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.
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