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There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the connection between the two is unexpected. The first example that comes to my mind is the proof by Furstenberg and Katznelson of Szemeredi's theorem on the existence of arbitrarily long arithmetic progressions in a set of integers which has positive upper Banach density, but using ergodic theory. Of course in the years since then, this idea has now become enshrined and may no longer be viewed as surprising, but it certainly was when it was first devised.

Another unexpected connection was when Kolmogorov used Shannon's notion of probabilistic entropy as an important invariant in dynamical systems.

So, what other surprising connections are there out there?

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  • $\begingroup$ It should be mentioned that the connection you refer to is due to Furstenberg (ams.org/mathscinet-getitem?mr=498471). Later Furstenberg and Katznelson together used this connection to derive other combinatorial results, including a multidimensional extension of Szemeredi's theorem and a density version of the Hales-Jewett's theorem. $\endgroup$ Aug 23, 2015 at 17:46

87 Answers 87

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The fact that the circumference of a unit circle is used to normalize the bell curve. Elementary compared to the other examples, yes, but how shocking was it when you first learned it?

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    $\begingroup$ To me, this isn't really shocking. It's a natural consequence of the cute (and, yes, maybe even surprising) fact that the square of $\int e^{x^2}\;dx$ is equal to $\int e^{x^2 + y^2}\;dx\;dy$, the integral of a function whose level sets are circles. $\endgroup$
    – Vectornaut
    Feb 1, 2012 at 21:45
  • $\begingroup$ @Vectornaut: your point is that this connection can be understood; but it still strikes me as initially surprising. $\endgroup$ Apr 12, 2015 at 11:33
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    $\begingroup$ @BenoîtKloeckner, I see. I was never surprised because, if I recall correctly, I never knew the normalization factor before being shown how to find it. $\endgroup$
    – Vectornaut
    Apr 12, 2015 at 21:29
  • $\begingroup$ I think James Stirling may have been the first to know this. $\endgroup$ Dec 15, 2021 at 18:12
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There exist two binary trees with rotation distance $2n-6$. The proof is unexpected and based on hyperbolic geometry (Sleator, Tarjan, Thurston (1988), "Rotation distance, triangulations, and hyperbolic geometry").

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    $\begingroup$ What is $n{}{}$? $\endgroup$ Jun 28, 2017 at 13:09
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Taniyama-Shimura-Weil connecting error terms counting number of points on an elliptic curve over finite fields and the Fourier coefficients of modular forms. It's less surprising these days because it's almost as famous as the two things it connects.

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The connection between homotopy groups of S2, Brunnian braids over the sphere, and Brunnian braids. This knocked me off my chair when I first heard about it. I know no conceptual explanation of this connection.

A. Berrick, F. R. Cohen, Y. L. Wong and J. Wu, Configurations, braids and homotopy groups, J. Amer. Math. Soc., 19 (2006), 265-326. Also available at http://www.math.nus.edu.sg/~matwujie/BCWWfinal.pdf (Wayback Machine) See also http://www.math.nus.edu.sg/~matwujie/cohen.wu.GT.revised.29.august.2007.pdf (Wayback Machine)

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    $\begingroup$ I'd like to find a more geometric proof of their result. There's a lot of geometric constructions that lead me to suspect such a result but I haven't found anything quite right. The main idea is to consider the closure of a Brunnian braid then look at things like the Koschorke invariants. mathoverflow.net/questions/234/… $\endgroup$ Feb 8, 2010 at 7:17
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  1. The Curry-Howard isomorphism linking various lambda calculi with intuitionistic logics; its extension to the classic logic via the concept of continuations.
  2. The conncetion between Borel hierarchy and arithmetical hierarchy.
  3. Fagin's theorem --- and later the whole branch of descriptive complexity --- linking well-known complexity classes with logics over finite models.
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  • $\begingroup$ Nice timing, I was just thinking about posting points one and three of your list :) $\endgroup$
    – user10891
    Mar 13, 2011 at 17:59
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Paul Vojta's discovery of the unexpected parallels between value distribution theory (Nevanlinna theory) in complex analysis and Diophantine approximation in number theory. See, e.g., Vojta's paper "Recent Work on Nevanlinna Theory and Diophantine Approximation". Serge Lang and William Cherry discuss the matter in their book Topics in Nevanlinna Theory.

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The analogy, still not understood to the full I think, between prime numbers and knots.

See Arithmetic topology in Wikipedia.

A most condensed picture is given by the Kapranov-Reznikov-Mazur dictionary

enter image description here

This is actually closely related to several answers here, and in fact initially I mentioned it in a comment to one of the answers but then still decided to make a separate entry.

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I don't know whether people will consider this surprising or not.

I think it may have been in the earliest part of the 20th century that it was shown that random walks in $n$ dimensions are recurrent if $n\le2$ and transient if $n\ge3.$

Then in the 1950s it was shown that the maxmimum-likelihood estimator of the expected value of a multivariate normal distribution in $n$ dimensions is an admissible estimator, in the decision-theoretic sense, when $n\le2$ but (a surprise) not when $n\ge3.$

Around 1990 or so, Morris L. (Joe) Eaton showed that these two propositions both say essentially the same thing.

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  • $\begingroup$ Interesting - could you provide references? $\endgroup$
    – R W
    Jun 24 at 10:58
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    $\begingroup$ @RW jstor.org/stable/2242007 $\endgroup$ Jun 26 at 4:59
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    $\begingroup$ @RW : "A Statistical Diptych: Admissible Inferences—Recurrence of Symmetric Markov Chains", Morris L. Eaton, The Annals of Statistics, Vol. 20, No. 3 (September, 1992), pp. 1147–1179 (33 pages) $\endgroup$ Jun 26 at 5:05
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Root systems, which are completely combinatorial objects, have a lot to do with topological objects, such as compact Lie groups, and linear algebraic objects, such as Lie algebras. Not just that, they classify semisimple ones among them!

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I agree with Zavosh that Jones' linking of Von Neumann algebras to knot theory is one of the great connections in modern times. Closer to home for me is Pisier's use of a theorem of Beurling on holomorphic semigroups to prove the duality of type and cotype of B-convex Banach spaces.

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There are several surprises regarding convex polytopes:

A) There are combinatorial types of polytopes that cannot be realized with rational coordinates (first discovered by Perles). This is not the case in three dimension but by now there are examples in every dimension greater equal 4. This adds to several examples on the wild combinatorial nature of convex polytopes in dim at least 4.

B) The applications of commutative algebra to the study of face-numbers of polytopes - Stanley proofs of the upper bound theorem using the Cohen-Macaulay argument and many subsequent results. Also surprising is the application of algebraic geometry: toric varieties, Hard Lefschetz theorem, intersection homology etc.

C) It is a special surprise that some proofs regarding the face number of polytopes applies only to polytopes with rational coordinates.

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That mechanical vibrations (mass-spring-dashpot systems) satisfy the same differential equations as electrical systems (inductor-resistor-capacitor circuits).

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  • $\begingroup$ Yes! This is a fundamental "surprise". :) $\endgroup$ Feb 4, 2016 at 0:34
  • $\begingroup$ Maxwell himself presented this connection before he dispensed with it in a later paper. $\endgroup$ Nov 16, 2021 at 22:06
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Another surprising connection is the Ax-Kochen theorem. Let $\mathcal{F}_{p,n,d}$ denote the set of homogeneous polynomials ("forms") in $n$ variables over the $p$-adics $\mathbb{Q}_p$ of degree $d$. The Ax-Kochen theorem is: For every positive integer $d$ there is a finite set $Y_d$ of "bad" prime numbers such that if $p$ is a "good" prime for $d$ (i.e. not in $Y_d$) then every $f \in \mathcal{F}_{p,d^2+1,d}$ has a non-trivial zero.

This was proved using model theory.

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    $\begingroup$ This would be easier to read as “for each degree $d$, for all sufficiently large primes $p$, any homogenous polynomial of degree $d$ in at least $d^2+1$ variables has a non-trivial zero in the $p$-adic numbers.” $\endgroup$
    – Matt F.
    Oct 28, 2021 at 16:01
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Being a physicist I'm still puzzled by the connection between:

  1. Wick theorem -- which is combinatorics (for me).
  2. Multivariate Gaussian integrals -- which is calculus (for me).
  3. Determinants and eigensystems -- which is linear algebra (for me).
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  • $\begingroup$ And these are intimately related to the Hermite polynomials whose moments are the perfect matchings of the vertices of the hypertetrahedra/hypertriangles/n-simplices with applications in combinatorics, geometry, analysis, and, of course, physics. $\endgroup$ Nov 16, 2021 at 20:28
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I'd like to share the very elementary fact (so elementary that I found surprising only after I taught Calculus course) that all the elementary functions are analytic in the global way. Of course, that's no surprise for polynomials. But I found no intuition why the trigonometric functions and the exponential functions, in the way they are originally considered by human, turn out to be equal to their Taylor expansions everywhere. Consider again the fact that Taylor expansion uses only the information on an infinitesimal neighborhood at a point, a function which is not originally defined by power series should be of extremely little chance to equal its Taylor expansion. I don't know if I'm right, but I finally told my students this is really a miracle.

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    $\begingroup$ You are completely correct, that this is in some way an astonishing thing. The downvotes are an expression of the absence of this astonishment from the official account of things... So, by accident, it is not surprising that you'd get downvotes. But I think you are perfectly correct... $\endgroup$ Feb 4, 2016 at 0:32
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    $\begingroup$ This is part of a larger miracle that complex numbers are so useful in maths, and mathematicians tend to forget how it is miraculous and non-trivial @paulgarrett $\endgroup$
    – reuns
    Oct 31, 2016 at 19:43
  • $\begingroup$ "Official" accounts consist of CURRICULA, in which it is (dishonestly) decreed that EVERYONE must, or should, follow the curriculum, and this practice anesthetizes most people against astonishment, by a mechanism that is blindingly obvious and that we all see in operation daily. That the story of the naked emperor is not exaggerated is seen in the fact that nearly all otherwise intelligent people don't see this. $\endgroup$ Dec 15, 2021 at 18:21
  • $\begingroup$ You seem to take advantage of the fact that elementary function can mean pretty much anything you want it to mean. For example, most of the "elementary functions" listed here are not entire: en.wikipedia.org/wiki/Elementary_function $\endgroup$ Jun 24 at 0:30
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The connection between rational homotopy theory and local algebra has been very useful, I was told. See Section 3 of this survey by Kathryn Hess and the references therein, especially Anick's counterexample to a conjecture of Serre.

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Another post reminded me of the following fact. The Poisson summation formula is a special case of the trace formula. Also the Frobenius reciprocity for finite groups follows from another special case of the trace formula, where the groups in question are finite. I find that these two theorems are related in such a way very surprising.

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    $\begingroup$ For me, the Frobenius reciprocity formula follows from $\left(A\otimes_R B\right)\otimes_S C\cong A\otimes_R\left(B\otimes_S C\right)$, where $R$ and $S$ are two unital (not necessarily commutative) rings, $A$ is a $\left(\mathbb Z,R\right)$-bimodule, $B$ is a $\left(R,S\right)$-bimodule, and $C$ is a $\left(S,\mathbb Z\right)$-bimodule. The "other" Frobenius formula is simply the trace of the former. Is this what you mean? But then I wouldn't really call it a connection. $\endgroup$ Feb 8, 2010 at 11:26
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    $\begingroup$ The connection that I was talking about is the following. The Arthur-Selberg trace formula is an identity of distributions for a pair of groups(with some conditions). When the groups are R and Z, then the trace formula reduces to Poisson summation. When the groups are finite, and with the right choice of a test function, the trace formula reduces to Frobenius reciprocity. $\endgroup$
    – MBN
    Feb 8, 2010 at 14:28
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    $\begingroup$ There is some subtlety in making the tensor-product associativity be the complete answer... for the topological vector space end of the analogy. Too technical, and maybe not immediately interesting, but P. Cartier's 1973/4 Sem. Bourb. talk/article explains how certain technical points (at a later "perfect" extreme the Dixmier-Malliavin theorem) make heuristics into theorems in such regards. Maybe the fact that the heuristics are "obvious" makes the actual surprise less? $\endgroup$ Feb 4, 2016 at 0:40
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Grothendieck's dessins d'enfants: the Galois group $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on certain graphs (with a decoration) on 2-dimensional topological surfaces.

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Stone duality usually refers to the equivalence between the category of Boolean algebras and the category of compact totally disconnected spaces. This duality intertwines the theory of Boolean algebras with general topology so much that Boolean algebras cannot be studied in depth without mentioning general topology and compact totally disconnected spaces cannot be studied in great detail without mentioning their relation with Boolean algebras. For example, the free Boolean algebras and free $\sigma$-complete Boolean algebras are normally represented not in terms of generators and relations, but as clopen sets (Baire sets) on the cantor cube $2^{I}$ for some set $I$.

Stone duality was originally a very surprising result, and it is probably a bit surprising to people seeing this result for the first time as well. Around 1937 when Marshall Stone formulated this duality it was difficult to imagine nice topological spaces that arose from algebraic structures rather than geometric or analytic structures.

Besides Stone duality, there are many dualities (equivalences of categories) similar in nature to Stone duality that relate different structures to each other and hence relate different areas of mathematics to each other (I have developed some of these dualities myself). For instance, one can relate topologies satisfying higher separation axioms with topologies that are not even $T_{1}$. One can also relate structures such as proximity spaces and uniform spaces with algebras of sets and Boolean algebras. There are also many dualities relating different in order theory to each other.

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  • $\begingroup$ Here's my terse expository account of Stone's duality: the totally disconnected compact Hausdorff space associated with a Boolean algebra $A$ is the space of all homomorphisms from $A$ into the two-element Boolean algebra, with the topology of pointwise convergence of nets of such homomorphisms. The Boolean algebra associated with a totally disconnected compact Hausdorff space is the set of all clopen subsets with the meet and join operations. To every homomorphism of Boolean algebras there naturally corresponds a continuous mapping beteween these spaces, going in the opposite direction. $\endgroup$ Sep 26 at 17:55
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The connection between the sphere packing problem and modular forms which was brought to light by recent breakthrough work of Viazovska (https://arxiv.org/abs/1603.04246) is very surprising, in my opinion.

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I believe the way R. Schoen solved Yamabe problem (after the contributions of Yamabe, Trudinger, Obata and Aubin) is truly impressive: after a long series of computations, he unexpectedly related the constant term in the expansion of certain Green functions associated to Yamabe problem (a Differential Geometry problem) with the so-called ADM mass in General Relativity (from Mathematical Physics); thus, he "reduced" the (remaining cases of) Yamabe problem to the infamous positive mass theorem, a result S.-T. Yau and himself proved (using Differential Geometry) to answer a (seemingly unrelated) central problem in General Relativity. See the survey of Lee and Parker for a nice account on this surprising connection between Differential Geometry and General Relativity.

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    $\begingroup$ Well but... isn't GR a part of DG in some sense? $\endgroup$
    – Qfwfq
    Sep 27, 2012 at 9:22
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The chromatic number of the Kneser graph $KG_{n,k}$ is equal exactly $2n-k+2$. There are very simple proof based on Borsuk-Ulam theorem.

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The surprising application of algebra into solving the problem of classification of manifolds or topological spaces, from which arose such concepts as fundamental group, homology groups, etc..

I think a lot of things will be "surprising" like this. I think the creations of most of the important topics or active areas of research in math arose out of some such "surprising" connection.

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    $\begingroup$ No, I don't think that everything is really surprising. There are a lot of theorems that have been based on hard work, but within the existing circle of ideas surrounding that result. What I'm after is when disparate parts of mathematics are brought together in unexpected ways. $\endgroup$ Feb 8, 2010 at 0:20
  • $\begingroup$ The fundamental group and homology were designed to study manifolds. So again this isn't remotely surprising that they'd be useful in the study of manifolds. $\endgroup$ Feb 8, 2010 at 0:23
  • $\begingroup$ Yes of course. But I meant, if you look at a manifold, who on earth would imagine that a group or module over a ring would help to classify them? When I took my first course in topology, this notion was a big surprise to me. $\endgroup$
    – Feb7
    Feb 8, 2010 at 0:27
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    $\begingroup$ Well personally it was surprising for me. I knew something about point set topology from one book, and from the next book I knew about groups, rings, linear algebra and so on. Then I go and sit in algebraic topology course because it was mandatory for some reason, and lo and behold! $\endgroup$
    – Feb7
    Feb 8, 2010 at 0:35
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    $\begingroup$ Hi Ryan, would you consider it obvious that the obstruction to promoting a homotopy equivalence to a simple homotopy equivalence should live in a group? And if so, could you have guessed which group? I think there are many places in topology where algebra is surprisingly effective. And for at least half a century mathematicians studied topological spaces, and manifolds in particular, before beginning to apply algebra to these questions. In 1942 the field was still referred to, at least by some, as "combinatorial topology" rather than "algebraic topology". $\endgroup$
    – Tom Church
    Feb 8, 2010 at 3:37
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The inverse calculus of a slope is the calculation of an area.

Barrow's Lemma: https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

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Application of ``thermodynamic formalism'' to questions of Analysis. Thermodynamic formalism have its origin in equilibrium statistical mechanics. First unexpected thing was its application to the theory of smooth dynamical systems, see beautiful paper of Ruelle, Is our mathematics natural? in BAMS. Later unexpected applications were discovered to problems of analysis which have nothing to do with dynamical systems, statistical mechanics or mathematical physics. One example is Astala's theorem on the area distortion under quasiconformal mappings. There is a very simple, self-contained proof of this theorem in MR1283548, using no dynamical considerations. But it is hard to imagine how could this proof be invented without dynamical and "thermodynamical" considerations.

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  • $\begingroup$ Of course (:-) the equilibrium statistical mechanics (general Ising models) is applied to the topological knot theory. $\endgroup$ Jan 16, 2015 at 5:37
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This is much fuzzier than many of the other answers, but the connections between graph theory, arithmetic, and geometry are breathtaking. (IMHO, anyone working anywhere even close to the intersection of these fields who hasn't read [at least some of] Serre's Trees needs to. Really everyone should read Trees though.)

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Stumbled on the following couple of days ago, when searching a good picture for a general 3-step filtration in an abelian category (in fact, there are similar structures in triangulated categories which I am ultimately up to):enter image description here

After feasting my eyes on it for a while I suddenly realized that what I am actually staring at is the Desargues configuration (in the form of five generic planes in 3-space)!

Not sure if this has any significance or whether one can do anything with it, but I certainly find it amusing.

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  • $\begingroup$ I guess it also solves the problem of placing 9 points in the plane so there are 9 lines with 3 points on a line. $\endgroup$ Mar 24, 2015 at 12:20
  • $\begingroup$ @GerryMyerson If I'm not mistaken that one is the Pappus configuration. Desargues' is $10_310_3$, i. e. ten points three on a line, ten lines three at a point (as five generic planes in 3-space produce $\binom53$ points and $\binom52$ lines intersecting like that) $\endgroup$ Mar 24, 2015 at 17:11
  • $\begingroup$ Yes, I missed a line. 10 it is. $\endgroup$ Mar 24, 2015 at 22:35
  • $\begingroup$ I don't know what the precise definition of $\text{“} 10_3 10_3 \text{”}$ is, but there exists at least one configuration of 10 lines and 10 points with each line passing through three of the 10 points and each point lying on three of the 10 lines, that is not incidence-isomorphic to the Desargues configuration, so I wonder whether that notation is enough to specify a particular configuration. $\endgroup$ Jun 29 at 2:46
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    $\begingroup$ @MichaelHardy Thank you! I should remember, definitely have seen it before. What I did not notice before is that Wikipedia says there are still seven more! $\endgroup$ Jun 30 at 5:36
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It seems that no one gave this one yet, although it probably hides behind many of previous answers.

The fact that in $\mathbb{C}$, product by a fixed complex number corresponds to a similarity is an incredible and far-reaching connection between algebra and geometry.

Among other things, it ties holomorphic functions with conformal maps of surfaces, so that for example one can identify a Riemann surface with a surface having a Riemannian metric of curvature $-1$, $0$ or $1$; more generally it allows for the use of complex analysis to study a number of problems in the geometry of surfaces.

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  • $\begingroup$ I originally learned this as the DEFINITION of multiplication of complex numbers, so when I encountered it presented as a theorem, I found it puzzling and wondered what the difference was between the theorem and the definition. $\endgroup$ Jun 29 at 3:13
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The amazing connection between $\eta$-identities and affine root systems, due to Macdonald and further elaborated upon by Kac! These identities encompass the Jacobi triple product identity, Euler's pentagonal number identity and many others. And these have connections to Complex simple Lie algebras.

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The connection between 'Electric-magnetic duality in string theory' and Langlands Program. See e.g. Witten-Kapustin. Not exactly a connection between two mathematics areas, but I think it nevertheless partially qualifies.

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