I dunno. There's a lot of conceptual metaphors and sometimes even structure from geometry in higher maths, but this is sometimes very abstract and merely makes

I dunno. There's a lot of conceptual metaphors and sometimes even structure from geometry in higher maths, but this is sometimes very abstract and merely makes reference to non-geometric ideas about geometry previously developed. For example, in the kind of stochastic calculus quant finance people learn, there's an isometry (like a transformation that preserves size in some sense) between two very different kinds of continuous, non-enumerable spaces. Geometric intuition is of no help there -- you did learn what an isometry was in high school so you know the word, but you can't see function spaces in any way, shape or fashion. OTOH what I'm working on for my dissertation is "geometric integrators" for certain kinds of differential equations where the isometries and references to geometry are more direct; basically, the most common numerical solvers for initial value problems in ODEs preserve certain invariants and are useful for many many problems, but sometimes you want computation to preserve some sense of volume -- clasically, in mechanics. So even though many problems are too abstract to be seen, the notion of volume conservation is the best way to acquire the basic notions of the field.