Lagrangian tangles are cobordisms between smooth links that generalize the classical Arnol'd theory of Lagrangian cobordism and the theory of Lagrangian cobordisms between Legendrian links. In this talk, we will explore the topology and (especially) the symplectic geometry of Lagrangian tangle links in the product of a surface with the complex numbers. The main tool is a novel Floer theory for Lagrangian tangles inspired by Morse theory for manifolds with gradient field tangent to the boundary. This is joint work with Ipsita Datta (ETH Zurich).
