My research focuses on the to derivation of structural results for the homological invariants that are obtained from generating families of Legendrian submanifold in order to better understand the relation between generating families and augmentations of the Chekanov-Eliashberg algebras.
The motivation for this topic arises from the classification of Legendrian submanifolds up to Legendrian isotopy, a knot theory flavoured problem in every odd dimensional manifold endowed with an additional geometric data: a contact structure.
Despite being of large codimension in contact manifolds, Legendrian submanifolds are of rigid nature: they are hard to convey in contact manifolds and carry a non-trivial information on the contact structure itself. This contradictory behaviour makes the classification of Legendrian submanifolds such a rich and deep question.
The generating families setup basically allows a Morse-Bott-Cerf approach to this classification problem, while the Chekanov-Eliashberg algebra setting uses symplectic field theory to tackle this question.
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