Chen and Ruan constructed symplectic orbifold Gromov-Witten
invariants more than 20 years ago. In ongoing work with Alex
Ritter, we show that moduli spaces of pseudo-holomorphic curves
mapping to a symplectic orbifold admit global Kuranishi
charts...
An inertial manifold is a positively invariant smooth
finite-dimensional manifold which contains the global attractor and
which attracts the trajectories at a uniform exponential rate. It
follows that the infinite-dimensional dynamical system is...
It is conjectured that every Reeb flow on a closed
three-manifold has either two, or infinitely many, simple periodic
orbits. I will survey what is currently known about this
conjecture. Then, I will try to explain some of the key ideas
behind...
Ratner's landmark equidstribution results for unipotent flows
have had dramatic applications in many mathematical areas. Recently
there has been considerable progress in the long sought for goal of
getting effective equidistribution results for...
(Joint with Samuel Grushevsky, Gabriele Mondello, Riccardo
Salvati Manni) We determine the maximal dimension of a compact
subvariety of the moduli space of principally polarized abelian
varieties Ag for any value of g. For g<16 the dimension is g−1,
while for g≥16, it is determined by the larged dimensional compact
Shimura subvariety, which we determine. Our methods rely on
deforming the boundary using special varieties, and functional
transcendence theory.
To study the asymptotic behavior of orbits of a dynamical
system, one can look at orbit closures or invariant measures. When
the underlying system has a homogeneous structure, usually coming
from a Lie group, with appropriate assumptions a wide...
