Brett Parker
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.
The following links are to an introductory lecture series I gave on this work: The first lecture here gives a geometric understanding of what holomorphic curves look like, and why tropical curves appear in many limits. The second lecture gives basic definitions of exploded manifolds and links them to log schemes; these talks were interspaced with lectures by Bernd Siebert on logarithmic Gromov--Witten invariants, found here, here, here and here. The third lecture discusses families of holomorphic curves in an exploded manifold, and how the moduli space of holomorphic curves itself has a regular exploded manifold structure, and is a nice stack over the category of exploded manifolds. The final lecture, here, explains the links between exploded manifolds and constructions of Ionel and Tehrani, and then quickly sketches how tropical gluing formulae can be used to count Gromov--Witten invariants in Calabi--Yau 3-folds. A more technical talk from 2016 discussing the ingredients of the tropical gluing formula is here.