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 brief description (partially rewritten March 2026) for non-experts of the content of the essay is as follows:

Given a compact oriented manifold $M$, we have an associated cohomology ring $H^*(M)$. Under Poincaré duality, we can think of the product operation in the cohomology ring as the intersection of transverse submanifolds. More precisely, we can write $$[A] \cdot [B] = \sum_C N(A,B,C)[C],$$ where $N(A,B,C)$ is an integer, representing the multiplicity. In particular, by transversality, we know that if $N(A,B,C)$ is non-zero, then $$\mathrm{codim}(A) + \mathrm{codim}(B) = \mathrm{codim}(C),$$ and we should think of $N(A, B, C)$ as the multiplicity of $[C]$ in $[A \cap B]$.

The quantum cohomology is defined to be a deformation of $H^*(M)$, where we deform the product operation as follows. $$[A] * [B] = \sum_\alpha \sum_C N_\alpha(A, B, C)q^\alpha[C].$$ Given a class $\alpha \in H_2(X)$, the number $$N_\alpha(A, B, C)$$ counts the number of genus 0 Riemann surfaces in $X$, in the class $\alpha$, intersecting $A, B$ and $C$. In particular, $N_0(A, B, C) = N(A, B, C)$. The formal variable $q^\alpha$ ensures that the sum is in fact finite. More generally, one should work in a Novikov ring, which is briefly discussed in the essay.

Recall that a smooth function $f \colon M \to \mathbb R$ on a smooth manifold $M$ is Morse if it has non-degenerate critical points. “Classical” Morse theory shows that one can construct a cell structure using the Morse function. A more modern approach involves counting flow-lines of the Morse function. More precisely, let $\mathrm{Crit}_k(f)$ denote the set of critical points of $f$ of index $k$, which is a finite set. We let $$C_k(f) = \bigoplus_{p \in \mathrm{Crit}_k(f)} Z \cdot [p]$$ be the free abelian group generated by $\mathrm{Crit}_k(f)$. From this, we can define a boundary operator by $$\partial p = \sum_q n_{p, q}q,$$ where $n_{p, q}$ is a signed count of flow-lines going from $p$ to $q$. One can then show that $(C_k(f), \partial)$ is a chain complex, and the homology recovers the singular homology of the manifold. In fact, one can recover the cohomology ring from this.

Floer theory is the generalisation of this to infinite dimensions. Given an infinite dimensional manifold, one can make sense of critical points of this manifold, and by considering flow lines between critical points, we obtain a homology theory. In the essay, we consider the special case of Hamiltonian Floer homology. In this, the critical points correspond to closed orbits under the Hamiltonian flow, and the product is given by counting pseudoholomorphic pairs-of-pants.

In the specific context of the essay, one has an isomorphism between quantum cohomology and Hamiltonian Floer homology², called the PSS isomorphism. The map from Hamiltonian Floer homology to quantum cohomology is given by “capping off” the pairs of pants, turning them into genus 0 Riemann surfaces.

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.