### University of Arkansas Topology Seminar: 9/19/2013

### Speaker: Dan Rutherford

### Title: Augmentations of the Legendrian contact homology
algebra

Abstract: Legendrian contact homology is an
invariant for Legendrian submanifolds of contact manifolds that enjoys
functorial properties with respect to certain Lagrangian cobordisms.
The invariant is given by a differential graded algebra (DGA) whose
differential is defined via counting holomorphic disks in the
symplectization of the given contact manifold. While the differential
is difficult to compute in general, for Legendrian knots in
ℝ^{3} the computation may be made in a combinatorial manner.
One way to obtain information from the DGA is to consider
augmentations which are homomorphisms from the DGA into a base field.
The presence of an augmentation allows for finite dimensional
linearized homology groups to be extracted from the infinite
dimensional DGA. Time permitting, we will discuss some geometric
constructions of augmentations from Lagrangian cobordisms and
generating families and consequences for the corresponding linearized
homology groups.

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