Weinstein defined a morphism between symplectic manifolds as
a lagrangian submanifold of the product equipped with a half
density. I will describe the algebra of composition and
where this concept came from. Essentially, in the case of
cotangent bundles it is a geometric version of the
composition algebra of Fourier integral operators on the
base manifolds. I will assume no familiarity with
FIOs and microlocal analysis.

Last time Mike introduced canonical relations (aka
Lagrangian correspondences) as a generalization of
symplectomorphisms and described why certain types of
transversality are necessary for composing them. In this
talk we will work in the setting of derived (algebraic)
geometry, where intersections are automatically transverse.
For the first half, we will get comfortable with
differential forms on derived schemes and introduce shifted
symplectic structures on derived (Artin) stacks
(Pantev-Toen-Vaquie-Vezzosi, 2013). We will describe the
derived notion of Lagrangians and Lagrangian
correspondences, and then follow Calaque (2015) in the
construction of a semiclassical TFT from the data of a
\(k\)-shifted symplectic derived stack \(X\): the monoidal functor
\(\text{Map}( -,X): \mathsf{Cob}_d \to
\mathsf{LagCorr}_{k-d}\).