Let M be a map on a connected, closed and orientable surface X. If B is a subset of the edge-set of M such that X\B is connected, then we say that B is a base of M. The collection of all bases of M form a delta-matroid, also known as a Lagrangian matroid. Analogously to matroids, there are two rich families of Lagrangian matroids: those that arise from embedded graphs, and those that arise from maximal isotropic subspaces of symplectic vector spaces.
Aside from the usual contraction and deletion operations, Lagrangian matroids admit twists; in the case of embedded graphs, twists of Lagrangian matroids correspond to the operation of partial duality, introduced by Chmutov in 2009. A partial dual of a map M is a map with only some of the edges dualised, and it can be interpreted as an intermediate step between M and its dual map M*.
In this talk I shall explain the relationship between maps, Lagrangian matroids, their twists, and partial duals. I shall also talk about a family of abstract tropical curves that arises from a map and its partial duals, and how it fits with the Galois-theoretic aspect of maps on surfaces (in the sense of Grothendieck's programme on dessins d'enfants).