¤t Hamburg, vol. 82, no. 1, pp. 43-81.
- Parker, B 2012, 'Exploded Manifolds', Advances in Mathematics, vol. 229, no. 6, pp. 3256-3319.