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There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the connection between the two is unexpected. The first example that comes to my mind is the proof by Furstenberg and Katznelson of Szemeredi's theorem on the existence of arbitrarily long arithmetic progressions in a set of integers which has positive upper Banach density, but using ergodic theory. Of course in the years since then, this idea has now become enshrined and may no longer be viewed as surprising, but it certainly was when it was first devised.

Another unexpected connection was when Kolmogorov used Shannon's notion of probabilistic entropy as an important invariant in dynamical systems.

So, what other surprising connections are there out there?

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I think it could be interpreted as restricting the breadth of the list. I would suggest removing the tag to avoid this. –  François G. Dorais Feb 8 '10 at 0:30
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The surprising applications of algebraic geometry to number theory, for instance evidenced in the work of Deligne in proving the Ramanujan conjectures.

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Or, Riemann's use of complex variables to prove the Prime number theorem. –  Feb7 Feb 8 '10 at 0:16
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Riemann did not prove the prime number theorem. –  S. Carnahan Feb 8 '10 at 4:50
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Although he came pretty close! –  Emerton Feb 8 '10 at 5:05
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Deligne's work was about counting solutions to equations over finite fields. Ramanujan's conjecture was about bounding the absolute values of the Fourier coefficients of a certain complex analytically defined function. How is the connection possibly tautological? –  Emerton Feb 8 '10 at 5:14
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Indeed, the connection between the Weil conjectures (and in particular the Riemann hypothesis, the proof of which is the work of Deligne being referred to) and Ramanujan's conjecture was only made some time after both conjectures were formulated (by Serre, I believe). –  Emerton Feb 8 '10 at 5:17
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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.

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No, I don't think that everything is really surprising. There are a lot of theorems that have been based on hard work, but within the existing circle of ideas surrounding that result. What I'm after is when disparate parts of mathematics are brought together in unexpected ways. –  Victor Miller Feb 8 '10 at 0:20
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Well personally it was surprising for me. I knew something about point set topology from one book, and from the next book I knew about groups, rings, linear algebra and so on. Then I go and sit in algebraic topology course because it was mandatory for some reason, and lo and behold! –  Feb7 Feb 8 '10 at 0:35
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Hi Ryan, would you consider it obvious that the obstruction to promoting a homotopy equivalence to a simple homotopy equivalence should live in a group? And if so, could you have guessed which group? I think there are many places in topology where algebra is surprisingly effective. And for at least half a century mathematicians studied topological spaces, and manifolds in particular, before beginning to apply algebra to these questions. In 1942 the field was still referred to, at least by some, as "combinatorial topology" rather than "algebraic topology". –  Tom Church Feb 8 '10 at 3:37
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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.

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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 $f_k$: $$N=\tfrac1{n!}V.$$

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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!!

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This is the observation that should have occurred to everyone first! (It didn't to me either.) It is so familiar we forget how amazing it is. –  SixWingedSeraph Feb 8 '10 at 3:02
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I totally agree. To put another exclamation point on this idea, Algebra and Geometry co-existed for around a thousand years before this observation was made mainstream by Descartes and Fermat. I wonder what other yet-unseen mathematics we'll weave, in a thousand years, into middle school education. –  Hiro Lee Tanaka Sep 29 '13 at 22:34
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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?

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

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If you evaluate (appropriately normalized) elliptic functions at points lying in quadratic imaginary fields, the values you obtain are algebraic numbers, lying in abelian extensions of said quadratic imaginary fields; and all such extensions can be obtained in this way. (Compare with: $e^{2\pi i z}$ evaluated at rational numbers gives algebraic numbers, which generate abelian extension of ${\mathbb Q}$, and all abelian extension of ${\mathbb Q}$ are obtained in this way. –  Emerton Feb 8 '10 at 5:03
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Another way of saying it is that (with some slight fiddling) coordinates of points of finite order on an elliptic curve with complex multiplication give abelian extensions of the appropriate quadratic imaginary field. This was Kronecker's Jugendtraum (dream of his youth). Only in few cases is this explicit description of abelian extensions possible. Why do I think it is surprising? Compare what Emerton said above, $\exp(2 \pi iz)$ generating abelian extensions of $\mathbb{Q}$. Tell this to someone and ask them to guess how you'd generalise! It really is surprising that it is possible at all. –  Sam Derbyshire Feb 8 '10 at 8:37
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Elliptic curves (over $\mathbb{C}$) have their origins in studying elliptic integrals. As analytic objects they are $\mathbb{C}$ modulo a lattice. It's not immediately obvious to me that this is an algebraic object, and that the Weierstrass $\mathcal{P}$ function, which is an infinite sum, should compute anything number theoretic. So perhaps the connection is between analysis and algebra/arithmetic from this point of view. –  Zavosh Feb 8 '10 at 21:16
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The Jones polynomial of knot theory and Feynman path integrals.

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I'm not sure why this is surprising. It was originally defined via subfactors, but the path integral formalism followed very closely behind. Also, I'm not sure that Feynman path integrals count as mathematics... –  Daniel Moskovich Feb 8 '10 at 5:07
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Well, you are right. I think "surprising" is a subjective property, perhaps an experience of facing one's own ignorance. I don't know much about subfactors and first saw this polynomial in the context of knot invariants, divorced from its origins. For this reason, the physics connection seemed like a big surprise. It appears that you are an expert and so it is not surprising to me that you are not surprised. –  Zavosh Feb 8 '10 at 15:50
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I always found it ironic that knot theory began with Lord Kelvins model of atoms as knots in ether (loosely speaking). After a 360 degree rotation (or make that 720 degree :-) we're at string theory now. –  Hauke Reddmann Jul 25 '11 at 12:00
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Special values of the Riemann zeta function and class numbers of cyclotomic fields.

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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.

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

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A caveat, this is not a theorem. Montgomery showed this for test functions whose Fourier transform had restricted support (in fact, support [-2,2] iirc.) Montgomery conjectured the same holds for more general test functions. Odlyzko's computations provided spectacular numerical evidence. And Katz-Sarnak proved an analogous statement for function fields. –  Stopple Mar 13 '11 at 20:02
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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.

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I got them from a paper of Odlyzko and Rains: dtc.umn.edu/~odlyzko/doc/incr.subseq.pdf (actually listening to Odlyzko speak on it) –  Victor Miller Feb 8 '10 at 5:19
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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).

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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.

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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.
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See also: "My Lunch with Arnol'd" which helps put this surprise in perspective by showing it through the eyes of an amateur mathematician... www.gomboc.eu/99.pdf –  David White May 19 '11 at 20:08
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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.

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The connection that I was talking about is the following. The Arthur-Selberg trace formula is an identity of distributions for a pair of groups(with some conditions). When the groups are R and Z, then the trace formula reduces to Poisson summation. When the groups are finite, and with the right choice of a test function, the trace formula reduces to Frobenius reciprocity. –  MBN Feb 8 '10 at 14:28
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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. :)

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@Ryan: is the fact that geometry is less useful in high dimensions an empirical observation, or is there more mathematical content to this? –  Jim Conant Mar 13 '11 at 17:44
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@Jim: From Hillman's "Four-manifolds, Geometries and Knots", up to homeomorphism there are only 11 geometric 4-dimensional manifolds with finite fundamental group. In dimension 4 a finite-volume hyperbolic manifold's volume is a function of its Euler characteristic. I see those as having a fair bit of content. Sorry for being slow to reply. –  Ryan Budney Sep 1 '11 at 6:40
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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.

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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

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I'd like to find a more geometric proof of their result. There's a lot of geometric constructions that lead me to suspect such a result but I haven't found anything quite right. The main idea is to consider the closure of a Brunnian braid then look at things like the Koschorke invariants. mathoverflow.net/questions/234/… –  Ryan Budney Feb 8 '10 at 7:17
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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.

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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.

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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.]

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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.

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

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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!

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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.

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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. alt text

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.

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Wow,that IS pretty whack,Csar.This example alone is a testament to the power of modern topology and geometry and the incredible connections it has uncovered. –  Andrew L Jul 15 '10 at 19:55
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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").

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Here is a copypaste of something I've already mentioned in this question: 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.

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Slides of a recent talk by Dehornoy on the history of this braid group problem may be found at math.unicaen.fr/~dehornoy/Talks/DyfShort.pdf –  John Stillwell Feb 19 '10 at 1:45
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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.

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