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Legendrian contact homology and generating families

Final Report Summary - CONTACTMATH (Legendrian contact homology and generating families)

Contact geometry is the study of geometric spaces equipped with particular structures called contact structures. Contact geometry is in particular the natural framework of geometric optics, as well as thermodynamics. This geometry exists in odd dimensions only; its even dimensional analogue is symplectic geometry, which is the natural framework of classical mechanics in its Hamiltonian formulation.

A contact structure corresponds to the data, at every point of space, of a hyperplane. These hyperplanes satisfy a condition equivalent to the minimization of the dimension of surfaces that are everywhere tangent to the hyperplanes. The tangent surfaces with maximal dimension are called Legendrian submanifolds. These remarkable and important objects in contact geometry correspond to wave fronts in optics.

Contact structures and Legendrian submanifolds are a priori very flexible objects. However, they do possess some rigidity properties as well. Those can be detected using specific techniques in contact and symplectic topology. Those techniques can also be used to construction invariants for contact structures and Legendrian submanifolds, allowing to distinguish inequivalent such objects.

The ERC project ContactMath studied the relationship between two sophisticated invariants for Legendrian submanifolds: linearized Legendrian contact homology and generating family homology. While the former is based on holomorphic curves, the latter is defined via Morse theory, using differential equations instead. This study also led to a better understanding of the geography of Legendrian submanifolds and provided new applications of these techniques in contact geometry.