| Summary: | <p>In this thesis, we generalise the Steenrod square on the cohomology of a topological space to a quantum Steenrod square on the quantum cohomology of a symplectic manifold. We prove quantum versions of the Cartan and Adem relations, and use the former to calculate the quantum Steenrod square for CP<sup>n</sup>, as well as more generally giving the means to compute the operation for all monotone toric varieties. We calculate the quantum Steenrod squares for two examples of blowups along a subvariety, in which a quantum correction of the Steenrod square on the blowup is determined by the classical Steenrod square on the subvariety. We then relate the quantum Steenrod square to Seidel's equivariant pair-of-pants product for open convex weakly monotone symplectic manifolds, using an equivariant version of the PSS isomorphism. We proceed similarly for Z/2-equivariant symplectic cohomology, using an equivariant version of the c*-map. We prove a symplectic Cartan relation, pointing out the difficulties in stating it. We will give some chain level calculations of the equivariant pair-of-pants product for T*S<sup>n</sup>. We calculate the quantum Steenrod square for the negative line bundles O(-1) → CP<sup>m</sup>, and use this to calculate the equivariant pair-of-pants square by extending previous results due to Ritter. We finish with a brief sketch of further ideas, involving a Z/2-equivariant version of the Chas-Sullivan product and the Viterbo isomorphism.</p>
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