My research mostly focusses on addressing mathematical questions
arising from duality symmetries in physics, especially mirror
symmetry. My papers of the past two or three years establish the
following results, with older results listed first.
1. Generalized notion of fiberwise duality to G_2 manifolds
in M-theory with K3 and T^4 fibrations (w/ Gukov, Yau),
including the construction of a new G_2 structure and metric.
2. Proved the existence of a metric describing a semi-flat
Calabi-Yau fibration with degeneracy locus a trivalent vertex
(w/ Loftin, Yau). This could be a local model for the most
interesting part of the Lagrangian torus fibration,
as found by M. Gross and W.-D. Ruan.
3. Studied estimators and random processes in probability,
including a proof that the measure on a finite group defined
through random multiplication approaches the uniform measure
(w/ Abrams, Landau, Landau, Pommersheim).
4. Provided the first rigorous CFT computation involving
coisotropic branes (w/ Aldi), including results and conjectures
about morphisms between these objects in the Fukaya category
and a verification of aspects of the mirror correspondence.
5. General-audience article on "Physmatics" based on public
lecture at The Fields Institute, Toronto.
6. Construction of the mirror map and mirror variety from
the Fukaya category, a realization of an idea by Seidel.