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Legendrian knots and constructible sheaves

Algebra & Discrete Mathematics

Speaker: Vivek Shende, UC Berkeley
Location: 2112 MSB
Start time: Mon, Feb 3 2014, 4:10PM

A legendrian knot is a curve in R^3 along which the form dz - ydx vanishes. Identifying R^3 with the lower half of the cosphere bundle of R^2 by stereographic projection, we consider the category of sheaves on R^2 with singular support in the knot. This category turns out to be a Legendrian invariant, and (more to the point of the present talk) the moduli spaces of its objects are in some case familiar spaces. When the knot is a positive braid closed up by gluing R^2 into a cylinder, the second page of a spectral sequence built from the pushforward of the constant sheaf from the moduli space of objects in our category to the moduli space of local systems on one boundary circle of the cylinder is the Khovanov-Rozansky homology of the knot. When the knot is a positive braid, whose braid closure is formed in the plane by 'cusping off the ends', the orbifold cardinality of the rank 1 objects in the category over F_q is given by the lowest order term in 'a' of the HOMFLY polynomial of the link. In the above situation, when the braid is the link of a plane curve singularity, we conjecture that this moduli space is the wild character variety which corresponds under nonabelian Hodge theory to a wild Hitchin system whose central fibre is the compactified Jacobian of the curve singularity. This talk presents joint work with David Treumann and Eric Zaslow.