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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

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Fact that something such well known as group of rotations SO(3) is connected but not simply connected and which is more it may be shown (!) by Dirac Belt or even by toying of cup of tee and a hand!

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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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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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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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    $\begingroup$ Weak homotopy type. $\endgroup$
    – Todd Trimble
    Mar 26, 2015 at 14:16
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    $\begingroup$ Or homotopy type of a CW-complex... $\endgroup$ Apr 12, 2015 at 9:45
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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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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, 2011 at 17:08
  • $\begingroup$ That's right of course. Thank you. The usual mistake of neglecting the trivial case... $\endgroup$
    – Abel Stolz
    Aug 17, 2011 at 12:28
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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, 2015 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). $\endgroup$ Jan 13, 2015 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, 2015 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). $\endgroup$ Jan 13, 2015 at 20:34
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    $\begingroup$ @ToddTrimble -- the above connection is basic since Kronecker to the specialists in algebraic number theory and algebraic geometers. But if you name Lagrange's interpolation and the Chinese theorem in one breath to, say, true (:-) experts in Analysis they will be most likely bewildered. $\endgroup$ Jan 16, 2015 at 5:44
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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. $\endgroup$ Feb 3, 2016 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. $\endgroup$ Feb 3, 2016 at 16:13
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This is another old connection, but the very idea of analytic number theory is counterintuitive - why should continuous tools give answers to integer-based questions? Two examples:

  • The Basel problem, $\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}.$
  • The Hardy-Ramanujan-Rademacher formula for the number of integer partitions of $n$.
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Using the Chinese remainder theorem for proving Gödel's incompleteness theorems.

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  • $\begingroup$ Why the downvotes? $\endgroup$ Nov 8, 2013 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. $\endgroup$ Jan 13, 2015 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. $\endgroup$ Jan 13, 2015 at 18:09
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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, 2018 at 13:17
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I find it a fascinating and productive perspective that the algebra of compositional and multiplicative inversion of formal power series are determined by the refined Euler characteristic / refined signed face partition polynomials of the associahedra (cf. MO-A1 and MO-A2) and the permutahedra (cf. MO-A3). These, in turn, are related to Lie infinitesimal generators, formal group laws, functional iteration, complex dynamics (Maupertuis action principal, Hamilton-Jacobi dynamics/geometric optics), the antipodes of combinatorial Hopf algebras, the calculus and group properties of the Sheffer polynomials, lattice paths and trees (and other combinatorial models), algebraic geometry, Koszul duality of operads, and scattering processes in quantum field theory among other areas of current research in math and physics. Compositional inversion via reciprocals of formal power series is also connected to noncrossing partitions (cf. OEIS A134264) and, consequently, the theory of free probability (and random matrices), a relation that can be derived from successive inversions via the permutahedra and associahedra.

I was certainly surprised with these revelations after first deriving the partition polynomials for the two types of inversion while exploring the Sheffer umbral/finite operator calculus and then subsequently finding the connections to permutahedra via Alford Arnold's OEIS entry A049019 and the associahedra via an article by Loday, and I believe the associations among the convex polytopes and the two types of inversion can be said surprising from a historical perspective as well. The three dimensional permutahedron is a Archimedean polytope--been around for a while--yet recognition of the explicit relation between the combinatorics of the faces of the permutahedra and multiplicative inversion seems relatively recent (probably this century only), and, despite Newton having derived at least the first few partition polynomials for compositional inversion of formal power series, Loday seems to be the first to have noticed the connection between associahedra (a 20'th century invention) and inversion.

In an interview by Quanta Magazine, Federico Ardila expressed his surprise at some of these connections. He collaborated with Marcelo Aguiar to produce some interesting perspectives on these relationships.

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  • $\begingroup$ I'm cheating slightly. The refined noncrossing partitions can be derived through multiplicative inversion (MI) of formal power series--an inversion related to refined Pascal partition polynomials--more quickly than by MI of formal Taylor series, or e.g.f.s, which is directly related to the combinatorics of permutahedra, but the MIs differ only by simple scaling factors of the indeterminates. $\endgroup$ Nov 16, 2021 at 21:04
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This is a bit old but I still find it surprising.

Fourier series were essentially invented by Brook Taylor and Daniel Bernoulli. The first noticed the rather obvious fact that sines and cosines represent the movement of a string pretty well, and the second added the observation that a sum of sines and cosines also represents a possible evolution of the string. Then D'Alembert put these discoveries in perspective, inventing the wave equation. But all this is quite natural: the connection between oscillations, sums of sines-cosines, and the wave equation is not too surprising.

Now, think of Fourier who discovered that the heat equation can be analyzed using sines and cosines too. The intuition that equations having nothing to do with oscillations can be solved using sines and cosines is quite deep and unexpected, and the impact on mathematics was dramatic.

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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! :-) $\endgroup$ Jan 13, 2015 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, 2015 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. $\endgroup$ Jan 14, 2015 at 0:14
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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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Category theory shows that products are dual to coproducts.

Aka multiplication is dual to addition.

That category theory could say something new about such simple concepts is what convinced me to study it.

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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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Complex numbers : Dual numbers : Double numbers :: Elliptic geometry : Euclidean geometry : Hyperbolic geometry

but also

Complex numbers : Dual numbers : Double numbers :: Euclidean geometry : Galilean geometry : Minkowski geometry

and also

Complex numbers : Dual numbers : Double numbers :: Hyperbolic geometry : Minkowski geometry : anti-de Sitter geometry

See: https://en.wikipedia.org/wiki/Laguerre_transformations

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

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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    $\begingroup$ What is shocking about these connections? $\endgroup$ Nov 2, 2021 at 22:31
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    $\begingroup$ From the question: "some part of mathematics is brought to bear in a fundamental way on another, where the connection between the two is unexpected" While I don't mean to disparage the result, I don't think strongness and extendibility are disparate in this way, so I don't think this is a good answer to the question. $\endgroup$ Dec 3, 2021 at 19:18
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