In 2015, Chantraine, Dimitroglou-Rizell, Ghiggini and Golovko (CDRGG) defined a Floer complex associated to a pair of transverse Lagrangian cobordisms. These cobordisms are oriented non-compact Lagrangian submanifolds with cylindrical Legendrian ends. The Floer theory that CDRGG developed has permitted to find new relations between the topology of a cobordism and some Legendrian isotopy invariant of its Legendrian ends, leading to new obstruction results for the existence of Lagrangian cobordisms.
In this talk, I will explain how to enrich this Floer complex with a product structure which in particular recovers the cup product in the case of Lagrangian cobordisms that are Lagrangian fillings (empty negative Legendrian end). Moreover, this product structure can be extended to an A-infini structure that reflects the the A-infini structure of the augmentation category associated to the positive Legendrian end of a cobordism.