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