Monge-Ampère equations, mirror symmetry and optimal transport

Mirror symmetry is a phenomenon whose importance have been highlighted by some of the most influential mathematicians and physicists of recent decades.

One of the main interpretations of mirror symmetry is centred around a long standing conjecture in complex differential geometry, the SYZ-conjecture, on existence of special Lagrangian torus fibrations in Calabi-Yau manifolds near large complex structure limits.

Recent breakthroughs in pluripotential theory reduces a modern variant of this conjecture, the metric SYZ conjecture, to structural properties for solutions to non-Archimedean Monge-Ampère equations.

The aim of this project is to establish these structural properties and thus prove the metric SYZ-conjecture in large generality as well as leveraging the information gained to answer more detailed geometric questions about the fibrations. Motivated by recent success by myself and co-authors, I will apply a framework built on optimal transport theory.

I will also use methods from complex differential geometry, pluripotential theory and tropical geometry, as well as numerical methods, in particular entropic regularisation of optimal transport.