In the cotangent bundle of a smooth sphere, there are lots of interesting Lagrangian submanifolds with non-zero Floer invariants. These local Lagrangian submanifolds naturally sit inside a neighborhood of a Lagrangian sphere. We study how Floer invariants change when the ambient space changes, from the cotangent bundle to an abstract symplectic manifold. The strategy is to use the local information to control the global information to some extent. Applications include the existence of continuum families of nondisplaceable Lagarangian tori near a chain of Lagrangian 2-spheres and a new estimate of the displacement energy of Lagrangian 3-spheres.

The complex Grassmannians contain a nondisplaceable monotone Lagrangian torus, which is a fibre of an integrable system introduced by Guillemin-Sternberg in the ‘80s. Usually, it does not generate the whole Fukaya category. I will talk about “exotic” Lagrangian tori that generate some of the missing parts. Their existence is a rather natural consequence of general ideas in mirror symmetry, cluster algebras and the theory of Okounkov bodies. Time permitting, I will mention how these ideas fit in a broader picture of mirror symmetry for coadjoint orbits of Lie groups.

We associate a closed Lagrangian submanifold L of C^n to each Delzant polytope. We prove that L is monotone if and only if the polytope P is Fano. It turns out that two Delzant polytopes P1 and P2 provide Hamiltonian isotopic Lagrangians if and only if P1 = gP2, where g is an element of SL(n, Z). Similar theorems can be proved for Lagrangians of CP^n and (CP^n)^k.
Using this construction we can construct a huge number of monotone Lagrangian submanifolds. Many of the constructed monotone Lagrangians have equal minimal Maslov numbers and are smoothly isotopic, but they are not Hamiltonian isotopic. Also, the method allows the construction of infinitely many non-monotone embedded Lagrangians, no two of which are related by Hamiltonian isotopies (but they are smoothly isotopic). All these Lagrangians have many other interesting properties.
If time permits, we will discuss applications to monotone Lagrangian cobordisms.

Gay and Kirby proved that every smooth 4-manifold admits a trisection--a decomposition into three pieces, each of which is a 1-handlebody. A Weinstein trisection is a trisection which is nicely compatible with a symplectic structure on the 4-manifold. We will explain this structure and show that every symplectic 4-manifold admits a Weinstein trisection. This is joint work with Peter Lambert-Cole and Jeffrey Meier.