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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

71 Answers 71

I'd like to share the very elementary fact (so elementary that I found surprising only after I taught Calculus course) that all the elementary functions are analytic in the global way. Of course, that's no surprise for polynomials. But I found no intuition why the trigonometric functions and the exponential functions, in the way they are originally considered by human, turn out to be equal to their Taylor expansions everywhere. Consider again the fact that Taylor expansion uses only the information on an infinitesimal neighborhood at a point, a function which is not originally defined by power series should be of extremely little chance to equal its Taylor expansion. I don't know if I'm right, but I finally told my students this is really a miracle.

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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).

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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.

This explores the analogy between prime numbers and knots.

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That's because I changed the link. –  David Corwin Apr 15 at 4:34

It seems that no one gave this one yet, although it probably hides behind many of previous answers.

The fact that in $\mathbb{C}$, product by a fixed complex number corresponds to a similarity is an incredible and far-reaching connection between algebra and geometry.

Among other things, it ties holomorphic functions with conformal maps of surfaces, so that for example one can identify a Riemann surface with a surface having a Riemannian metric of curvature $-1$, $0$ or $1$; more generally it allows for the use of complex analysis to study a number of problems in the geometry of surfaces.

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The analogy, still not understood to the full I think, between prime numbers and knots.

See Arithmetic topology in Wikipedia.

A most condensed picture is given by the Kapranov-Reznikov-Mazur dictionary

enter image description here

This is actually closely related to several answers here, and in fact initially I mentioned it in a comment to one of the answers but then still decided to make a separate entry.

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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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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)

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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.

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Weak homotopy type. –  Todd Trimble Mar 26 at 14:16

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

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Stumbled on the following couple of days ago, when searching a good picture for a general 3-step filtration in an abelian category (in fact, there are similar structures in triangulated categories which I am ultimately up to):enter image description here

After feasting my eyes on it for a while I suddenly realized that what I am actually staring at is the Desargues configuration (in the form of five generic planes in 3-space)!

Not sure if this has any significance or whether one can do anything with it, but I certainly find it amusing.

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

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Minor correction: $N=n! V$. Otherwise it's an excellent example. –  MKO Jan 13 at 11:42

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\ universal\,\ f:X\rightarrow I^n\ $ (for every completely regular space $\ X$).

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Grothendieck's dessins d'enfants: the Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on certain graphs (with a decoration) on 2-dimensional topological surfaces.

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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.

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I don't see why this is surprising. –  Todd Trimble Jan 13 at 14:46
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I don't have a real opinion on that since I don't know what you are alluding to there. Offhand, it sounded interesting. –  Todd Trimble Jan 13 at 20:18

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.

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That mechanical vibrations (mass-spring-dashpot systems) satisfy the same differential equations as electrical systems (inductor-resistor-capacitor circuits).

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Stone duality usually refers to the equivalence between the category of Boolean algebras and the category of compact totally disconnected spaces. This duality intertwines the theory of Boolean algebras with general topology so much that Boolean algebras cannot be studied in depth without mentioning general topology and compact totally disconnected spaces cannot be studied in great detail without mentioning their relation with Boolean algebras. For example, the free Boolean algebras and free $\sigma$-complete Boolean algebras are normally represented not in terms of generators and relations, but as clopen sets (Baire sets) on the cantor cube $2^{I}$ for some set $I$.

Stone duality was originally a very surprising result, and it is probably a bit surprising to people seeing this result for the first time as well. Around 1937 when Marshall Stone formulated this duality it was difficult to imagine nice topological spaces that arose from algebraic structures rather than geometric or analytic structures.

Besides Stone duality, there are many dualities (equivalences of categories) similar in nature to Stone duality that relate different structures to each other and hence relate different areas of mathematics to each other (I have developed some of these dualities myself). For instance, one can relate topologies satisfying higher separation axioms with topologies that are not even $T_{1}$. One can also relate structures such as proximity spaces and uniform spaces with algebras of sets and Boolean algebras. There are also many dualities relating different in order theory to each other.

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Application of ``thermodynamic formalism'' to questions of Analysis. Thermodynamic formalism have its origin in equilibrium statistical mechanics. First unexpected thing was its application to the theory of smooth dynamical systems, see beautiful paper of Ruelle, Is our mathematics natural? in BAMS. Later unexpected applications were discovered to problems of analysis which have nothing to do with dynamical systems, statistical mechanics or mathematical physics. One example is Astala's theorem on the area distortion under quasiconformal mappings. There is a very simple, self-contained proof of this theorem in MR1283548, using no dynamical considerations. But it is hard to imagine how could this proof be invented without dynamical and "thermodynamical" considerations.

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"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.

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There are several surprises regarding convex polytopes:

A) There are combinatorial types of polytopes that cannot be realized with rational coordinates (first discovered by Perles). This is not the case in three dimension but by now there are examples in every dimension greater equal 4. This adds to several examples on the wild combinatorial nature of convex polytopes in dim at least 4.

B) The applications of commutative algebra to the study of face-numbers of polytopes - Stanley proofs of the upper bound theorem using the Cohen-Macaulay argument and many subsequent results. Also surprising is the application of algebraic geometry: toric varieties, Hard Lefschetz theorem, intersection homology etc.

C) It is a special surprise that some proofs regarding the face number of polytopes applies only to polytopes with rational coordinates.

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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.)

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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?

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...or between exponential and trigonometric functions generally. –  Michael Hardy Apr 23 '11 at 1:05
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In my opinion, this becomes a lot less mysterious as soon as you think of the exponential and trigonometric functions as eigenfunctions of the differentiation operator (respectively, its square), which is really the reason they're both so important. The basic properties and interrelationships of these functions - including the above identity - are natural consequences of this formulation. –  Robin Saunders Jul 29 '11 at 3:06
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Well yes, but that's the way it is with all these surprising results, isn't it? They all indicate a connection that no one had suspected, but is undoubtedly important. Once that connection is chewed over enough and becomes something you learn as a matter of course, then the original surprising result becomes "understandable", or sometimes even "trivial". But it certainly wasn't originally, and often isn't even to people first encountering these things today. –  Carl Offner Sep 30 '13 at 0:54

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.

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

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@TimothyChow -- I didn't down-vote. Nevertheless I feel that the Chinese here is quite arbitrary in the Gödel's context, and relatively very trivial. One could use other schemes as well with the same success. –  Włodzimierz Holsztyński Jan 13 at 2:18
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@WłodzimierzHolsztyński : What you say is true, but when exponentiation is not directly available, the options for a simple encoding are limited. –  Timothy Chow Jan 13 at 18:09

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.

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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.

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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!

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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.

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

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

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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!

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protected by François G. Dorais Sep 30 '13 at 0:52

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