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?
 A: Quillen's result in Elementary proofs of some results of cobordism theory using Steenrod operations that the ring of cobordism classes of (stably) complex manifolds is isomorphic to Lazard's ring (i.e. the universal ring classifying formal group laws). This seems so mysterious to me. Why should cobordism classes of complex manifolds have anything to do with the algebraic geometry of formal group laws? Nevertheless this has been one of the most important observations for modern homotopy theory. It is the driving force behind Chromatic Stable Homotopy which tries to build a dictionary between the algebraic geometry of FGLs and structures present in the stable homotopy category. It is shocking how successful this has been.
A: From an essay of Arnol'd:
Jacobi noted, as mathematics' most fascinating property, that in it one and the same function controls both the presentations of a whole number as a sum of four squares and the real movement of a pendulum.
A: The surprising application of algebra into solving the problem of classification of manifolds or topological spaces, from which arose such concepts as fundamental group, homology groups, etc..
I think a lot of things will be "surprising" like this. I think the creations of most of the important topics or active areas of research in math arose out of some such "surprising" connection.
A: The inverse calculus of a slope is the calculation of an area.
Barrow's Lemma: https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
A: 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. 
A: 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.

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.
A: This is much fuzzier than many of the other answers, but the connections between graph theory, arithmetic, and geometry are breathtaking. (IMHO, anyone working anywhere even close to the intersection of these fields who hasn't read [at least some of] Serre's Trees needs to. Really everyone should read Trees though.)
A: 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.
A: Stumbled on the following couple of days ago, when searching a good picture for a general 3-step filtration in an abelian category (in fact, there are similar structures in triangulated categories which I am ultimately up to):
After feasting my eyes on it for a while I suddenly realized that what I am actually staring at is the Desargues configuration (in the form of five generic planes in 3-space)!
Not sure if this has any significance or whether one can do anything with it, but I certainly find it amusing.
A: 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.
A: 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.
A: 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.


*

*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.

*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.
A: 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.
Galois Groups and Fundamental Groups by Szamuely looks deeply into the relationships between Galois groups and fundamental groups and eventually develops a theory which covers both.
An Invitation to Arithmetic Geometry by Lorenzini explores the beautiful relation between algebraic curves and algebraic number theory.
This post ("Mazur's knotty dictionary" on the neverendingbooks.org blog) explores the analogy between prime numbers and knots.
A: Classification of symmetric spaces by using classification of simple lie algebras due to cartan is my favourite one!
A: 
Traveling wave solutions to the KdV Equation for any speed
  and whose profiles look like the graph of the $\wp$-function for any elliptic
  curve. 

More precisely, if $u(x, t)$ is a solution of the KdV Equation that has the form
$$u(x, t) = w(x + ct)$$ then
$$u(x,t)=-2\wp(x + ct + \omega; k_1, k_2)+2c/3.$$
(See e.g. Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear PDEs by Alex Kasman)
A: 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 $n$.

A: 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?
A: The ubiquity of Littlewood-Richardson coefficients. Given three partitions $\lambda, \mu, \nu$ each with at most $n$ parts, there is a combinatorial definition for a number $c^\nu_{\lambda, \mu}$ which is nonzero if and only if any of the following statements are true:


*

*There exist Hermitian matrices $A, B, C$ whose eigenvalues are $\lambda, \mu, \nu$, respectively and $A + B = C$ (one can also replace Hermitian by real symmetric)

*The irreducible representation of ${\bf GL}_n({\bf C})$ with highest weight $\nu$ is a subrepresentation of the tensor product of those irreducible representations with highest weights $\lambda$ and $\mu$. 

*Indexing the Schubert cells of the Grassmannian ${\bf Gr}(d,{\bf C}^m)$ (where $d \ge n$ and $m-d$ is at least as big as any part of $\lambda, \mu, \nu$) by $\sigma_\lambda$ appropriately, the cycle $\sigma_\nu$ appears in the intersection product $\sigma_\lambda \sigma_\mu$. 

*There exists finite Abelian $p$-groups $A,B,C$ and a short exact sequence $0 \to A \to B \to C \to 0$ such that $B \cong \bigoplus_i {\bf Z}/p^{\nu_i}$, $A\cong \bigoplus_i {\bf Z}/p^{\lambda_i}$, and $C\cong \bigoplus_i {\bf Z}/p^{\mu_i}$.


And probably many more things.
A: Connection:


*

*The Langrange polynomial interpolation formula;

*The Chinese Remainder Theorem.


Formulations:


*

*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 $\ \forall_{x\in A}\ f(x)=\phi(x)$.

*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:


*

*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 $\ \deg(f)=|A|-1$).

*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):


*

*$\ \forall_{t\in K}\ f_b(t):=\frac{L_b(t)}{L_b(b)},\ $ where $\ L_b(t):=\prod_{a\in A\setminus\{b\}}\ (t-a)$

*$\ 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.

A: The existence of Nash equilibria is an example that connects elementary aspects of game theory, probability, geometry, and algebraic topology.
A: 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.
A: Ehud Hrushovski's proof, using model theory, of the geometric Mordell-Lang conjecture in algebraic geometry.
A: 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 $f_k$:
$$N=(n!)\cdot V.$$
A: Here is a copypaste of something I've already mentioned in another question.
The fastest known solution of the word problem in braid groups originated from research on large cardinal axioms; the proof is independent of the existence of large cardinals, although the first version of the proof did use them. See Dehornoy, From large cardinals to braids via distributive algebra, Journal of knot theory and ramifications, vol. 4 (1995), no. 1, 33–79 (MR).
To me this is an absolute mystery! Large cardinals are usually considered an esoteric subject situated on the border of the observable universe. So why should they have any relevance to braids, a very down to earth part of mathematics? Let alone give an algorithm for distinguishing braids, and what's more, the fastest algorithm known.
A: 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!
A: 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.
A: 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).
A: Another point that hasn't been mentioned yet: To prove the nonexistence of scissors congruences, one typically uses algebraic $K$-theory.
A: I think the disparity between the world-views in low-dimensional topology versus high-dimensional topology are surprising.  Even after you learn the reasons why, IMO they should still be surprising.   Examples:
1) Teichmuller space exists, yet hyperbolic manifolds in dimension $3$ and larger are rigid.   There are many interesting connections here such as the link between conformal geometry, complex analysis and hyperbolic geometry in dimension 2. 
2) Exotic smooth structures on $\mathbb R^4$ but not on $\mathbb R^n$ for $n\neq 4$.
3) Why the Poincare conjecture/hypothesis is "hard" in dimensions $3$ and $4$ yet relatively "easy" in other dimensions. 
4) Geometry being particularly relevant to $2$ and $3$-dimensional manifolds yet less so in higher dimensions. 
I could go on.  Some of these are connections, some I suppose are disconnections.  But a connection is only a surprise if you have reason to think otherwise.  :)
A: 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.
A: McKay's observation that the special fiber in the desingularization of du Val singularities is a bunch of $\mathbb P^1$s linked according to the Dynkin diagram corresponding to the group of the singularity.
A: This is probably not the most serious of applications, but I found the equivalence (in game theory) of the determinacy of Nash's board game Hex with the Brouwer Fixed Point theorem to be a surprising, if somewhat lighthearted, connection.
You can read David Gale's paper.
A: The pair correlation function between Riemann zeta function zeros is the same as the pair correlation function between eigenvalues of random Hermitian matrices. 
A: Complex multiplication of elliptic curves and the explicit construction of the maximal abelian extension of a quadratic imaginary number field. 
A: It is possible to compute the Betti numbers of a smooth complex variety $X(\mathbb{C})$ by computing the cardinality of $X(\mathbb{F}_{p^n})$ for a prime $p$ with good reduction and a finite number of positive integers $n$; in other words, by brute force.
The above claim is wrong, so I'll phrase it the other way around.  The Betti numbers of a smooth complex variety control the behavior of the number of points on $X(\mathbb{F}_{p^n})$; for example, for a smooth projective curve of genus $g$ we have $|\text{Card}(X(\mathbb{F}_q))| - q - 1| \le 2g \sqrt{q}$.
Generally I find the relationship between the arithmetic and topological properties of varieties surprising, although maybe this is a temporary kind of surprise that arithmetic geometers are used to.  Another example: if $X$ is a curve, then whether the curvature of $X(\mathbb{C})$ is positive, zero, or negative determines whether $X(\mathbb{Q})$ is rationally parameterizable, a finitely generated group, or finite (unless it's empty).
A: 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.)
A: The Jones polynomial of knot theory and Feynman path integrals.
A: Special values of the Riemann zeta function and class numbers of cyclotomic fields.
A: 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 $\mathbb{Z}$. 
A: 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! 
A: "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.
A: 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.
A: 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.
A: 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.
A: Disclaimer: This is a little bit of self-promotion, but when I discovered it, I was very fascinated by the relationship:
There are positive definite kernels, hence a reproducing kernel Hibert space over pitches of musical notes which can approximately capture the perceived consonance of two musical notes. Details are described here:
http://orges-leka.de/knn-music/Measuring_note_similarity_with_positive_definite_kernels.pdf
One manifestation of this relationship, to create relaxing study music, can be found here:
https://www.youtube.com/watch?v=Uc7D3Q_6baU
I find it intriguing that something like a Hilbert space shows up in the perceived consonance of musical notes. It opens up new possibilities for application of geometric intuition to music.
A: Ulam's problem on determining the length of the longest increasing subsequence of a random permutation. The solution and the full description of the answer brought together ideas from integrable systems, combinatorics, representation theory, probability (appearing in the form of polynuclear growth model for instance), and random matrix theory.
A: 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. $\delta = \text{(some sign)} \star d\star$ and $\star$ 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.
A: 
Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe. (The shortest path between two truths in the real domain passes through the complex domain.)
— Jacques Hadamard Paul Painlevé

It is often credited to Hadamard, but in fact Painlevé said this exact quote, and Hadamard in one of his books merely paraphrased it: "it has been written that..." See https://philosophy.stackexchange.com/a/61924
Hadamard clearly had his proof of the prime number theorem, along the approach of Riemann, in mind. The ubiquity of complex numbers may deserve a full answer in itself; but here we might highlight its use in number theory: Riemann zeta function and other L-functions, modular forms (which would be a whole answer in itself).
A: Goppa’s construction of error-correcting codes from curves, leading to the Tsfasman–Vladut–Zink bound (the first improvement over the Gilbert–Varshamov bound; see Modular curves, Shimura curves, and Goppa codes, better than Varshamov–Gilbert bound).  An error-correcting code may be regarded as a combinatorial structure, and I think that this is a surprising connection between algebraic geometry and combinatorics.
A: 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
$$ f=P_0+\cfrac{1}{P_1+\cfrac{1}{P_2+\ddots}}$$
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.
A: 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?
A: The surprising applications of algebraic geometry to number theory, for instance evidenced in the work of Deligne in proving the Ramanujan conjectures.
A: 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.
A: Using the Chinese remainder theorem for proving Gödel's incompleteness theorems.

*

*Arithmetization of syntax

*Gödel's beta function

*Gödel's numbering / using the Chinese remainder theorem
A: 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.
A: As well known as the connection is, I am constantly amazed by the power of analytical geometry (developed by Descartes and Fermat) to make connections between geometrical ideas and algebraic ideas. It seems remarkable to me that so much geometrical information (as for example in the case of the conic sections) can be represented so succinctly (via quadratic equations in two variables). The geometry suggests things to think about in algebra and the algebra suggests things to think about in geometry. It is just amazing!!
A: There exist two binary trees with rotation distance $2n-6$. The proof is unexpected and based on hyperbolic geometry (Sleator, Tarjan, Thurston (1988), "Rotation distance, triangulations, and hyperbolic geometry").
A: The connection between homotopy groups of S2, Brunnian braids over the sphere, and Brunnian braids. This knocked me off my chair when I first heard about it. I know no conceptual explanation of this connection.
A. Berrick, F. R. Cohen, Y. L. Wong and J. Wu, Configurations, braids and homotopy groups, J. Amer. Math. Soc., 19 (2006), 265-326. Also available at http://www.math.nus.edu.sg/~matwujie/BCWWfinal.pdf (Wayback Machine)
See also http://www.math.nus.edu.sg/~matwujie/cohen.wu.GT.revised.29.august.2007.pdf (Wayback Machine)
A: 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.
A: *

*The Curry-Howard isomorphism linking various lambda calculi with intuitionistic logics; its extension to the classic logic via the concept of continuations.

*The conncetion between Borel hierarchy and arithmetical hierarchy.

*Fagin's theorem --- and later the whole branch of descriptive complexity --- linking well-known complexity classes with logics over finite models.

A: The analogy, still not understood to the full I think, between prime numbers and knots.
See Arithmetic topology in Wikipedia.
A most condensed picture is given by the Kapranov-Reznikov-Mazur dictionary

This is actually closely related to several answers here, and in fact initially I mentioned it in a comment to one of the answers but then still decided to make a separate entry.
A: Monstrous Moonshine.
I mean why should the Fourier series of the $j$-invariant have coefficients related to the dimensions of the representations of the largest sporadic simple group? And why should the proof of this fact drag in mathematics from String Theory?
A: 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.
A: I don't know whether people will consider this surprising or not.
I think it may have been in the earliest part of the 20th century that it was shown that random walks in $n$ dimensions are recurrent if $n\le2$ and transient if $n\ge3.$
Then in the 1950s it was shown that the maxmimum-likelihood estimator of the expected value of a multivariate normal distribution in $n$ dimensions is an admissible estimator, in the decision-theoretic sense, when $n\le2$ but (a surprise) not when $n\ge3.$
Around 1990 or so, Morris L. (Joe) Eaton showed that these two propositions both say essentially the same thing.
A: 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!
A: I agree with Zavosh that Jones' linking of Von Neumann algebras to knot theory is one of the great connections in modern times.  Closer to home for me is Pisier's use of a theorem of Beurling on holomorphic semigroups to prove the duality of type and cotype of B-convex Banach spaces. 
A: Another surprising connection is the Ax-Kochen theorem.  Let $\mathcal{F}_{p,n,d}$ denote the set of homogeneous polynomials ("forms") in $n$ variables over the $p$-adics $\mathbb{Q}_p$ of degree $d$.  The Ax-Kochen theorem is: For every positive integer $d$ there is a finite set $Y_d$ of "bad" prime numbers such that if $p$ is a "good" prime for $d$ (i.e. not in $Y_d$) then every $f \in \mathcal{F}_{p,d^2+1,d}$ has a non-trivial zero.
This was proved using model theory.
A: Being a physicist I'm still puzzled by the connection  between:

*

*Wick theorem -- which is combinatorics (for me).

*Multivariate Gaussian integrals -- which is calculus (for me).

*Determinants and eigensystems  -- which is linear algebra (for me).

A: 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.
A: That mechanical vibrations (mass-spring-dashpot systems) satisfy the same differential equations as electrical systems (inductor-resistor-capacitor circuits). 
A: Grothendieck's dessins d'enfants: the Galois group $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on certain graphs (with a decoration) on 2-dimensional topological surfaces.
A: Another post reminded me of the following fact. The Poisson summation formula is a special case of the trace formula. Also the Frobenius reciprocity for finite groups follows from another special case of the trace formula, where the groups in question are finite. I find that these two theorems are related in such a way very surprising.
A: The connection between rational homotopy theory and local algebra has been very useful, I was told. See Section 3 of this survey by Kathryn Hess and the references therein, especially Anick's counterexample to a conjecture of Serre.
A: 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.
A: I'll recycle one I mentioned in a thread last week, connecting an elementary problem about polynomials to the classification of finite simple groups:
Definition: A polynomial $f(x) \in \mathbb{C}[x]$ is indecomposable if whenever $f(x) = g(h(x))$ for polynomials $g$, $h$, one of $g$ or $h$ is linear.
Theorem. Let $f, g$, be nonconstant indecomposable polynomials over $\mathbb C$. Suppose that $f(x)-g(y)$ factors in $\mathbb{C}[x,y]$. Then either $g(x) = f(ax+b)$ for some $a,b \in \mathbb{C}$, or
$$\operatorname{deg} f = \operatorname{deg} g = 7, 11, 13, 15, 21, \text{ or } 31,$$
and each of these possibilities does occur.
The proof uses the classification of the finite simple groups [!!!] and is due to Fried ["Exposition on an arithmetic-group theoretic connection via Riemann's existence theorem", 1980, in the proceedings of the 1979 Santa Cruz conference on finite groups], following a the reduction of the problem to a group/Galois-theoretic statement by Cassels [1970]. [W. Feit, "Some consequences of the classification of finite simple groups," 1980.]
A: My favorite surprise, which is perhaps the record-holder for the longest time
it took for the two ideas to be brought together, is the connection between
regular n-gons and Fermat primes. The Greeks knew how to construct regular
n-gons by ruler and compass for n=3,4,5,6. Fermat introduced numbers of the
form $2^{2^m}+1$ around 1640 in the mistaken belief they were prime for all m.
Then in 1796 Gauss discovered how to construct the regular 17-gon, and a few 
years later showed that the n in a constructible n-gon is the product of some
power of 2 by distinct Fermat primes.
A: The chromatic number of the Kneser graph $KG_{n,k}$ is equal exactly $2n-k+2$. There are very simple proof based on Borsuk-Ulam theorem.
A: I believe the way R. Schoen solved 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 for a nice account on this surprising connection between Differential Geometry and General Relativity.
A: 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. 
A: The connection between the sphere packing problem and modular forms which was brought to light by recent breakthrough work of Viazovska (https://arxiv.org/abs/1603.04246) is very surprising, in my opinion.
A: 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\text{ universal } f:X\rightarrow I^n\ $ (for every completely regular space $\ X$).
======================================================
======== 2022-10-15 ==========
====
Mathematicians that worked on the topological dimension theory and on the the fixed point property are among the greatest. The dimension theory is an integral part of topology, and a bridge between the general and algebraic topology. The f.p.p. appears in the theory of differential and integral equations, functional analysis, ...
The theory of universal maps (and morphisms) is shocking rather than a surprising bridge or rather a common roof over the dimension theory and f.p.p. Many related theorems get or should get farther clarification and a generalization.

The following theorem features dimension and fixed points only (and not universal maps at all!) but its proof is simple and natural only after involving universal maps:
Theorem (Włodzimierz Holsztyński) Let $\ X\ $ be a Hausdorff compact space such that the product $\ X\times\mathbb I^n\ $ of $\ X\ $ and the finite-dimensional cube $\ \mathbb I^n,\ $ has the fixed point property. Then, for any continuous mapping $\ f:X\times\mathbb I^n\to X\ $ there exists $\ x\in X\ $ such that
$$ \dim\{t\in\mathbb I^n: f(x,t)=x\}\ \ge n. $$

This theorem was followed by another theorem about $\ f:X\times\mathbb I^n\to X\ $; this time the theorem is about a narrower class of spaces, namely for Hausdorff compact ANR spaces, it applies the Lefschetz number in its assumption, and the proof takes advantage of a generalized Hurewicz dimension theory theorem by B.Pasynkov, but the thorem still manages to arrive at the same conclusion as in the theorem above.
And there are theorems about universal maps explicitly that generalize pairs of theorems, one from the dimension theory, and one from the f.p.p. theory. The theory of universal maps and morphisms is neglected for no rational reason.
A: I remember the first time I heard about quadratic reciprocity, I thought it was very "strange". If $p$ and $q$ are two odd primes, why is the question of whether or not $p$ is a quadratic residue mod $q$ related to the different question of whether or not $q$ is a quadratic residue mod $p$?
I remember reading some proofs and yet, not feeling that the proofs "explained" what was really going on under the hood (well, of course, they were proofs, and I did not have doubts about them, but they did not seem to explain the full story).
Then I was excited to learn about Artin's work and of course the Langlands program.
That being said, I remember watching an interview with Langlands where he remarked something along the lines that when he first learned about quadratic reciprocity, he just thought it was some kind of curiosity or something, but he didn't attach much importance to it (sorry for paraphrasing, if someone knows the exact quote, I can include it here, instead of my paraphrase!).
A: 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} \, d\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.
A: 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)
A: We all know how the limiting Fibonacci ratio $(1+\sqrt5)/2$ is tied in with the geometry and construction of the regular pentagon. But what about the connection between the neusis construction of the regular hendecagon and the _tribonacci _ constant, the latter defined as the limiting ratio of the sequence $1,1,1,3,5,9,17,...$ (each term after the third is the sum of the previous three)?
We begin with the existence of the referenced neusis construction. The neusis constructibility of regular $n$-gons is guaranteed by the existence of a (or many) neusis trisection of an angle, if the Euler totient function of $n$ has no prime factors greater than $3$. Now, the number $11$ is the smallest natural number failing to meet this criterion, as its Euler totient is a multiple of $5$, and no neusis angle quintisection is yet known. However, Benjamin and Snyder[1] demonstrate a neusis construction for the regular hendecagon. In the method they present, the solubility of a quintic equation to determine parameters that can be put into a neusis construction depends on a resolving seventh-degree equation, which seems to be a step backwards. However, for the specific quintic equation for $2\cos(2k\pi/11)$ with $k\not\equiv0\bmod 11$:
$x^5+x^4-4x^3-3x^2+3x+1=0$
"a miracle occurs"; the seventh-degree equation is reducible and the required  equation is reduced to a cubic factor, rendered in the work as
$u^3+2u^2+2u+2=0.$
This root of a cubic equation is neusis constructible, and the parameters required to retrieve the regular hendecagon are derived in terms of this root.
The above cubic equation is not the tribonacci ratio equation, which is instead
$t^3-t^2-t-1=0;$
but it turns out that $t=-1/(1+u)$ and the distance from the pole of the neusis to the catch line for one of the marks may be rendered as $1/t(=-(1+u))$. So where might the tribonacci ratio have come from in this solution of a seemingly unrelated quintic equation?
If you are familiar with quadratic Gauss sums, you have probably seen the result
$\sin(2\pi/11)-\sin(4\pi/11)+\sin(6\pi/11)+\sin(8\pi/11)+\sin(10\pi/11)=\sqrt{11}/2,$
which is just the imaginary part of the quadratic sum with eleventh roots of unity. In a rather common homework problem in this area, the above sum is multiplied through by $\cos(8\pi/11)$, various manipulations are made with the trigonometric sum-product relations, then the multiplier $\cos(8\pi/11)$ is divided back out to get the sine-tangent relation
$4\sin(2\pi/11)-\tan(8\pi/11)=\sqrt{11}$
or something similar. We could have just as well used the multiplier $\cos(2k\pi/11)$ for any $k\in\{1,2,3,4,5\}$ to get a group of five of these relations which may be symmetrically expressed as
$4\sin(6k\pi/11)-\tan(2k\pi/11)=(k|11)\sqrt{11}$
where $(k|11)$ is the Legendre symbol of residue $k\bmod11$. This applies for all integers $k$, including $0$ (which trivially gives $0=0$).
Suppose we square the above symmetric relation and render $x=2\cos(2k\pi/11)$, the quintic roots for which we (and Benjamin and Snyder) intended to solve. We can use multiple-angle trigonometric formulae and the Pythagorean relation $\sin^2\theta+\cos^2\theta=1$ to obtain a rational-function equation for $x$, from which fractions may be cleared to obtain a polynomial equation. The net result, however, isn't the quintic equation we expect but an octic one:
$x^8-6x^6-x^5+9x^4+5x^3-x^2-4x-1=0.$
What happened to the quintic equation for the trigonometric roots? It's actually there, as a factor of the octic. And guess what the complementary cubic factor is:
$(x^5+x^4-4x^3-3x^2+3x+1)\color{blue}{(x^3-x^2-x-1)}=0.$
So the cubic factor found by Benjamin and Snyder, which enables the neusis construction of the regular hendecagon, is not entirely an accident. In the form of the tribonacci constant, and therefore one of the distance parameters in Benjamin and Snyder's construction, it is adjoined to the quintic roots through the Gauss sum!
Reference

*

*E. BENJAMIN and C. SNYDER (2014). On the construction of the regular hendecagon by marked ruler and compass . Mathematical Proceedings of the Cambridge Philosophical Society, 156, pp 409-424 doi:10.1017/S0305004113000753

A: The Gauss-Bonnet theorem. It only uses concepts from classical differential geometry of 2D surfaces and can be explained to an undergraduate, but it connects geometric notions of curvature to a purely topological concept (the Euler characteristic), thereby relating two very different levels of mathematical structure. I still find the result pretty amazing.
A: 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
A: 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.
