My Part III Essay (more commonly called a masters dissertation at other universities) was titled Quantum Cohomology and the Seidel Representation, and this was set by Jack Smith. A copy of my essay can be found here1.
A (work-in-progress)2 description of my essay is as follows:
Suppose $M$ is a manifold, then we have an associated ring, $H^*(M)$, called the cohomology ring. Under Poincaré duality, we can think of the product operation as the intersection of (transverse) submanifolds. The first part of the essay discusses pseudoholomorphic curve theory, so that we can define (genus 0) Gromov-Witten invariants, under certain technical assumptions (and black-boxing all of the analysis involved). Using this, we can define the (small) quantum cohomology $QH^*(M)$, which we can think of as a deformation of the ordinary cohomology ring, with higher order terms coming from computing Gromov-Witten invariants.
Next, we define some basic notions in Hamiltonian Floer Theory, which is defined as an infinite-dimensional version of Morse theory, performed on $C^\infty(M \times S^1)$. This gives us another homology theory, where the chain complex is generated by closed orbits under the Hamiltonian action. The associated homology theory is called Hamiltonian Floer homology, and we have a product structure given by pairs-of-pants. The final part is a description of an isomorphism, called the PSS isomorphism, between quantum cohomology and Floer homology3.
The second part of the title is to do with the Seidel representation, which is a map $\pi_0(\tilde G) \to QH^*(M)$, where $\tilde G$ is a covering of the free loop space on $\mathrm{Ham}(M)$. The way that we construct the action is to first use the PSS isomorphism above, so we need to instead construct an action on Hamiltonian Floer homology. In this case, there is a natural action on the chain complex, which can be interpreted through the PSS isomorphism as counting pseudoholomorphic sections of an appropriate symplectic fibre bundle, constructed using the clutching construction. This is where the action enters, as we have a natural $S^1$ action, which we can use to glue the two hemispheres together.
The final part of the essay consists of two applications of the theory. The first is showing that $\mathrm{Ham}(\Sigma_2)$ has an element of infinite order, where $\Sigma_2 = \mathbb{P}(\mathcal{O} \oplus \mathcal{O}(2))$ is a Hirzebruch surface. The second application is to do with Hamiltonian $S^1$-actions. In particular, we use the Seidel homomorphism to compute the Quantum cohomology ring of certain toric surfaces.
Note the warning about mistakes in the essay, which have not been fixed. The version here is the submitted version, and some day I might fix the issues (if I’ll need holomorphic curve theory). ↩︎
Yes, I’m aware that right now this is mostly incomprehensible unless if you already know the content. ↩︎
At this point, one might (rightly) object and say that the grading on the two are different, one is a homology theory and the other is a cohomology theory. There is a natural Poincaré duality map for both, and so it does not affect the results. The main reason for this choice is that in McDuff-Salamon’s book, they use cohomology and in Seidel’s paper, they use homology, and so I decided to keep the parts closest to the source as-is, and so we have a switch in convention at this point. ↩︎