Abstract：
Eliashberg, Givental and Hofer's Symplectic Field Theory is a large project aiming to subsume under a unified topological field theoretical approach several techniques from symplectic topology (Floer homology, contact homology and more). Similarly to what happens in Gromov-Witten theory, at its core we find holomorphic curve counting. The general target manifold considered in SFT is a symplectic cobordism between contact manifolds (or more generally between stable Hamiltonian structures). When the cobordism is just a cylinder from a contact manifold to itself, the corresponding operator in SFT is, in particular, a collection of mutually commuting quantum Hamiltonians in a Weyl algebra.
These ideas were behind the introduction, by Buryak and myself, of the quantum double ramification hierarchy, which can be seen as a transposition of the SFT approach to the algebraic category together with several enhancements. I will introduce the double ramification hierarchy with an eye to its origins in Symplectic FIeld Theory and showcase some examples that we were able to fully compute.
