Your favorite surprising connections in mathematics

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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

Answer 4

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)

Answer 5

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.

Answer 6

The existence of Nash equilibria is an example that connects elementary aspects of game theory, probability, geometry, and algebraic topology.

Answer 7

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

Answer 8

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.

Answer 9

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.

Answer 10

"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.

Answer 11

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!

Answer 12

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 $Y\times\mathbb{R}$.

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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!

Answer 13

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.

Answer 14

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.

Answer 15

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.

Answer 16

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.

Answer 17

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

Answer 18

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).

Answer 19

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.

Answer 20

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.

Answer 21

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 $1,1,1,3,5,9,17,\ldots$ (each term after the third is the sum of the previous three)? It turns out that the tribonacci constant is connected with the cosines of the hendecagonal angles via the Gauss sum.

Neusis construction of the regular hendecagon

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, $11$ is the smallest natural number failing to meet this criterion, as its Euler totient is a multiple of $5$. 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?

The 11th-order Gauss sum

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

$$4\sin(6k\pi/11)-\tan(2k\pi/11)=(k\mid11)\sqrt{11}$$

where $(k\mid11)$ is the Legendre symbol of residue $k\bmod11$. This applies for all integers $k$, including $0$ (which trivially gives $0=0$).

The tribonacci constant emerges from the Gauss sum

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

1. 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

Answer 22

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.

Answer 23

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.

Answer 24

Complex analysis and Brownian motion

Here there have a been a variety of results eg. conformal invariance of Brownian motion, various proofs of known complex analysis results. It has been used for intuition purposes eg. see answers by B.Thurston Does Riemann map depend continuously on the domain?. The connections are also still being explored in the study of SLE curves (defined in terms of a 1d-Brownian motion) in various statistical models.

A priori this bridge is surprising even if one just starts with the original construction using the heat equation.

Random matrices and Partial differential equations

Here the main connections come through integrability theory. The main equation that stands out here is the KPZ equation and the KPZ universality class. The height function has beautiful formulas in terms of Airy processes and Fredholm determinant quantities that show up in random matrix theory. In particular, when dealing with boundary conditions, we find different types of random matrix families. So the connections run deep.

eg. see survey by PL Ferrari "From interacting particle systems to random matrices".

This bridge is surprising given the fact that KPZ originates from the study of interfaces and the Burgers equation which in turn was studied as a model for turbulence.

Renormalization, Regularity structures and SPDEs

This is more of a recent development. But I personally think it is quite surprising that so many of the tools/concepts developed in the specific setting of quantum fields such as Feynman-diagrams, BPHZ renormalization turn out to have analogues that apply to large families of stochastic differential equations. And that the generalization of rough paths was the natural framework to formalize those notions.

Answer 25

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)

Answer 26

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!).

Answer 27

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 $\ X$).

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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.

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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

$$ \dim\{t\in\mathbb I^n: f(x,t)=x\}\ \ge n. $$

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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.

Answer 28

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.

Answer 29

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