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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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I think it could be interpreted as restricting the breadth of the list. I would suggest removing the tag to avoid this. –  François G. Dorais Feb 8 '10 at 0:30

61 Answers 61

Definitely not a pure interplay between two subfields of math, but the omnipresence of quaternions in physics is stunning. I even figured out a few years ago that writing down the equivalent of Cauchy-Riemann equations for functions of a quaternionic variable in a matricial form and multiplying this matrix on the left by the metric tensor of special relativity directly gives rise to continuity equation and wave equation, which is truly intriguing and amazingly beautiful.

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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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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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In topology, the connection between the fixed point property and the topological dimension (the covering dimension)--the following two theorems are equivalent:

  • the cube $\ I^n\ $ has the fixed point property;
  • there exists a normal topological space $\ X\ $ such that $\ \dim X\ \ge\ n$.

for every $\ n=0\ 1\ ...$

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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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"Vojta's analogy" between Nevanlinna theory and Diophantine approximation. Nevanlinna theory studies holomorphic curves from the affine line $C$ to complex projective space $P^n$ (or to other complex manifolds). The main characteristic of such a curve is called the Nevanlinna characteristic. It was introduced by H. Cartan (for the case of projective space) in 1929. Almost simultaneously, heights was introduced to number theory by Weil and Siegel (1928).

Some people noticed the similarity of these two notions, but only in 1987, Vojta started to explore this similarity systematically. The result was very profitable for both theories.

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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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Connection between the typical number of isolated nonzero solutions ($N$) of a system of equations $$f_1=f_2=\cdots=f_n=0,$$ where each $f_k$ is a polynomial in $n$ complex variables, and the mixed volume ($V$) of the Newton polytopes of $f_k$: $$N=\tfrac1{n!}V.$$

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The beautiful analogy between number fields and function fields (and in general, algebra and geometry) that one learns about in arithmetic algebraic geometry.

Some specific examples:

The idea that a Galois group and a fundamental group (one algebro-number theoretic, the other geometric/topological) are two instances of the same thing.

The use of the term ramification in both number theory and geometry. Describing $\mathbb{Z}$ as simply connected because $\mathbb{Q}$ has no unramified extensions.

The appearance of integral closure in both algebraic geometry and algebraic number theory. The integral closure, in the former case, actually corresponds to a distinct geometric idea: non-singularity.

The idea of considering a prime number to be a point; then viewing localization at that prime, -adic completion at that prime, and the residue field of that prime as if they were the corresponding geometric objects. In particular, using the term "local" in number theory, as if we were talking about geometry! This idea is built into scheme theory.

There are many more examples.

This book looks deeply into the relationships between Galois groups and fundamental groups and eventually develops a theory which covers both.

This book explores the beautiful relation between algebraic curves and algebraic number theory.

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I was recently amazed at a quick connection between two facts I've known since high school. The Euler characteristic of a sphere, thought of as #vertices + #faces - #edges on a polyhedron, buckyball, etc., is 2; I re-deduced this from the fact that the derivative of $f(x)=1/x$ is $f'(x)=-1/x^2$.

The steps of the proof are as follows: construct the Riemann sphere using two complex charts, both C, with the holomorphic transition map $f(z)=1/z$ on each neighborhood minus its origin. Now we want to look at the Chern class of the cotangent bundle, which in standard orientation is the negative of the Euler class of the tangent bundle, i.e. the sphere. Well, assuming complex analysis, look at $df=\frac{-1}{z^2}dz$ to see the effect of the transition map on the cotangent bundles: as a ``holomorphic'' 1-form, that has a double pole at one point and no zeros. Thus we know that a section of the cotangent bundle of the sphere has divisor degree $-2$. So $\chi(S^2)=2$ and I now cannot separate this fact from $f'(x)=-1/x^2$ in my mind. It seem somehow more mysterious, ridiculous, and delightful that this connection is so short.

(Everyone I've mentioned this to prefers their own proof and perhaps it's better to do this slightly more directly to get a self-intersection 2 for a section of the tangent bundle, i.e. vector fields vanish twice, which gives the Euler class in $H^2(S^2)$ as a multiple of the orientation class.)

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How about something simple: $e^{i\pi}=-1$.

Like when you first hear that, what the hell does the ratio of circumference to diameter of circles have to do with the square root of negative one and the base of the natural exponent?

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...or between exponential and trigonometric functions generally. –  Michael Hardy Apr 23 '11 at 1:05
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In my opinion, this becomes a lot less mysterious as soon as you think of the exponential and trigonometric functions as eigenfunctions of the differentiation operator (respectively, its square), which is really the reason they're both so important. The basic properties and interrelationships of these functions - including the above identity - are natural consequences of this formulation. –  Robin Saunders Jul 29 '11 at 3:06
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Well yes, but that's the way it is with all these surprising results, isn't it? They all indicate a connection that no one had suspected, but is undoubtedly important. Once that connection is chewed over enough and becomes something you learn as a matter of course, then the original surprising result becomes "understandable", or sometimes even "trivial". But it certainly wasn't originally, and often isn't even to people first encountering these things today. –  Carl Offner Sep 30 '13 at 0:54

My favorite connection in mathematics (and an interesting application to physics) is a simple corollary from Hodge's decomposition theorem, which states:

On a (compact and smooth) riemannian manifold $M$ with its Hodge-deRham-Laplace operator $\Delta,$ the space of $p$-forms $\Omega^p$ can be written as the orthogonal sum (relative to the $L^2$ product) $$\Omega^p = \Delta \Omega^p \oplus \cal H^p = d \Omega^{p-1} \oplus \delta \Omega^{p+1} \oplus \cal H^p,$$ where $\cal H^p$ are the harmonic $p$-forms, and $\delta$ is the adjoint of the exterior derivative $d$ (i.e. $\delta = \text{(some sign)} \*d\*$ and $\*$ is the Hodge star operator). (The theorem follows from the fact, that $\Delta$ is a self-adjoint, elliptic differential operator of second order, and so it is Fredholm with index $0$.)

From this it is now easy to proof, that every not trivial deRham cohomology class $[\omega] \in H^p$ has a unique harmonic representative $\gamma \in \cal H^p$ with $[\omega] = [\gamma]$. Please note the equivalence $$\Delta \gamma = 0 \Leftrightarrow d \gamma = 0 \wedge \delta \gamma = 0.$$

Besides that this statement implies easy proofs for Poincaré duality and what not, it motivates an interesting viewpoint on electro-dynamics:

Please be aware, that from now on we consider the Lorentzian manifold $M = \mathbb{R}^4$ equipped with the Minkowski metric (so $M$ is neither compact nor riemannian!). We are going to interpret $\mathbb{R}^4 = \mathbb{R} \times \mathbb{R}^3$ as a foliation of spacelike slices and the first coordinate as a time function $t$. So every point $(t,p)$ is a position $p$ in space $\mathbb{R}^3$ at the time $t \in \mathbb{R}$. Consider the lifeline $L \simeq \mathbb{R}$ of an electron in spacetime. Because the electron occupies a position which can't be occupied by anything else, we can remove $L$ from the spacetime $M$.

Though the theorem of Hodge does not hold for lorentzian manifolds in general, it holds for $M \setminus L \simeq \mathbb{R}^4 \setminus \mathbb{R}$. The only non vanishing cohomology space is $H^2$ with dimension $1$ (this statement has nothing to do with the metric on this space, it's pure topology - we just cut out the lifeline of the electron!). And there is an harmonic generator $F \in \Omega^2$ of $H^2$, that solves $$\Delta F = 0 \Leftrightarrow dF = 0 \wedge \delta F = 0.$$ But we can write every $2$-form $F$ as a unique decomposition $$F = E + B \wedge dt.$$ If we interpret $E$ as the classical electric field and $B$ as the magnetic field, than $d F = 0$ is equivalent to the first two Maxwell equations and $\delta F = 0$ to the last two.

So cutting out the lifeline of an electron gives you automagically the electro-magnetic field of the electron as a generator of the non-vanishing cohomology class.

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I am always impressed how countability conditions and topological properties interact, like in the following cases.

Assume there is a topological group $P$ which is, as an abstract group, isomorphic to a direct product of groups $G$ and $H$. Assume all groups to be Hausdorff and locally compact. Then $P$ is isomorphic as a topological group to $G\times H$ in the product topology if $G$ and $H$ are sigmacompact.

And another example: Every non-discrete locally compact totally disconnected group has uncountable cardinality.

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Here is one of my favorite, that I learned from A. G. Khovanskii: let $f$ be a univariate rational function with real coefficients. Then, you can think of $f$ as inducing a continuous self-map of $\mathbb{RP}^1 \cong S^1$, in particular, it has a topological degree, say $[f]$, and if $f$ happens to be a polynomial, it is obvious that $[f]=0$ if $\deg(f)$ is even, and that $[f]=\pm 1$ if $\deg(f)$ is odd (depending on the sign of the main coefficient).

If the decomposition of $f$ in continued fraction is $$ f=P_0+\frac{1}{P_1+\frac{1}{P_2+\ddots}}$$ Then one can prove easily that $[f]$ is the (finite) sum: $[f]=\sum_{i \geq 0} (-1)^i[P_i]$. (Khovanskii himself taught this to high-schoolers in Moscow.)

The interesting connection for me follows: for any real polynomial $P$, the topological degree of the fraction $P'/P$ is clearly the (negative of the) number of real roots of $P$. Thus, the computation formula above applied to $[P'/P]$ allows us to recover Sturm's theorem.

I don't know if it really qualifies as a new proof of the theorem, but it's definitely a different point of view on that proof.

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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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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 inverse calculus of a slope is the calculation of an area.

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

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Although it is not all that spectacular, since it does not really relate two different fields of mathematics, it has always been surprising to me that the gradient flow equation for the Chern-Simons functional on a (closed, oriented) 3-manifold $Y$ turns out to be the ASD(=Yang-Mills) equation on the cylinder $Y\times\mathbb{R}$.


The next thing is not really a connection, but definitely one of my favorite surprises in mathematics. By the work of Michael Freedman, the classification of closed, oriented, simply-connected topological 4-manifolds is basically equivalent to the classification of unimodular, symmetric bilinear forms (uSBFs) over the integers. As nice as this is, it comes with the grain of salt that the classification of uSBFs is not an easy task. Specifically, the classification of definite uSBFs is a hard problem and far from being solved.

And now comes the surprise: Simon Donaldson tells us that if we look at smooth 4-manifolds, then the only definite uSBFs that can occur are the trivial ones!

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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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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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I believe the way R. Schoen solved Yamabe problem "http://en.wikipedia.org/wiki/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 "http://www.ams.org/journals/bull/1987-17-01/S0273-0979-1987-15514-5/" for a nice account on this surprising connection between Differential Geometry and General Relativity.

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Here is one of my favorites. If you consider a singular node of an algebraic curve locally it looks like the curves $xy=0$ in $\mathbb{C}^2$, or $x^2+y^2=0$. This consists of two smooth arcs intersecting to each other transversally (reducible in particular).

Now, one step further, if we consider a cusp which is analytically equivalent to the origin in the curve $y^2+x^3=0$ in $\mathbb{C}^2$, it is locally irreducible. However, here comes the interesting point, if we intersect the singularity with a small ball $$[(x,y)\in \mathbb C^2:\ |x|^2+|y|^2=\epsilon]\cong S^3$$ what we've got is that such an intersection is $$(ae^{2i\theta},a^{3/2}e^{3i\theta})\subset S^1\times S^1\subset S^3$$ which is contained in a torus winding two times in one direction in the torus and three times in the other direction, in other words, we have an trefoil knot. alt text

Now in the case of surfaces, all these facts give rise to an amazing relation between topology and algebraic geometry. The underlaying space topological space in $\mathbb C^4$ of $$x^2+y^2+z^2+w^3=0$$ is a manifold!! (note it is singular at the origin in the context of AG!). As far as I know, if one intersects a small ball with the singularity, as I did above, one gets a topological sphere whose differential structure is NOT the standard one. Even more, considering in $\mathbb C^5$ the following hypersurface $$x^2+y^2+z^2+w^3+t^{6k-1}=0$$ and carrying out the intersection with a small sphere around the origin, for $k=1,2,\ldots 28$ one may get all the 28 possible exotic differential structures on the 7-sphere that Milnor found.

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Wow,that IS pretty whack,Csar.This example alone is a testament to the power of modern topology and geometry and the incredible connections it has uncovered. –  Andrew L Jul 15 '10 at 19:55

My personal favorite is Multiple Zeta Values $$ \zeta(s_1,\ldots,s_d) = \sum_{n_1>\ldots>n_d} \frac{1}{n_1^{s_1}\ldots n_d^{s_d}} $$ They appears in relation with

  • Quantum groups (they are coefficient of Drinfeld's KZ associator)
  • Deformation quantization (Kontsevich's formula for the affine space)
  • Feynmann diagrams (a large class of diagrams evaluate to MZV's)
  • Kashiwara-Vergne conjecture (representation theory of Lie groups)
  • Modular forms (Zagier noticed that the space of relations in depth 2 is canonically isomorphic to the space of cusp forms on $SL_2$ through their period polynomials)
  • Moduli spaces of curves of genus 0 $\mathcal{M}_{0,n}$

the list goes on and on... the reason for all this lies in the theory of mixed Tate motives over $\mathbb{Z}$.

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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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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 shortest path between two truths in the real domain passes through the complex domain. Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe.Jacques Hadamard

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Why down vote? I thought Jacques Hadamard expressed in his quote that in his time the connection of prime numbers to the zeroes of the Riemann zeta function was surprising and much of a shortcut to proving the Prime Number Theorem. –  Unknown Jun 2 '10 at 1:05

The work of Nabutovsky and Weinberger applying computability theory (a.k.a. recursion theory) to differential geometry. For example one of their results is that if you consider the space of Riemannian metrics on a smooth compact manifold $M$ of dimension at least 5 and sectional curvature $K\le 1$, then there are infinitely many extremal metrics. This is a purely geometric statement, but the only known proof uses concepts from computability theory. Moreover the results from computability theory that are used in their work are very deep; prior to their work, some skeptics regarded this area of computability theory as being overly specialized and having no hope of being connected to other areas of mathematics. See the exposition of Robert Soare (available on his website) for more information.

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One more. The application of string theory (mirror symmetry) to solving the Clemens conjecture in enumerative geometry, by finding the generating function for the number of rational curves which pass through a certain number of points. The coefficients are Gromov-Witten invariants. This is the work of Candelas, et al.

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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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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. –  Vectornaut Feb 1 '12 at 21:45

protected by François G. Dorais Sep 30 '13 at 0:52

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