ECH capacities have found many applications to symplectic
embedding problems, most of which in the toric setting. I will
discuss a new application of ECH to studying optimal embeddings for
non-toric rational surfaces. The key convex geometric...
An Anosov flow Φ on a closed 3-manifold M gives rise to a
non-Weinstein Liouville structure on V:=[−1,1]×M. Building upon the
work of Hozoori, we establish a homotopy correspondence between
Anosov flows and certain pairs of contact forms. Moreover...
I will discuss an idea of constructing a category associated
with a pair of holomorphic Lagrangians in a hyperkahler manifold,
or, more generally, a manifold equipped with a triple of almost
complex structures I,J,K satisfying the quaternionic...
Most work on Lagrangian fillings of Legendrian knots to date has
concentrated on orientable fillings, but instead I will present
some first steps in constructions of and (especially) obstructions
to the existence of (decomposable exact) non...
Because of the presence of non-trivial automorphisms of stable
maps, Gromov-Witten invariants of a general symplectic manifold are
usually rational-valued. Realizing a proposal of Fukaya-Ono back in
the 1990s, I will explain how to construct integer...
Given a trivalent plane graph embedded in the Euclidean plane
(up to isotopy), Treumann and Zaslow constructed and studied a
certain associated Legendrian surface embedded in standard contact
R5, nowadays referred to as a Legendrian 2-weave. Using...
I will discuss a recent work constructing quasimorphisms on the
group of area and orientation preserving homeomorphisms of the
two-sphere. The existence of these quasimorphisms answers a
question of Entov, Polterovich, and Py. As an
If G is a Lie group whose adjoint representation preserves a
nondegenerate symmetric bilinear form on its Lie algebra (e.g. a
semisimple group) and F is the fundamental group of a closed
oriented surface S, then the spaces of equivalence classes
I will discuss some recent work showing that a generic
area-preserving diffeomorphism of a closed surface has an
equidistributed sequence of periodic orbits. The proof uses several
properties of spectral invariants from periodic Floer