Important open exposition problems?

Timothy Chow, in his article A beginner's guide to forcing, defines an open exposition problem as a certain concept or topic in mathematics that has yet to be explained "in a way that renders it totally perspicuous."

What are some open exposition problems in your field, particularly ones that you think would help interested mathematicians break into it?

For instance, there is no shortage of references on chromatic homotopy theory -- Ravenel's Green and Orange books, Lurie's course notes, and Hopkins' COCTALOS notes. Nonetheless, there does not seem to be a complete, smooth, and carefully-put-together exposition of the chromatic story that assumes neither too much homotopy theory nor algebraic geometry.

I'd nominate the theory of Macdonald polynomials (and associated topics). This is an extremely important area of algebraic combinatorics. Even if we restrict to type A, there are certain features of the subject that are intrinsically complicated, but nevertheless, learning this subject is currently a lot harder than it needs to be IMO. Haglund's monograph is certainly helpful because it gives a clear account of a lot of the purely combinatorial side of the theory, but his goal was not to give a full account of Macdonald polynomials.

I recently discovered the slides for a talk by Ole Warnaar that give a very nice introduction. They solve an important aspect of the exposition problem, which is to construct an engaging story that can serve as the backbone for a more complete exposition. Of course the slides themselves only scratch the surface.

• Schubert polynomials and all the related objects (key polynomials, slide polynomials, Grothendieck polynomials, Kohnert tableaux, etc.) are also worthy of a textbook nowadays. Though it seems that new results are coming up from Assaf and Searles and it's worth waiting for them to be fully established. Commented Sep 25, 2017 at 4:30

This is an interesting question.

Responding sideways, I can give a closed exposition problem, namely the Kepler conjecture. Here is a talk by Thomas Hales:

Lessons learned from the Formal Proof of the Kepler Conjecture

I think it makes for interesting watching. Part of the jist of talk is that the first proof, the one submitted to Annals of Mathematics and which took seven years to referee, was difficult even for specialists. This wasn't just down to the immense amount of computational data that the proof relied on, but was also down to the nature of the proof itself. When Hales came to rewrite it in order to make it amenable to formal treatment, he was surprised to find out how much time he spent in math land (to use his words). New insights were had, in fact other outstanding conjectures were settled as a by-product of this work. The resulting new proof Hales calls the blueprint proof, and in all respects it is a very different one to the original. It is more structured, shorter, and simpler.

If someone suggest tackling the original proof I would run a mile. But the new, blueprint proof? I might read some of it one day and I am hopeful that its lower reaches would not be beyond me. In short, I'm not intimidated by it. So I would say that the Kepler conjecture used to be an open exposition problem but now, because of the blueprint proof, it's closed!

• I recall Gonthier saying something similar about the formalized proof of the 4-color theorem: In the process of formalizing the proof, lots of simplifications were made. This is perhaps not that surprising, seeing that proof checkers these days aren't really able to understand the full spectrum of reasoning that we humans consider mathematically sound, and so one needs to "dumb down" a proof on the mathematical side to formalize it. Commented Sep 25, 2017 at 4:33

A combinatorial study of Pfaffians, as requested by Peter Heinig on MathOverflow #280362. Should include things like the minor summation formula and analogues of various determinantal identities.

I have two such open exposition questions which I hope to address some day (alas, I have a long list of both mathematical research and pedagogical problems which are all worthy of attention - but this kind of question is fun because it is at the intersection of these areas). One is giving a cochain-level treatment of Poincaré duality through intersection theory and then using that to give geometric, cochain level refinements of much of "intermediate" algebraic topology: wrong-way maps, characteristic classes, Thom isomorphism, Steenrod operations, Eilenberg-Moore spectral sequence. A second is to treat loop spaces and classifying spaces in a more unified way, from scratch (course title: "Loop, de-loop.") This would expand on my little expository paper on Koszul duality.