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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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6  
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

66 Answers 66

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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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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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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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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Using the Chinese remainder theorem for proving Gödel's incompleteness theorems.

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1  
@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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protected by François G. Dorais Sep 30 '13 at 0:52

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