# What problem in pure mathematics required solution techniques from the widest range of math sub-disciplines?

(This is a restatement of a question asked on the Mathematics.SE, where the solutions were a bit disappointing. I'm hoping that professional mathematicians here might have a better solution.)

What are some problems in pure mathematics that require(d) solution techniques from the broadest and most disparate range of sub-disciplines of mathematics? The difficulty or importance or real-world application of the problem is not my concern, but instead the breadth of the range of sub-disciplines needed for its solution. The ideal answer would be a problem that required, for instance, number theory, group theory, set theory, formal logic, homotopy theory, graph theory, combinatorics, geometry, and so forth.

Of course, most sub-branches of mathematics overlap with other sub-branches, so just to be clear, in this case you should consider two sub-branches as separate if they have separate listings (numbers) in the Mathematics Subject Classification at the time of the result. (Later, and possibly in response to such a result, the Subject Classifications might be modified slightly.)

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. Is this also the case in research mathematics?

I'm not asking for an opinion—this question is fact-based, or at minimum a summary of the quantification of the expert views of research mathematicians, mathematics journal editors, mathematics textbook authors, and so forth. The issue can minimize the reliance on opinion by casting it as an objectively verifiable question (at least in principle):

What research mathematics paper, theorem or result has been classified at the time of the result with the largest number of Mathematics Subject Classification numbers?

Moreover, as pointed out in a comment, the divisions (and hence Subject Classification numbers) are set by experts analyzing the current state of mathematics, especially its foundations.

The ideal answer would point to a particular paper, a result, a theorem, where one can identify objectively the range of sub-branches that were brought to bear on the proof or result (as, for instance, might be documented in the Mathematics Subject Classification or appearance in textbooks from disparate fields). Perhaps one can point to particular mathematicians from disparate sub-fields who collaborated on the result.

• I think a more genuine, sophisticated, professional version of "number theory" may use/require the greatest range of other bits of mathematics for substantial success. (Part of the point is that an entry-level or elementary notion of "number theory" is typically 200 years out of date, or based on inaccurate if popular premises... seeming to make the subject a special case of elementary abstract algebra and elementary combinatorics... which will not get anyone very much farther than Euler 250 years ago...) Is such a response of interest? – paul garrett Jun 10 '17 at 0:58
• I disagree with the statement that this question has any objective meaning: all possible answers will be based on an opinion. – Alexandre Eremenko Jun 10 '17 at 7:28
• @alexandreeremenko: A perfectly reasonable objective interpretation of this question is "What math paper's citation list has the largest total number of distinct arxiv subject tags?" I don't expect anyone will answer the question this way, but it is rooted in something factual. – Paul Siegel Jun 10 '17 at 7:43
• I disagree with the claim that this question is fact-based in any meaningful way, since quantitative measures would rely on rather arbitrarily chosen divisions between mathematical areas. On the other hand, we could just use it as an opportunity to list our favorite theorems whose proofs involve a large number of unexpected techniques. For this purpose, I am imposing "community wiki" mode. – S. Carnahan Jun 10 '17 at 9:01
• I think this is a good question and should stay open, but it would be much better if it were formulated without reference to "the most msc or arxiv subject tags" (which, taken literally, seems unlikely to be an interesting or useful measure of mathematical breadth). – John Pardon Jun 11 '17 at 18:56

The proof of the Ramanujan conjecture by Deligne. It uses:

• number theory

• algebraic geometry

• topology

• representation theory

• commutative algebra

• complex analysis

• Didn't the Wiles Theorem require all of this too? – Deane Yang Jun 10 '17 at 12:27
• Probably. So another exmple to single out. – Libli Jun 10 '17 at 17:31
• Is there a real world example of the usage of Ramanujan Conjecture in practical applications? Wiki states application as for the construction of "Ramanujan Graphs" but no real world example. – I should change my Username Jun 12 '17 at 5:14
• Ramanujan graphs are special classes of expander graphs (googling will give you some "real world applications" of these). More specifically, you can apply Ramanujan graphs in cryptography, see e.g. here: whitman.edu/Documents/Academics/Mathematics/2014/maricqaj.pdf – Lennart Meier Jun 12 '17 at 7:56
• I want to stress that the application to "real-world" problems or "practical applications" is irrelevant to the question. – David G. Stork Jun 12 '17 at 17:21

I'm not qualified to certify optimality, but I've always thought that the Mostow rigidity theorem is a good candidate. The theorem says that every isomorphism between the fundamental groups of two finite volume hyperbolic manifolds of dimension at least 3 is induced by a unique isometry. Mostow's original proof (for the compact case) used:

• Riemannian geometry
• Conformal geometry
• Geometric group theory
• Representation theory
• Ergodic theory
• A dash of number theory

For generalizations to symmetric spaces you need algebraic geometry and more serious number theory as well.

• Why number theory and representation theory? – ThiKu Jun 10 '17 at 20:02
• @ThiKu It's hard to give a lot of detail without really getting into the proof, but roughly speaking one lifts the fundamental group of a hyperbolic manifold to a lattice in a Lie group (viewing hyperbolic space as a Riemannian symmetric space). The theorem (and especially its generalizations) uses arithmetic properties of this lattice and the representation theory of the Lie group. – Paul Siegel Jun 10 '17 at 23:52
• There is a number of different proofs of Mostow rigidity now. This master thesis: wiki.epfl.ch/grtr/documents/lucker2010.pdf reviews proofs by Thurston (via analysis and ergodic theory), by Gromov (via homological methods and geometry of hyperbolic simplices) and by Besson-Courtois-Gallot (via Riemannian geometry and entropy). – Lennart Meier Jun 12 '17 at 7:41
• The first proof is actually Mostow's even though it appears in Thurston's lecture notes. – ThiKu Jun 12 '17 at 17:58

One relatively recent result that comes to mind is the Kadison-Singer problem, which was originally formulated in 1959 as a question in $C^*$-algebra theory, but was successively reduced to more tractable and accessible questions in other fields by several mathematicians (including the MathOverflow member Nik Weaver). It was solved in 2013 by the computer scientists Marcus, Spielman and Srivastava using properties of random polynomials.

The Banach-Ruciewicz problem: Is the Lebesgue measure the only finitely additive measure on the Lebesgue sets in $S^n$ that is invariant under the rotation action by $O(n+1)$ and has total measure $1$?

The answer was shown to be negative for $n=1$ by Banach. While this is ostensibly a problem in measure theory, the case of $n\geq 4$ was affirmatively solved by Margulis and Sullivan using mainly infinite-dimensional representation theory (property T), but also a bit of number theory and algebraic group theory. The case of $n=2,3$ was affirmatively solved by Drinfeld. His use of representation theory was more hardcore and actually used crucially Deligne's solution of Ramanujan's conjecture and the Jacquet-Langlands correspondence. In particular, all the topics entering in Deligne's solution of the Ramanujan conjecture (which uses in turn the Weil conjectures) also enter into the solution of the Banach-Ruciewicz problem.

It should be said that the techniques are very similar to those to construct expander graphs and Ramanujan graphs. See Lubotzky's book Discrete Groups, Expanding Graphs and Invariant Measures.

The Smith conjecture; Morgan and Bass could write in 1984 that "the Smith conjecture stands in the first rank of mathematical problems when measured by the amount and depth of new mathematics required to solve it." See https://en.wikipedia.org/wiki/Smith_conjecture for the statement. You get convinced of the huge variety of techniques used in the proof, just by looking at the table of contents of the book: Morgan, J. W. and Bass, H. (Eds.). The Smith Conjecture (Papers Presented at the Symposium Held at Columbia University, New York, 1979. Orlando, FL: Academic Press, 1984)

• I must say that this is the best justified answer so far, including reference to a book that touts the wide range of techniques. I will study the Smith Conjecture now to appreciate the disciplinary breadth. – David G. Stork Jun 12 '17 at 17:53

Kronheimer, P.B.; Mrowka, T.S., Witten’s conjecture and Property P, Geom. Topol. 8, 295-310 (2004). ZBL1072.57005.

This paper is rather deep in the field of 3-manifold topology, using most of the major developments of low-dimensional topology from the previous 30 years. The main theorem states that a homotopy 3-sphere cannot occur as non-trivial surgery on a knot. Of course, this also follows now from the geometrization theorem and the knot complement problem. However, at the time of publication the geometrization theorem had not been vetted or published. Tracing back the proofs of theorems that this relies on involves the fields of

• Riemannian geometry (e.g. used in instanton homology)
• Algebraic geometry (featuring heavily in the proof of the cyclic surgery theorem)
• Complex analysis (used in Thurston's proof of geometrization of Haken 3-manifolds, as well as pseudo-holomorphic curves I suppose)
• Dynamics (used in Thurston's proof again, e.g. in Sullivan rigidity, a generalization of Mostow rigidity)
• Analysis and PDEs (for gauge theory)

• Mathematical Physics, in the guise of gauge theory, but specifically the work on Witten's conjecture of the equivalence between Seiberg-Witten and Donaldson invariants. This conjecture was motivated by ideas from string theory, so is not rigorous mathematics.

• and of course Topology, with quite a few specialties involved (foliations, symplectic and contact structures, 3- and 4-dimensional manifolds, Kleinian groups, Morse Theory).

I should also comment that there are now shorter proofs of this and related theorems independent of the Poincaré conjecture available that don't use quite as much gauge theory. And one can substitute Perelman's proof of geometrization for Thurston's, which substitutes Riemannian geometry and PDEs for complex analysis and dynamics.

• Excellent answer. Thanks. I'll look up the paper and see how much I can understand before I award the reputation points. – David G. Stork Jun 13 '17 at 5:13

In a sense this question is ill-posed. Every instance of coordinated use of several topics to prove a single result gives an evidence that these topics are strongly interconnected, so if these topics have been classified as separate, the classification must be revised to take into account these interconnections.

The mathematical subject classification can never be finalized I think. After all, in ancient times there was even no clear distinction between music, physics and mathematics.

• What, objective, then would you say is the difference between a result such as Gödel's Incompleteness Theorem (which relied nearly entirely on formal logic), and the Ramanujan conjecture by Deligne, noted above? How would you modify or edit the question to capture such differences and avoid it being "ill-posed"? What if the question restricted consideration to a single point in time, admitting that later the fields thought to be rather disparate were in fact closer than originally thought? What about a question forcusing on the greatest consolidation of fields previous thought disparate? – David G. Stork Jun 12 '17 at 17:08
• This should be a comment imo. That's not to say I disagree with you. – Wojowu Jun 12 '17 at 17:15
• @DavidG.Stork I honestly don't know how to formulate such question, although I must say your suggestions are very interesting for me. Seems like most mathematicians switch between two opposite regimes. Me myself I am like this - I can try to avoid other topics for years, when digging for an answer to some particular deep question. And then for years I might try to broaden my understanding, driven by strong feeling that the whole mathematics is one unified body of knowledge that we are challenged to grasp in total. The truth is more likely somewhere in between, as it usually happens... – მამუკა ჯიბლაძე Jun 12 '17 at 17:15
• @Wojowu I agree that definitely it is not an answer. On the other hand, it contains a conceptual viewpoint on the content of the question, so maybe it is not a comment either. Besides, this is cw anyway :D – მამუკა ჯიბლაძე Jun 12 '17 at 17:17
• @DavidG.Stork let me also add that if I created impression of negative attitude, I regret it. The aim was rather to bring in some paradoxical aspects inherent in your question. Something like what happens in quantum mechanics where observing implies altering the state of the observed, which makes the observation obsolete, so that one needs some new non-classical approaches because of that. – მამუკა ჯიბლაძე Jun 12 '17 at 17:22

The Beilinson regulator is $\frac{1}{2}$ times the Borel regulator.

The Beilinson regulator is a map from algebraic K-theory to Deligne cohomology and the above equality generalizes Borel's theorem on the algebraic K-theory of number rings, which in turn generalizes the class number formula from algebraic number theory. A complete proof is contained in this book. To get an impression of the range of involved fields one may just look at its Table of Content, from which I copy the names of chapters:

• Simplicial and Cosimplicial Objects
• H-spaces and Hopf Algebras
• The Cohomology of the General Linear Group
• Lie Algebra Cohomology and the Weil Algebra
• Group Cohomology and the van Est Isomorphism
• Small Cosimplicial Algebras
• Higher Diagonals and Differential Forms
• Borel's regulator
• Beilinson's Regulator

The classical statistical mechanics (on lattices) already feels as big as the entire mathematics. It relates to analysis, measure theory, algebra (especially the commutative algebra), combinatorics, ... There is a promise in it that should connect to algebraic geometry and number theory. No wonder that the technique of the classical statistical mechanics has contributed to a breakthrough in the knot theory (geometric topology strongly connected to the general groups which as a rule are not abelian).

A huge problem is the theory of phase transitions in the temperatures which are neither high nor low. The high temperature case is easy, while it took a very long time to basically solve the low temperatures for the ferromagnetic systems (and nearly ferromagnetic); actually, there is still a lot of thong to do there. The problem by definition has an analytical character. In the case of low temperatures, everything got reduced to algebra--you may say that the low temperatures froze analysis into algebra. However the intermediate temperatures present a huge problem which will require analysis (including dynamic systems and ergodicity considerations), algebra, combinatorics, ... Even the non-ferromagnetic systems in low temperatures still present a challenge.

The quantum statistical mechanics is still much richer then the classical. However one infinity versus two infinities... The classical case is already overwhelming.

PS. It's been long years since I was active in this topic (the classical case). I am sure that there were some breakthroughs during that time. But I am equally sure that it is still very far from fully meeting the challenge which I have mentioned above).