Your favorite surprising connections in Mathematics

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

"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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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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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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Classification of symmetric spaces by using classification of simple lie algebras due to cartan is my favourite one!

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The homotopy hypothesis, namely that two concepts, one that of a (weak) $n$-groupoid arising in higher category theory, and the other that of (the homotopy type of) a topological space $X$ with $\pi_i\left(X\right)=0$ for all $i > n,$ are the same thing.

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Weak homotopy type. –  Todd Trimble Mar 26 at 14:16

A recent connection between macroeconomics and electrodynamics by Maldacena: see Appendix here that derives Maxwell equations from the condition that currency exchange banks don't go bust (in a certain made-up but not too unreasonable currency exchange setup).

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Connection:

1. The Langrange polynomial interpolation formula;
2. The Chinese Remainder Theorem.

Formulations:

1. Let $\ K\$ be a field of characteristic $\ 0.\$ Let $\ \phi:A\rightarrow K\$ be an arbitrary function, where $\ A\$ is a non-empty finite subset of $\ K.\$ Then there exists an exactly polynomial $\ f:K\rightarrow K\$ of degree $\ n < |A|,\$ such that $\ \forall_{x\in A}\ f(x)=\phi(x)$.
2. Let $\ A\$ be a nonempty finite set of positive integers such that $\ \gcd(a\ b)=1\$ for every two different $\ a\ b\in A.\$ Let $\ \phi:A\rightarrow\mathbb Z\$ be arbitrary. Then there exists $\ f\in\mathbb Z\$ such that $\ \forall_{a\in A}\ f\equiv \phi(a) \mod a.\$ The integer $\ f\$ is unique in the following sense: $$\ \forall_{a\in A}\ g\equiv\phi(a)\!\!\!\mod a\quad \Rightarrow\quad g\equiv f\!\!\!\mod \prod A$$

Crucial special (basic) cases:

1. There exists exactly one polynomial $\ f_b:K\rightarrow K\$ of degree $\ n < |A|,\$ such that $\ f_b(b)=1,\$ and $\ \forall_{x\in A\setminus\{b\}}\ f_b(x)=0\,\$ for every $\ b\in A\$ (actually $\ \deg(f)=|A|-1$).
2. There exista exactly one integer $\ f_b\!\!\mod\prod A\$ such that $\ f_b\equiv 1\!\!\mod b,\$ and $\ \forall_{a\in B\setminus\{b\}}\ f_b\equiv 0\!\!\mod a\$ for every $\ b\in A$.

Once we get the basic elements $\ f_b,\$ then $\ f\$ is uniquely obtained as the respective linear combination of elements $\ f_b\$ both in the Lagrange and in the Chinese cases.

Construction (of the basis elements):

1. $\ \forall_{t\in K}\ f_b(t):=\frac{L_b(t)}{L_b(b)},\$ where $\ L_b(t):=\prod_{a\in A\setminus\{b\}}\ (t-a)$
2. $\ f_b\ := C_b\cdot d_b,\$ where $\ C_b:=\prod_{a\in A\setminus\{b\}} a,\$ and $\ d_b\cdot C_b\equiv 1\mod b$.

We see that $\ C_b\$ corresponds to $\ L_b,\$ and $\ d_b\$ to $\ \frac 1{L_b(b)}$.

This connection, when generalized, unites the algebraic number theory and the theory of algebraic functions.

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I don't see why this is surprising. –  Todd Trimble Jan 13 at 14:46
I don't have a real opinion on that since I don't know what you are alluding to there. Offhand, it sounded interesting. –  Todd Trimble Jan 13 at 20:18

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 $\ n=0\ 1\ ...$

The connection is my notion of the universal function (or universal morphism in general). The beginning of the story is:

THEOREM $\ \dim X \ge n\ \Leftrightarrow\ \exists\ universal\,\ f:X\rightarrow I^n\$ (for every completely regular space $\ X$).

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I have been studying a paper recently called "Pointwise Fourier Inversion: a Wave Equation Approach" by Mark Pinsky and Michael Taylor. Even though Fourier analysis and PDEs have close connections, a particular connection I like is that the solution to the standard wave equation \begin{equation*} (\partial^2_t-\Delta)u=0 \end{equation*} with initial velocity $0$ and initial position $f(x)$ is used to establish pointwise convergence of the partial Fourier Integrals \begin{equation*} S_Rf(x)=\frac{1}{(2\pi)^n}\int_{|\xi|\leq R}\hat{f}(\xi)e^{ix\cdot\xi} \end{equation*} in $L^2$ as $R\to\infty$ so that we have "concrete" statements to deal with rather than arbitrary information and distributions.

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@TimothyChow -- I didn't down-vote. Nevertheless I feel that the Chinese here is quite arbitrary in the Gödel's context, and relatively very trivial. One could use other schemes as well with the same success. –  Włodzimierz Holsztyński Jan 13 at 2:18
@WłodzimierzHolsztyński : What you say is true, but when exponentiation is not directly available, the options for a simple encoding are limited. –  Timothy Chow Jan 13 at 18:09

That the Jeffreys' prior in the pole-zero parameterization of a transfer function is the hyperbolic transfinite diameter of the support of the poles and zeros.

It's my favorite because I just discovered it last month. I like laughing at my own jokes.

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

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