The quantum differential equation (qDE) is a rich object attached to a smooth projective variety X. It is an ordinary differential equation in the complex domain which encodes information of the enumerative geometry of X, more precisely its Gromov-Witten theory. Furthermore, the asymptotic and monodromy of its solutions conjecturally rules also the topology and complex geometry of X. These differential equations were introduced in the middle of the creative impetus for mathematically rigorous foundations of Topological Field Theories, Supersymmetric Quantum Field Theories and related Mirror Symmetry phenomena. Special mention has to be given to the relation between qDE's and Dubrovin-Frobenius manifolds, the latter being identifiable with the space of isomonodromic deformation parameters of the former. The study of qDE’s represents a challenging active area in both contemporary geometry and mathematical physics: it is continuously inspiring the introduction of new mathematical tools, ranging from algebraic geometry, the realm of integrable systems, the analysis of ODE’s, to the theory of integral transforms and special functions. This talk will be a gentle introduction to the analytical study of qDE’s, their relationship with derived categories of coherent sheaves (in both non-equivariant and equivariant settings), and a theory of integral representations for its solutions. The talk will be a survey of the results of the speaker in this research area.
