*(Partly joint work with Ivan Smith)* See arXiv:1106.3975 and arXiv:1201.5880.

Faced with the monumental task of introducing (wrapped and unwrapped)

Floer homology, symplectic homology and Fukaya categories as well as

telling us about his theorem (all in the final hour before dinner),

Alex rose to the challenge with a beautiful set of highly illustrated

beamer slides.

The spaces Alex was talking about are the total spaces of line bundles

over symplectic manifolds (mostly \(\mathbf{CP}^n\)) such that the first

Chern class is a negative multiple of the cohomology class of the

symplectic form. For instance, \(\mathcal{O}(-1)\) over

\(\mathbf{CP}^1\). These are noncompact symplectic manifolds with

symplectic fibres and symplectic base and they are convex in the sense

that a sequence of holomorphic curves cannot escape to infinity. The

first theorem he proved was that the symplectic homology of such a

space is a quotient of the quantum homology (symplectic homology is a

Floer theory counting periodic orbits of a Hamiltonian which gets very

big quite quickly in the noncompact end of the manifold, quantum

homology just counts compact holomorphic spheres!). In particular you

quotient by the kernel of a certain map: the quantum cup product with

a high power of the first Chern class. When there are no spheres with

positive symplectic area in the base (and hence in the total space)

the quantum and classical cup products agree and hence a sufficiently

large power of the first Chern class vanishes, to the whole quantum

cohomology is in its kernel, which recovers an older result of Oancea

(that assuming there are no spheres with positive symplectic area the

symplectic homology vanishes). The idea of the proof was the

following: symplectic homology is defined as a limit of Floer

homologies for a sequence of Hamitonians. For a suitable choice of

these Hamiltonians you can ensure that each of these Floer homologies

is isomorphic to quantum cohomology (roughly speaking you rotate the

fibre in such a way as to ensure that all closed orbits lie in the

zero section) and the maps in the sequence are precisely

multiplications by the first Chern class.

Not satisfied with this, Alex raised the stakes algebraically and

introduced the “open-closed string map” (one of the more complicated,

though increasingly central, aspects of the Fukaya/Floer story). This

is a map from the Hochschild cohomology of the Fukaya category to the

quantum homology. I think (hope) I'm right in saying the

following. For a single Lagrangian \(L\) it takes a collection of cycles

(“inputs” – the Hochschild cohomology having as its $n$th chain group

the tensor product of \(n\) copies of the Floer chain group of \(L\)) to

an ambient cycle. The ambient cycle is traced out by a marked point on

a holomorphic disc with as many marked points as there are inputs

where each point point is required to be mapped to the corresponding

cycle. Mad. And then he raised the stakes yet more by introducing the

analogous map from the wrapped Fukaya category to the symplectic

homology.

Why? Well Abouzaid recently proved a criterion for when a Lagrangian

(or collection of Lagrangians) split-generates the Fukaya category (or

some part of it) by looking at the image of this open-closed string

map. Ritter and Smith have adapted this to the monotone setting they

need for these negative line bundles. Using this criterion (namely

that the image should contain some invertible element) they prove that

you only need a single Lagrangian to split-generate the wrapped Fukaya

category of a negative line bundle over \(\mathbf{CP}^n\) (for suitably

low Chern class of the line bundle). The Lagrangian in question is the

circle bundle living over the Clifford torus (which, when taken with

various flat connections, generates all the various parts of the

Fukaya category). In particular, the wrapped category is generated by

a compact Lagrangian, so this proves that all the potential

infinite-dimensionality introduced by wrapped Floer cohomology is

actually only finite-dimensional (in the same way that the symplectic

cohomology reduced to a quotient of quantum cohomology).

At this point, Alex brought the discussion back down to earth by

discussing the equator in the sphere (one of the most instructive

Lagrangian submanifolds, well worth your contemplation). Then we went

for tea.