3,139 research outputs found
Reeb orbits and the minimal discrepancy of an isolated singularity
Let A be an affine variety inside a complex N dimensional vector space which
has an isolated singularity at the origin. The intersection of A with a very
small sphere turns out to be a contact manifold called the link of A. Any
contact manifold contactomorphic to the link of A is said to be Milnor fillable
by A. If the first Chern class of our link is torsion then we can assign an
invariant of our singularity called the minimal discrepancy, which is an
important invariant in birational geometry. We define an invariant of the link
up to contactomorphism using Conley-Zehnder indices of Reeb orbits and then we
relate this invariant with the minimal discrepancy. As a result we show that
the standard contact 5 dimensional sphere has a unique Milnor filling up to
normalization proving a conjecture by Seidel.Comment: 67 Pages, 4 figures. I have added an introduction to some of the
tools used in the paper and a sketch of the proof for cone singularities for
non-experts. I have also corrected some minor errors and expanded some of the
more technical proofs as well as motivating the
Scald of barley
Scald is a common foliar disease in Victorian barley crops as the majority of current varieties are susceptible. Scald severity varies greatly from crop to crop depending on variety resistance, paddock history and local climate. Scald is more likely to be a problem when a susceptible barley variety is sown into infected stubble from a previous crop or when infected barley grass is present. Scald can be managed using an integrated approach that includes avoiding susceptible varieties, delaying early sowing, using seed dressings and not sowing into infected crop residues
Singularities and Semistable Degenerations for Symplectic Topology
We overview our recent work defining and studying normal crossings varieties
and subvarieties in symplectic topology. This work answers a question of Gromov
on the feasibility of introducing singular (sub)varieties into symplectic
topology in the case of normal crossings singularities. It also provides a
necessary and sufficient condition for smoothing normal crossings symplectic
varieties. In addition, we explain some connections with other areas of
mathematics and discuss a few directions for further research.Comment: 18 pages, 2 figure
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