Your favorite surprising connections in Mathematics - MathOverflow

Your favorite surprising connections in Mathematics

2010-02-08T00:10:18Z

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 by Feb7 for Your favorite surprising connections in Mathematics

2010-02-08T00:12:19Z

The surprising applications of algebraic geometry to number theory, for instance evidenced in the work of Deligne in proving the Ramanujan conjectures.

Answer by Feb7 for Your favorite surprising connections in Mathematics

2010-02-08T00:15:24Z

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.

Answer by John Stillwell for Your favorite surprising connections in Mathematics

2010-02-08T00:38:05Z

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 by Anton Petrunin for Your favorite surprising connections in Mathematics

2010-02-08T00:59:26Z

Connection between the typical number of 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 $f_k$: $$N=\tfrac1{n!}V.$$

Answer by Joseph Malkevitch for Your favorite surprising connections in Mathematics

2010-02-08T01:03:24Z

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 by Dan Piponi for Your favorite surprising connections in Mathematics

2010-02-08T01:34:27Z

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 by Zavosh for Your favorite surprising connections in Mathematics

2010-02-08T02:25:06Z

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

Answer by Zavosh for Your favorite surprising connections in Mathematics

2010-02-08T03:06:39Z

The Jones polynomial of knot theory and Feynman path integrals.

Answer by Hunter Brooks for Your favorite surprising connections in Mathematics

2010-02-08T03:40:28Z

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

Answer by Bill Johnson for Your favorite surprising connections in Mathematics

2010-02-08T03:51:31Z

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.

Answer by Alex R. for Your favorite surprising connections in Mathematics

2010-02-08T03:56:25Z

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

Answer by Gjergji Zaimi for Your favorite surprising connections in Mathematics

2010-02-08T04:07:24Z

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 by Qiaochu Yuan for Your favorite surprising connections in Mathematics

2010-02-08T04:10:13Z

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 by Mariano Suárez-Alvarez for Your favorite surprising connections in Mathematics

2010-02-08T04:49:32Z

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 by Qiaochu Yuan for Your favorite surprising connections in Mathematics

2010-02-08T05:13:54Z

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.

Answer by MBN for Your favorite surprising connections in Mathematics

2010-02-08T05:21:47Z

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 spacial 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.

Answer by Ryan Budney for Your favorite surprising connections in Mathematics

2010-02-08T05:50:30Z

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 by Steven Sam for Your favorite surprising connections in Mathematics

2010-02-08T05:54:46Z

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 by Daniel Moskovich for Your favorite surprising connections in Mathematics

2010-02-08T07:07:10Z

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 See also http://www.math.nus.edu.sg/~matwujie/cohen.wu.GT.revised.29.august.2007.pdf

Answer by Adrian Petrescu for Your favorite surprising connections in Mathematics

2010-02-08T07:09:07Z

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 by unknown (google) for Your favorite surprising connections in Mathematics

2010-02-08T14:06:53Z

Goppa’s construction of error-correcting codes from curves, leading to the Tsfasman-Vladut-Zink bound (the first improvement over the Gilbert-Varshamov 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.

Answer by Graham Leuschke for Your favorite surprising connections in Mathematics

2010-02-08T14:37:04Z

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 [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 by Chris Schommer-Pries for Your favorite surprising connections in Mathematics

2010-02-08T15:24:45Z

Quillen's result 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 by Zavosh for Your favorite surprising connections in Mathematics

2010-02-08T16:36:31Z

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

Answer by kakaz for Your favorite surprising connections in Mathematics

2010-02-09T15:03:30Z

Fact that something such well known as group of rotations S(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 by Victor Miller for Your favorite surprising connections in Mathematics

2010-02-11T00:33:39Z

Another surprising connection: The Ax-Kochen theorem that for each positive integer $d$ there is a finite set $Y_d$ of prime numbers, such that if $p$ is any prime not in $Y_d$ then every homogeneous polynomial of degree $d$ over the $p$-adic numbers in at least $d^2+1$ variables has a nontrivial zero.

This was proved using model theory.

Answer by Csar Lozano Huerta for Your favorite surprising connections in Mathematics

2010-02-16T22:35:57Z

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 by akopyan for Your favorite surprising connections in Mathematics

2010-02-16T23:37:41Z

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").

Answer by algori for Your favorite surprising connections in Mathematics

2010-02-17T00:06:54Z

Here is a copypaste of something I've already mentioned in this question: http://mathoverflow.net/questions/12804/large-cardinal-axioms-and-grothendieck-universes

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, 4, 1, 33-79.

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 by Marko Amnell for Your favorite surprising connections in Mathematics

2010-02-17T00:26:42Z

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.

Answer by akopyan for Your favorite surprising connections in Mathematics

2010-02-17T00:59:35Z

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.

Answer by Hailong Dao for Your favorite surprising connections in Mathematics

2010-02-17T01:33:10Z

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.

Answer by Andrew Mullhaupt for Your favorite surprising connections in Mathematics

2010-02-18T21:56:09Z

That the Jeffreys' prior in the pole-zero parameterization of a transfer function is the hyperbolic transfinite diameter of the support of the poles and zeros.

It's my favorite because I just discovered it last month. I like laughing at my own jokes.

Answer by Harrison Brown for Your favorite surprising connections in Mathematics

2010-02-19T09:23:03Z

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.)

Answer by Chad Groft for Your favorite surprising connections in Mathematics

2010-02-22T02:17:27Z

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 by Jeremybub for Your favorite surprising connections in Mathematics

2010-03-05T03:47:42Z

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 by SandeepJ for Your favorite surprising connections in Mathematics

2010-04-14T17:07:40Z

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 by Jamie Weigandt for Your favorite surprising connections in Mathematics

2010-05-11T22:55:37Z

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.

Answer by Timothy Chow for Your favorite surprising connections in Mathematics

2010-05-27T03:14:09Z

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 by To be cont'd for Your favorite surprising connections in Mathematics

2010-05-27T07:31:06Z

The shortest path between two truths in the real domain passes through the complex domain. Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe.Jacques Hadamard

Answer by David Corwin for Your favorite surprising connections in Mathematics

2010-07-15T15:59:29Z

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.

This book looks deeply into the relationships between Galois groups and fundamental groups and eventually develops a theory which covers both.

This book explores the beautiful relation between algebraic curves and algebraic number theory.

Answer by Frank for Your favorite surprising connections in Mathematics

2011-03-13T16:39:31Z

I'll take a risk and provide a slightly off-topic connection (feel free to downvote).

How come mathematics can describe physical phenomenons so accurately.

I faced this in the article by Eugene Wigner "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".

Answer by Elizabeth S. Q. Goodman for Your favorite surprising connections in Mathematics

2011-03-13T17:24:44Z

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 by Michal R. Przybylek for Your favorite surprising connections in Mathematics

2011-03-13T17:28:14Z

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.

Answer by YBL for Your favorite surprising connections in Mathematics

2011-03-13T19:11:22Z

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 by Abel Stolz for Your favorite surprising connections in Mathematics

2011-03-14T09:21:39Z

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 by Kostya for Your favorite surprising connections in Mathematics

2011-03-14T11:38:42Z

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).

Answer by Matheus for Your favorite surprising connections in Mathematics

2011-03-14T15:37:03Z

I believe the way R. Schoen solved Yamabe problem "http://en.wikipedia.org/wiki/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 "http://www.ams.org/journals/bull/1987-17-01/S0273-0979-1987-15514-5/" for a nice account on this surprising connection between Differential Geometry and General Relativity.

Answer by DJC for Your favorite surprising connections in Mathematics

2011-03-14T15:56:05Z

I would say Gy. Elekes' beautiful and simple argument utilizing incidence theory to prove a sum-product result.

Answer by Stefan Behrens for Your favorite surprising connections in Mathematics

2011-07-25T11:51:05Z

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!

Answer by mikitov for Your favorite surprising connections in Mathematics

2011-07-25T12:33:16Z

The inverse calculus of a slope is the calculation of an area.

Barrow's Lemma: http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

Answer by Shripad for Your favorite surprising connections in Mathematics

2011-07-25T15:57:17Z

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 by Shripad for Your favorite surprising connections in Mathematics

2011-07-26T13:48:19Z

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!

Answer by Thierry Zell for Your favorite surprising connections in Mathematics

2011-07-26T15:53:49Z

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 $$ f=P_0+\frac{1}{P_1+\frac{1}{P_2+\ddots}}$$ 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 by kostja for Your favorite surprising connections in Mathematics

2011-08-29T18:27:17Z

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. $\delta = \text{(some sign)} *d*$ and $*$ 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 by Stephan Wehner for Your favorite surprising connections in Mathematics

2012-01-06T22:20:53Z

Using the Chinese remainder theorem for proving Gödel's incompleteness theorems.