Abstract: This is joint work with Itai Ben Yaacov of Lyon, focused on the model theory of Gurarij's separable universal Banach space. It has been known for awhile that in continuous model theory the Gurarij space is a very nice Fraisse limit---it's theory is separably categorical and has quantifier elimination. We have written clear and simple proofs of all those basic facts, and also have been studying the type spaces for this theory, over an arbitrary set of parameters, which we may as well take to be a separable Banach space $E$. Our main achievement is to completely characterize isolated types in the type spaces over $E$, using tools from convex analysis. This lets us derive a lot of information about the situation in which the set of isolated types is dense (for the logic topology) and hence there is an embedding $T$ of $E$ into the Gurarij space $G$ such that the structure $(G,T(e) : e \in E)$ is atomic; this is the same as saying there is a generic orbit in the Polish space of all such embeddings, under the action of ${\rm Aut}(G)$. For example this happens for any $E$ of dimension $\leq 3$, for any finite dimensional E that is smooth or polyhedral, but not for all $E$---we give an $E$ of dimension $4$ such that the isolated types over $E$ are not dense, and we show that over many familiar infinite dimensional spaces $E$, there are no isolated types except for the obvious ones (coming from elements of $E$). The tools from convex analysis that we develop should help answer several open questions about the Gurarij space, such as: how complex is the space of orbits of the action of ${\rm Aut}(G)$ on the unit sphere of $G$? (We can derive a new result, that there are infinitely many orbits, but this is far from giving the final story.)