Dusa McDuff
Monodromy in Hamiltonian Floer theory.
Thu, 02/18/2010 - 16:01 — SomNambulAuthors: Dusa McDuff
Subjects: Symplectic Geometry
AbstractSchwarz showed that when a closed symplectic manifold (M,\om) is
symplectically aspherical (i.e. the symplectic form and the first Chern class
vanish on \pi_2(M)) then the spectral invariants, which are initially defined
on the universal cover of the Hamiltonian group, descend to the Hamiltonian
group Ham (M,\om). In this note we describe less stringent conditions on the
Chern class and quantum homology of M under which the (asymptotic) spectral
invariants descend to Ham (M,\om).The embedding capacity of 4-dimensional symplectic ellipsoids, I.
Fri, 12/04/2009 - 16:01 — SomNambulAuthors: Dusa McDuff, Felix Schlenk
Subjects: Symplectic Geometry
AbstractThis paper calculates the function $c(a)$ whose value at $a$ is the infimum
of the size of a ball that contains a symplectic image of the ellipsoid
$E(1,a)$. (Here $a \ge 1$ is the ratio of the area of the large axis to that of
the smaller axis.) The structure of the graph of $c(a)$ is surprisingly rich.
The volume constraint implies that $c(a)$ is always greater than or equal to
the square root of $a$, and it is not hard to see that this is equality for
large $a$.Displacing Lagrangian toric fibers via probes.
Fri, 10/30/2009 - 16:01 — SomNambulAuthors: Dusa McDuff
Subjects: Symplectic Geometry
AbstractThis note studies the geometric structure of monotone moment polytopes (the
duals of smooth Fano polytopes) using probes. The latter are line segments that
enter the polytope at an interior point of a facet and whose direction is
integrally transverse to this facet. A point inside the polytope is
displaceable by a probe if it lies less than half way along it. Using a
construction due to Fukaya-Oh-Ohta-Ono, we show that every rational polytope
has a central point that is not displaceable by probes.Symplectic embeddings and continued fractions: a survey.
Tue, 09/01/2009 - 12:47 — SomNambulAuthors: Dusa McDuff
Subjects: Symplectic Geometry
AbstractAs has been known since the time of Gromov's Nonsqueezing Theorem, symplectic
embedding questions lie at the heart of symplectic geometry. After surveying
some of the most important ways of measuring the size of a symplectic set,
these notes discuss some recent developments concerning the question of when a
4-dimensional ellipsoid can be symplectically embedded in a ball. This problem
turns out to have unexpected relations to the properties of continued fractions
and of exceptional curves in blow ups of the complex projective plane.
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