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

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!!

Answer 2

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?

Answer 3

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.]

Answer 4

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.

Answer 5

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.

(Source: On teaching mathematics, V. Arnol’d, 1997, trans. A. Goryunov.)

Answer 6

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.

Answer 7

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.

Answer 8

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.

Answer 9

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?

Answer 10

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.

Answer 11

Ehud Hrushovski's proof, using model theory, of the geometric Mordell-Lang conjecture in algebraic geometry.

Answer 12

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 $f_k$: $$N=(n!)\cdot V.$$

Answer 13

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.

Answer 14

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. :)

Answer 15

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.

Answer 16

The pair correlation function between Riemann zeta function zeros is the same as the pair correlation function between eigenvalues of random Hermitian matrices.

Answer 17

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.

Answer 18

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.

Answer 19

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.)

Answer 20

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.

Answer 21

Complex multiplication of elliptic curves and the explicit construction of the maximal abelian extension of a quadratic imaginary number field.

Answer 22

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 $\mathbb{Z}$.

Answer 23

The Jones polynomial of knot theory and Feynman path integrals.

Answer 24

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).

Answer 25

Special values of the Riemann zeta function and class numbers of cyclotomic fields.

Answer 26

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.

Answer 27

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).

Answer 28

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.

Answer 29

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?

Answer 30

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.