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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. $\endgroup$ – Joel Moreira Aug 23 '15 at 17:46

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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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"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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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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    $\begingroup$ Weak homotopy type. $\endgroup$ – Todd Trimble Mar 26 '15 at 14:16
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    $\begingroup$ Or homotopy type of a CW-complex... $\endgroup$ – David Roberts Apr 12 '15 at 9:45
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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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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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  • $\begingroup$ Abel, you'd better rule out discrete groups in your last sentence. $\endgroup$ – Todd Trimble Jul 25 '11 at 17:08
  • $\begingroup$ That's right of course. Thank you. The usual mistake of neglecting the trivial case... $\endgroup$ – Abel Stolz Aug 17 '11 at 12:28
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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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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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  • $\begingroup$ I am not exactly sure that some non-trivial macroeconomics can be extracted from the connection. I think it is more of an illustrative thing, though I might be wrong $\endgroup$ – user74900 May 22 '18 at 13:17
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Using the Chinese remainder theorem for proving Gödel's incompleteness theorems.

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  • $\begingroup$ Why the downvotes? $\endgroup$ – Timothy Chow Nov 8 '13 at 16:13
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    $\begingroup$ @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. $\endgroup$ – Włodzimierz Holsztyński Jan 13 '15 at 2:18
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    $\begingroup$ @WłodzimierzHolsztyński : What you say is true, but when exponentiation is not directly available, the options for a simple encoding are limited. $\endgroup$ – Timothy Chow Jan 13 '15 at 18:09
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The existence of Nash equilibria is an example that connects elementary aspects of game theory, probability, geometry, and algebraic topology.

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Another point that hasn't been mentioned yet: To prove the nonexistence of scissors congruences, one typically uses algebraic $K$-theory.

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  • $\begingroup$ Can you more explain or give some references?Thank you. $\endgroup$ – Ali Taghavi Feb 3 '16 at 11:05
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    $\begingroup$ There is a nice book by Johan Dupont. The connection goes via group cohomology, as far as I recall. $\endgroup$ – Sebastian Goette Feb 3 '16 at 16:13
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I also guess the links between differential geometry and geometric analysis on one hand and algebraic topology on the other were rather surprising when they were found.

  1. The Pontryagin-Thom construction. Smooth cobordism is described by homotopy groups of the Thom spectrum (which seems to have forgotten the smooth structures entirely). On the way, it gives one one of the most geometric motivations to study homotopy theory and spectra.

  2. The Atiyah-Singer index theorem allows one to guess the dimension of solution/moduli spaces of (sometimes even nonlinear) partial differential equations using characteristic classes that do not involve any hard analysis at all. Because the topological formula for the index also has a geometric interpretation, one gets applications to curvature questions in Riemannian geometry as a bonus. The surprise continues when one compares the different proofs of this theorem using either abstract $K$-theory (Atiyah-Singer) or the heat equation (Atiyah-Bott-Patodi, Getzler, Bismut, and others) or the geometry and representation theory of Lie groups (Berline-Vergne). The combination of these methods is still leading to new insights not only in differential topology.

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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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    $\begingroup$ I don't see why this is surprising. $\endgroup$ – Todd Trimble Jan 13 '15 at 14:46
  • $\begingroup$ @ToddTrimble -- what about the topological dimension and the fixed point property? (I am just curious, see above--and you're welcome to down-vote it too, fair is fair, it should be a two-way street). $\endgroup$ – Włodzimierz Holsztyński Jan 13 '15 at 20:03
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    $\begingroup$ I don't have a real opinion on that since I don't know what you are alluding to there. Offhand, it sounded interesting. $\endgroup$ – Todd Trimble Jan 13 '15 at 20:18
  • $\begingroup$ @ToddTrimble -- (me and sophisticated alluding? :-); there is my answer in the same surprising connection thread just a couple places above this answer on which we are commenting right now (well, I am more diverting than commenting). $\endgroup$ – Włodzimierz Holsztyński Jan 13 '15 at 20:34
  • $\begingroup$ You mean here: mathoverflow.net/a/143549/2926 Yeah, I knew which answer you meant, but I still don't know what that connection is about. Sorry I can't respond more intelligently. Perhaps you could explain just a bit more there? $\endgroup$ – Todd Trimble Jan 13 '15 at 20:42
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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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  • $\begingroup$ @ToddTrimble -- done! :-) $\endgroup$ – Włodzimierz Holsztyński Jan 13 '15 at 23:01
  • $\begingroup$ Thanks. I have to keep reminding myself what it means for $f$ to be "universal" (for every $g: X \to I^n$ there exists $x \in X$ such that $f(x) = g(x)$). By the way, just so you know: even if you type @name, the intended recipient won't receive notification unless he/she commented before (or was the one who posted). I just happened to notice you made an edit, so I took a look and saw you tried to reach me. $\endgroup$ – Todd Trimble Jan 13 '15 at 23:28
  • $\begingroup$ @ToddTrimble -- yes, about the system of notifications, thank you for the info. And this time indeed, the system reacted to at-T immediately, expanded it to your full name. $\endgroup$ – Włodzimierz Holsztyński Jan 14 '15 at 0:14
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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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The Selberg trace formula, relating chaotic geodesic motion on a compact space of negative curvature with the eigenvalues of the Laplacian operator on that space. (someone mentioned trace formulas already, but from a different perspective)

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The following amazing connection is a special case of a theorem by Sato Kentaro and another theorem by Normal Perlummter.

Theorem: The following are equivalent for regular $\kappa$:

(i) For every function $f: \kappa\rightarrow\kappa$, there is some $\alpha\lt\kappa$ such that $f``\alpha\subseteq\alpha$ and there is some $j: V\prec M$ with critical point $\alpha$, and $V_{j^2(f)(j(\alpha))}\subseteq M$.

(ii) For every function $f: \kappa\rightarrow\kappa$, there is some $\alpha\lt\kappa$ such that $f``\alpha\subseteq\alpha$ and there is some $j: V\prec M$ with critical point $\alpha$, and $M^{j(f)(\alpha)}\subseteq M$.

(iii) For every $rank(S)=\kappa$, there is some $\mathfrak M$,$\mathfrak N\in S$, and a $j: \mathfrak M\prec\mathfrak N$.

Proof. $(i)\leftrightarrow (iii)$ follows from a theorem in Sato Kentaro's Double helix in large large cardinals and iteration of elementary embeddings, and $(ii)\leftrightarrow (iii)$ from Norman Perlmutter's The large cardinals between supercompact and almost-huge.◼

Another amazing theorem is this:

Theorem: The following are equivalent:

(i) For every $\gamma$, there is some $j: V\prec M$ with critical point $\kappa$, and $V_{j(\gamma)}\subseteq M$.

(ii) For every $\lambda$, there is some $j: V\prec M$ with critical point $\kappa$, $M^{\lambda}\subseteq M$ and $V_{j(\kappa)}\subseteq M$.

(iii) $\kappa$ is extendible.

Proof. $(i)\leftrightarrow (iii)$ follows from a theorem in Sato Kentaro's Double helix in large large cardinals and iteration of elementary embeddings, and $(ii)\leftrightarrow (iii)$ from Konstantinos Tsaprounis' Elementary chains and $C^{(n)}$-cardinals.◼

These all highlight the shocking connection between strongness and extendibility.

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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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