From a complex analytic perspective Teichmüller space - the universal
cover of the moduli space of Riemann surfaces - is a contractible
bounded domain in a complex vector space. Likewise, Bounded Symmetric
domains arise as the universal covers of locally symmetric varieties
(of non-compact type). In this talk we will study isometric maps
between these two important classes of bounded domains equipped with
their intrinsic Kobayashi metric.

We consider classes of diffeomorphisms of Euclidean space with partial asymptotic expansions at infinity; the remainder term lies in a weighted Sobolev space whose properties at infinity fit with the desired application. We show that two such classes of asymptotic diffeomorphisms form topological groups under composition. As such, they can be used in the study of fluid dynamics according to the method of V. Arnold. Specific applications have been obtained for the Camassa-Holm equation and the Euler equations.

I'll report on joint work with Nick Sheridan (Princeton/IAS) about mirror symmetry for Calabi-Yau (CY) manifolds. Kontsevich's homological mirror symmetry (HMS) conjecture proposes that the Fukaya category of a CY manifold (viewed as a symplectic manifold) is equivalent to the derived category of coherent sheaves on its mirror. We show that if one can prove an apparently weaker fragment of this conjecture, for some mirror pair, then one can deduce HMS for that pair. We expect this fragment  to be amenable to proof for the mirror pairs constructed in the Gross-Siebert program, for example.  We also show that the "closed-open string map" is an isomorphism, thereby opening a channel for proving the "closed string" predictions of mirror symmetry.

A well-known question in differential geometry is to prove the
isoperimetric inequality under intrinsic curvature conditions. In
dimension 2, the isoperimetric inequality is controlled by the integral of
the positive part of the Gaussian curvature. In my recent work, I prove
that on simply connected conformally flat manifolds of higher dimensions,
the role of the Gaussian curvature can be replaced by the Branson's
Q-curvature. The isoperimetric inequality is valid if the integral of the
Q-curvature is below a sharp threshold. Moreover, the isoperimetric
constant depends only on the integrals of the Q-curvature. The proof
relies on the theory of A_p weights in harmonic analysis.