— where the above contour integral winds once around a neighborhood of in which remains holomorphic — defines a continuous (and holomorphic) function of z even into the possible singularity. This is the only proof I know, actually, and it doesn’t discuss analyticity, but just like analyticity of holomorphic functions, it follows almost immediately from the representation formula. What would be interesting would be to see a proof of these facts about holomorphic functions without relying on the explicit representation formula, maybe something to do with the notion of capacity in the case of removable singularities.

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]]>Perhaps something in algebraic combinatorics would ideal? There are lots of topics to cover, and most are fairly concrete, but it gets a bit theoretical at points as well. Also there is lots of room for examples, both in terms of figures and in terms of specific applications of more general techniques.

Of course, there is a lot out there already, like Alon’s treatise on Combinatorial Nullstellensatz, so there is the question of how much it would benefit from a polymath approach.

We could also pick some fairly new topic for which it is not even obvious how to organize the existing material, and perhaps a polymath approach will more quickly make it clear how to present it. This has the nice advantage of having a direct impact on current research, and may perhaps even inspire new avenues of inquiry. The disadvantage is that there could be fairly high barriers to entry if we pick the wrong field.

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