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I've edited and reposted a version of this question on mathOverflow (for professional mathematicians) here, which may prove interesting to others.

Some mathematical problems require solution techniques from a single branch (sub-discipline) of mathematics. For instance, most problems in formal logic can be addressed by the methods of formal logic without the need for other methods.

Some problems require solution techniques from two branches of mathematics. For instance, there are many problems in plane geometry that resist purely geometric solution techniques, but also require algebra.

Yet other problems may require the branches geometry and algebra and combinatorics and graph theory, etc.

What are some abstract problems that require a broad range of solution techniques, i.e., from disparate sub-disciplines? The ideal response to this question is a short, crisply posed problem whose solution requires methods from as many and as widely disparate branches of mathematics as possible.

Of course, most branches of mathematics overlap with a number of other branches, but just to be clear, in this case you should consider two branches as separate if they have separate listings (numbers) in the Mathematics Subject Classification.

Real-world problems, such as "design an airplane," are not acceptable answers.

One of the reasons I'm interested in this problem is by analogy to technology. More and more problems in technology require a range of disciplines, e.g., electrical engineering, materials science, perceptual psychology, optics, thermal physics, and so forth. Consequently, the best leaders of technical teams should have such broad range of disciplinary knowledge.

What about the case for mathematics? Is it that problems posed in formal logic will need techniques from differential topology (say) for their solution? Or can mathematical problems be more "placed in silos" or "disciplinarily isolated" than in other disciplines? Having a few good examples of problems that require a broad range of disciplines would be relevant to understand this issue. Descartes' brilliant bridging of algebra and geometry (formerly considered rather disparate disciplines) allowed many new problems to be solved (and posed).

Incidentally, while the documentary, cited below, of Wiles' proof of Fermat's Last Theorem indeed cites the work of many mathematicians and many different results, it is not clear quite how broad the disciplinary "reach" of the proof is. Surely it is large... but might there be other succinctly stated problems that (while not as hard or celebrated as FLT) nevertheless require techniques from a wider range of sub-disciplines?

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    $\begingroup$ How wide-ranging is Wiles' proof of Fermat's Last Theorem? $\endgroup$
    – DylanSp
    Mar 16, 2016 at 20:24
  • $\begingroup$ I don't know the range of branches needed in Wiles' proof, but if you can list/quantify them, yours might be a great solution to my question. Fermat's Last Theorem is certainly crisply posed! $\endgroup$
    – David G. Stork
    Mar 16, 2016 at 20:27
  • $\begingroup$ Check out this, particularly near the end. There are mentions of the different areas of Mathematics used in Wiles' proof: vimeo.com/18216532 $\endgroup$
    – pshmath0
    Mar 16, 2016 at 20:37
  • $\begingroup$ Are you using the word 'simple' when you actually mean 'succinct'? Because proving Fermat's last theorem was far from simple! $\endgroup$
    – Nij
    Mar 16, 2016 at 20:38
  • $\begingroup$ In my opinion "simply stated" is acceptable and clearly not to be confused with "simply solved". By definition no "simply solved" problem will require multiple areas of math to solve. $\endgroup$
    – fleablood
    Mar 16, 2016 at 22:28

2 Answers 2

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What is the maximum volume bounded by 16 points, where no two points are further than 1 apart?

This is the biggest little polyhedron problem. I used monte carlo methods, optimization methods, extremal combinatorics, graph theory, geometry, iterative methods, algebraic numbers, field theory, simplification methods and group theory to solve it.

blp16

I haven't fully solved 12-15 and 17+ points. These are requiring even more branches of mathematics. For example, I can get a solution for 12 points down to a polynomial that fills 2 pages and which is currently uncrackable by my current tools.

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    $\begingroup$ Excellent answer (and nice blog). +1. Let's see what other answers arise... $\endgroup$
    – David G. Stork
    Mar 16, 2016 at 22:29
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You have an unlimited number of irregularly burning 1-minute fuses. What is the smallest amount of time over 4 minutes that can be measured?

This is the fusible number problem. It may be the fastest growing function, since it relies on the first number missed by an infinite number of infinite sums. Most other gigantic number systems build up.

Jeff Erickson claims fuse(3) is $3 + 2^{-1541023937}$.
Junyan Xu claims fuse(3) is larger than 2↑↑↑↑↑↑↑↑↑16, in Knuth's up-arrow notation.

There aren't enough branches of mathematics yet to even approach fuse(4). But it can be studied with many different methods.

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    $\begingroup$ The fusable number problem is cool, but I don't see why its solution somehow requires a large number of branches of mathematics. Surely the size of the solution is irrelevant. After all calculating $2 \uparrow \uparrow \ldots \uparrow 16$ requires just one: simple arithmetic. Likewise Ackermann's function (which gives humongous numbers) involves just one—or at most two—branches of math. $\endgroup$
    – David G. Stork
    Mar 16, 2016 at 23:19
  • $\begingroup$ The actual number isn't known. It's not a computation. The problem starts by computing an infinite number of infinite series, then adding subsets. But that's currently not possible. Instead, symmetry-breaking techniques must be used to make the problem approachable. $\endgroup$
    – Ed Pegg
    Mar 16, 2016 at 23:37
  • $\begingroup$ Ed Pegg: Again, whether the actual number is known or not and its size are all irrelevant: As far as I see, just one discipline (branch) of math is needed for its solution (apparently number theory). The ideal solution to my request would be a problem that requires number theory, topology, graph theory, set theory, analysis, group theory, formal logic and complexity theory (for instance). $\endgroup$
    – David G. Stork
    Mar 16, 2016 at 23:41

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