The pursuit of absolute truth and perfection is the art of science, and within it mathematics holds a special place. Some see it as the means of expressing ideas true for eternity, others as the tool that explains every other branch of science. In either case it gives students the ability to bridge the gap between an idea held in the mind and a materialised result, one that can then be used to materialise further ideas, and so on without end. There is a beauty in this. It is almost magical that mathematics should permit such a passage from imagination to reality.
Learning how simple differentiation leads to covariant differentiation, and from there into the broader field of differential geometry and the results of the index theorems, or how a simple perspective on the additive properties of the integers, carried through linear algebra, yields insight into the behaviour of topological spaces and opens into the vast field of algebraic topology, is among the greatest treasures a student can gain.
A glimpse of what that ascent looks like in practice may help. Gradient, curl and divergence are the derivative operators every physicist meets early, and they are usually learnt as three separate rules to be memorised. Seen properly they are three faces of a single operator, the exterior derivative, acting on objects of different dimension, and the familiar identities that the curl of a gradient vanishes and the divergence of a curl vanishes turn out to be one statement, that applying this operator twice always yields nothing. From that single fact grows the machinery of homology and cohomology, and with it the generalised Stokes theorem, which gathers the fundamental theorem of calculus, Green's theorem, the divergence theorem and Stokes theorem into one line. What such machinery measures is obstruction, the ways in which a space can fail to be as simple as it first appears, and physics is full of these failures made visible, from the field circulating around a current-carrying wire to the phase a charged particle acquires travelling around a solenoid it never touches.
That a simple phenomenon can be abstracted far enough to express the properties of gigantic, infinite-dimensional spaces is simply beautiful, and this is the journey one embarks upon at FARII.