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It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)

I'd love to learn about further basic or central examples and I think such examples serve as good invitations to various areas. (Which is why a bounty was offered.)

Related MO questions: What-are-your-favorite-instructional-counterexamples, Cannonical examples of algebraic structures, Counterexamples-in-algebra, individual-mathematical-objects-whose-study-amounts-to-a-subdiscipline, most-intricate-and-most-beautiful-structures-in-mathematics, counterexamples-in-algebraic-topology, algebraic-geometry-examples, what-could-be-some-potentially-useful-mathematical-databases, what-is-your-favorite-strange-function; Examples of eventual counterexamples ;

To make this question and the various examples a more useful source there is a designated answer to point out connections between the various examples we collected.

In order to make it a more useful source, I list all the answers in categories, and added (for most) a date and (for 2/5) a link to the answer which often offers more details. (~year means approximate year, *year means a year when an older example becomes central in view of some discovery, year? means that I am not sure if this is the correct year and ? means that I do not know the date. Please edit and correct.) Of course, if you see some important example missing, add it!

Logic and foundations: $\aleph_\omega$ (~1890), Russell's paradox (1901), Halting problem (1936), Goedel constructible universe L (1938), McKinsey formula in modal logic (~1941), 3SAT (*1970), The theory of Algebraically closed fields (ACF) (?),

Physics: Brachistochrone problem (1696), Ising model (1925), The harmonic oscillator,(?) Dirac's delta function (1927), Heisenberg model of 1-D chain of spin 1/2 atoms, (~1928), Feynman path integral (1948),

Real and Complex Analysis: Harmonic series (14th Cen.) {and Riemann zeta function (1859)}, the Gamma function (1720), li(x), The elliptic integral that launched Riemann surfaces (*1854?), Chebyshev polynomials (?1854) punctured open set in C^n (Hartog's theorem *1906 ?)

Partial differential equations: Laplace equation (1773), the heat equation, wave equation, Navier-Stokes equation (1822),KdV equations (1877),

Functional analysis: Unilateral shift, The spaces $\ell_p$, $L_p$ and $C(k)$, Tsirelson spaces (1974), Cuntz algebra,

Algebra: Polynomials (ancient?), Z (ancient?) and Z/6Z (Middle Ages?), symmetric and alternating groups (*1832), Gaussian integers ($Z[\sqrt -1]$) (1832), $Z[\sqrt(-5)]$,$su_3$ ($su_2)$, full matrix ring over a ring, $\operatorname{SL}_2(\mathbb{Z})$ and SU(2), quaternions (1843), p-adic numbers (1897), Young tableaux (1900) and Schur polynomials, cyclotomic fields, Hopf algebras (1941) Fischer-Griess monster (1973), Heisenberg group, ADE-classification (and Dynkin diagrams), Prufer p-groups,

Number Theory: conics and pythagorean triples (ancient), Fermat equation (1637), Riemann zeta function (1859) elliptic curves, transcendental numbers, Fermat hypersurfaces,

Probability: Normal distribution (1733), Brownian motion (1827), The percolation model (1957), The Gaussian Orthogonal Ensemble, the Gaussian Unitary Ensemble, and the Gaussian Symplectic Ensemble, SLE (1999),

Dynamics: Logistic map (1845?), Smale's horseshoe map(1960). Mandelbrot set (1978/80) (Julia set), cat map, (Anosov diffeomorphism)

Geometry: Platonic solids (ancient), the Euclidean ball (ancient), The configuration of 27 lines on a cubic surface, The configurations of Desargues and Pappus, construction of regular heptadecagon (*1796), Hyperbolic geometry (1830), Reuleaux triangle (19th century), Fano plane (early 20th century ??), cyclic polytopes (1902), Delaunay triangulation (1934) Leech lattice (1965), Penrose tiling (1974), noncommutative torus, cone of positive semidefinite matrices, the associahedron (1961)

Topology: Spheres, Figure-eight knot (ancient), trefoil knot (ancient?) (Borromean rings (ancient?)), the torus (ancient?), Mobius strip (1858), Cantor set (1883), Projective spaces (complex, real, quanterionic..), Poincare dodecahedral sphere (1904), Homotopy group of spheres, Alexander polynomial (1923), Hopf fibration (1931), The standard embedding of the torus in R^3 (*1934 in Morse theory), pseudo-arcs (1948), Discrete metric spaces, Sorgenfrey line, Complex projective space, the cotangent bundle (?), The Grassmannian variety,homotopy group of spheres (*1951), Milnor exotic spheres (1965)

Graph theory: The seven bridges of Koenigsberg (1735), Petersen Graph (1886), two edge-colorings of K_6 (Ramsey's theorem 1930), K_33 and K_5 (Kuratowski's theorem 1930), Tutte graph (1946), Margulis's expanders (1973) and Ramanujan graphs (1986),

Combinatorics: tic-tac-toe (ancient Egypt(?)) (The game of nim (ancient China(?))), Pascal's triangle (China and Europe 17th), Catalan numbers (18th century), (Fibonacci sequence (12th century; probably ancient), Kirkman's schoolgirl problem (1850), surreal numbers (1969), alternating sign matrices (1982)

Algorithms and Computer Science: Newton Raphson method (17th century), Turing machine (1937), RSA (1977), universal quantum computer (1985)

Social Science: Prisoner's dilemma (1950) (and also the chicken game, chain store game, and centipede game), the model of exchange economy, second price auction (1961)

Statistics: the Lady Tasting Tea (?1920), Agricultural Field Experiments (Randomized Block Design, Analysis of Variance) (?1920), Neyman-Pearson lemma (?1930), Decision Theory (?1940), the Likelihood Function (?1920), Bootstrapping (?1975)

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I think that this should be community wiki. – Loop Space Nov 11 '09 at 7:55
@Jose: Hard to say exactly. My instinct is that the kind of answers that this question will garner are those that didn't involve much actual thought, and the votes up or down will be more an assessment of whether the voter liked the example rather than whether the voter liked the answer (which, ideally, should contain an explanation of why that example shaped the discipline); both of these indicate that the answerers should not gain reputation for their answers, hence community wiki. – Loop Space Nov 11 '09 at 9:50
I've hit this with the wiki hammer. – Scott Morrison Nov 11 '09 at 19:34
I can't imagine a counterexample to the following rule: Any question whose purpose is to produce a sorted list of resources (i.e. the question includes, or should include, "one per post please") should be community wiki. – Anton Geraschenko Nov 12 '09 at 8:03
Why does this question have a bounty anyway? – Kevin H. Lin Nov 21 '09 at 17:33

136 Answers 136

The Harmonic series 1+1/2+1/3+1/4+1/5+... and for that matter the Riemann zeta function.

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$$2^{\aleph_0} = \aleph_1$$

The Continuum Hypothesis is an example of an undecidable statement par excellence. It is an example of a problem that is:

  • natural;
  • historied -- it was first asked by Cantor himself;
  • celebrated -- it was Hilbert's first of his famous 23 problems; and
  • undecidable in a very deep way -- it's independent of all current large cardinal axioms, and in fact something deeper than this can be said, but that is currently beyond my understanding.

Moreover, the desire to "solve" CH was the main motivation, or at least one of the main motivations, for:

  • Godel's description of the Constructible Universe, which has already been listed as a fundamental example in logic and foundations, and the beginnings of inner model theory; and
  • Cohen's invention of the method of forcing which is now indispensable and ubiquitous in modern set theory, and also won Cohen a Fields medal.
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Recursion theory has the halting problem. From it, you obtain the first example of an interesting Turing degree, and generalizing it, you get the Turing jump operator.

The halting problem also plays an important role in (the boundary of) computational complexity theory, since it's one of the main tools for demonstrating the insolubility of computational problems.

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In the theory of holomorphic functions of several variables, Hartogs's theorem that any holomorphic function on a punctured open set of $\mathbb C^n$ ($n\geqslant 2$) can holomorphically be continued through the deleted point really started the subject and is still the most spectacular divide between $n \geqslant 2$ and $n=1$ [where it is completely false: on $\mathbb C^\ast$ look at 1/z or, worse, exp(1/z): these functions clearly can't be continued holomorphically through zero]

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It is interesting, but this is a theorem, not an example right? – Gil Kalai Nov 11 '09 at 8:56
Perhaps this answer refers to a class of examples, the punctured open sets of C^n when n>1, of domains which are not domains of holomorphy. – Jonas Meyer Nov 11 '09 at 9:52
Dear Gil and Jonas:thanks for the comments and, yes, you are both right. Hartogs's example-theorem shows that there exist in C^n domains which are not regions of holomorphy,a phenomenon impossible in dimension one.This launched the notion of pseudoconvex domains (which exclude Hartogs-type extensions of holomorphic functions) and ultimately led to the concept of Stein manifolds, central in complex geometry (the analogues of affine varieties in algebraic geometry) – Georges Elencwajg Nov 11 '09 at 10:28
I see, I guess it is a concrete example! (and even a popular one.) – Gil Kalai Nov 11 '09 at 14:09

The full matrix rings of any order over another ring (and their direct union of row-finite, column-finite matrices) are a fundamental example for Noncommutative Ring Theory: they are simple enough to be easily understood "in a glimpse" but complex enough to highlight many interesting concepts of the theory.

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Dirac's Delta function

It was introduced by 1927 by Paul Dirac. Can be regarded as an important example towards the theory of distribution.

For examples of distributions to keep in mind see this question.

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In operator theory, the unilateral shift. This operator is not only the fundamental example of an isometry on a Hilbert space, it can also be shown that this operator contains all other operators, in some sense.

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Hyperbolic geometry, which led to the understanding that there are non Euclidean geometries and to many other developments.

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sl2 is the fundamental example of a finite dimensional simple Lie algebra. It forms the basis of the Cartan-Killing classification of complex semisimple Lie algebras, since all of the others can be made by gluing (in some sense) copies of sl2 together. Its representation theory is both straightforward and illuminating, since it points the way to the general theory of highest-weight representations.

For the same reason, SU(2) is the fundamental example of a nonabelian compact Lie group. By Borel-Weil, its irreducible representations can be geometrically realized as the spaces of sections of complex line bundles on the 2-sphere (somewhat easier to handle than a general flag variety).

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Various families of orthogonal polynomials can probably be considered fundamental in some way, but here I want to single out the Chebyshev polynomials, which seem to be the most miraculous. They have explicit formulas (T_n(x) = cos(n arccos(x)), for example), and they are closed under composition. Also, the polynomial of a given degree that has given bounds on an interval and grows as fast as possible outside the interval is a translated and scaled Chebyshev polynomial. The Chebyshev polynomials are fundamental in numerical analysis. Among their uses:

  • Accurate numerical integration
  • Accurate polynomial interpolation (if one is allowed to choose the evaluation points)
  • Sometimes the mere existence of polynomials with the boundedness/growth properties of Chebyshev polynomials is useful, e. g., in analyzing the convergence of the method of conjugate gradients

(A little bit more on the relation to conjugate gradients: n steps of CG can be interpreted as optimizing something over the space of polynomials of degree n. The optimality of the Chebyshev polynomials in the sense of being as small as possible on [-1, 1] (stated above in a different form) is not quite equivalent to what CG requires, but it's close enough to imply that the optimal polynomial in the sense of CG must be very good (because it is at least as good as the Chebyshev polynomial).)

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$\operatorname{SL}_2(\mathbb{Z})$ and its action on the hyperbolic plane. It is the "minimal" example of mapping class groups and arithmetic groups. And one can already see a lot of the general behavior.

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In combinatorics, the (Pascal) Binomial Triangle more or less started the entwining of combinatorics, probability and algebra.

The other two most seminal and ubiquituous triangles of numbers are the Stirling (duals between 1st and 2nd sort, permutation world and set world, the most often generalized family of numbers) and the Eulerian numbers (permutations seen as words on ordered alphabet), the reference statistics on permutations.

For links between combinatorics and number theory, I think the first prize would be the Bernoulli numbers.

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The Heisenberg group: the group of 3 by 3 upper triangular matrices of the form

$$\begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}.$$

When making a first step to non-commutativity? Or beyond planar models? Try the Heisenberg group.

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I'm quite fond of this group, but surely the first step to non-commutativity (in groups) is $S_3$ or $D_8$ or $2\times2$ real invertible matrices. – Gerry Myerson Jun 10 at 12:39

Gödel's Constructible Universe, L, is the fundamental example of inner model theory. L was the first inner model, and is the minimal one. One could argue that the point of inner model theory is to build L-like models that do things L cannot.

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Tsirelson space (see the Wiki entry for quick facts) in Banach space theory was seminal, in that it generated a stream of further refined and specialized counterexamples (most notably in the work of Casazza, Odell, Schlumprecht, and culminating in the famous examples of hereditarily indecomposable spaces by Gowers and Maurey (apologies for the countless others not mentioned here, but the list would be too long). Several long-standing conjectures (even dating back to Banach) have been proved or disproved using these examples, and the flow is not over even now.

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The wiki entry is . See also a discussion on Gowers's blog… – Gil Kalai Nov 13 '09 at 7:12

Fundamental examples of coalgebraic structures:

  • Homology-of-a-space Hopf algebra
  • Hopf algebra representing a linear algebraic group
  • Group rings
  • Spheres as cogroups in the homotopy category of spaces
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The torus is THE example in many branches of math. Algebraic Topology: it is the example where you can compute explicitly its fundamental group as well as its covering spaces, universal cover and everything else. Algebraic geometry: provided it is smooth, is a cubic which is not rational. It is a compact Lie group (and then a non so trivial example of a trivial tangent bundle). It is compact in $\mathbb{R}^4$ and has zero curvature (Riemannian geometry). It is a Riemann Surface. Actually one can compute very explicitly the moduli space of such Riemann surfaces as well as (very explicitly) the mapping class group (which is a beautiful example). It is an elliptic curve, an abelian variety... and it has all those properties I don't know about...meaning, an endless amount of properties.

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But in what way has the torus shaped any of these subjects? It's certainly a good example demonstrating many of the features, but I don't see (from your answer) that it has played a significant role in shaping them. – Loop Space Nov 11 '09 at 9:51
"Algebraic Topology: it is the example where you can compute explicitly its fundamental group as well as its covering spaces, universal cover and everything else." I don't see how this distinguishes the torus from any other closed surface, from the algebro-topological point of view. Indeed, if you're interested in the fundamental group, the fact that it's abelian in this case makes it highly unrepresentative. – HJRW Nov 11 '09 at 17:36

Some people may disagree that this is an example per se, but I'd put up the Feynman path integral (see Wikipedia), because:

  • it provided a completely new physical picture of quantum mechanics

  • it led to systematic development of quantum field theory and string theories, both of which have had led to enormous synergistic growth in mathematics

  • it uncovered a fundamental similarity between of stochastic processes and deterministic quantum dynamics

  • it used the connection between Lie algebras and Lie groups in new and unexpected ways

  • questions about the path measure have stimulated much development in measure theory and analysis

  • tricks like the Wick rotation not only relate statistical mechanics and quantum mechanics (and the corresponding field theories) to each other, but also have stimulated further research in applications of analytic continuation

...and probably more that I am unaware of.

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I agree that basic mathematical examples which are fundamental in physics or other sciences should be counted. (And often they become fundamental in mathematics as well.) We can add the Poincare group along with the group SU(3) x SU(2) x U(1) ( – Gil Kalai Nov 23 '09 at 19:01

Theory of Algebraically Closed Fields (ACF for short) is an important example in Model Theory. This was a motivation example to develop theory of stable and simple theories (leading, eventually, to a beautiful proof of Mordell-Lang conjecture by Ehud Hrushovski).

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Turing machines (1937) and Boolean circuits: the primary models for digital computers.

Universal Quantum computers, and quantum Turing machines (Deutsch, 1985).

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The ring of Gaussian integers Z[i] is a fundamental example of a ring of integers extending Z. Many early results in number theory were motivated by understanding and generalizing properties exhibited by Z[i] and its relationship with Z.

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The Riemann zeta function is the fundamental example of a Dirichlet L-series. It is central in analytic number theory.

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(I think we had it this example before.) – Gil Kalai Jan 13 '10 at 19:59
Ah, I see it now in the real and complex analysis section. It seems a curious omission in the number theory section, as well as the modular function $j(\tau)$ or the Dedekind eta function. Some of my suggestions for this question are not designed to be new examples to many mathematicians. They indicate that I think those examples are fundamental, which might not be as obvious to mathematicians outside those fields. – Douglas Zare Jan 14 '10 at 9:21
Explaining how examples already mentioned are fundamental can be very useful! Note that you can freely edit existing answers. – Gil Kalai Jan 15 '10 at 8:40
Actually, I can't edit them. – Douglas Zare Jan 15 '10 at 9:41

In geometric/combinatorial group theory, Grigorchuk's 2-group is a fundamental example. It originally was constructed as a particularly elegant infinite finitely generated torsion group. Then Grigorchuk showed it had intermediate growth, answering Milnor's problem. For a time it was seen as the universal counterexample in group theory. But now it has spawned a theory of groups acting on rooted trees, self-similar groups and branch groups. Pierre de la Harpe has an entire chapter of his book devoted to this group.

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I think the seven bridges of Königsberg initiated a lot of modern discrete mathematics.

Answered by Tomaž Pisanski

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I think I read at some point the 'game' of finding a Hamiltonian path on an dodecahedron was quite fashionable (ah, the pre-facebook days!) Is that contemporary to the time when solving the Königsberg problem was in fashion? – Mariano Suárez-Alvarez May 10 '10 at 17:36
They are over a century apart. Euler published his paper on the bridges of Koenigsberg in 1735 while Hamilton invented the "icosian game" in 1857. – Tomaž Pisanski May 10 '10 at 21:59

The complex projective space is the fundamental example in toric geometry, symplectic and GIT quotients,...

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See my comment on the torus and SU_3! – Loop Space Nov 11 '09 at 9:53
I agree, complex projective planes (and spaces), real projective planes and spaces, and those over finite field are very important examples. – Gil Kalai Nov 11 '09 at 18:38

In convex geometry, the Euclidean ball. In fact (as I think Gil knows but many other readers here probably don't) a huge portion of (high-dimensional) convex geometry consists of results that show that arbitrary high-dimensional convex bodies behave like the Euclidean ball in various ways.

And if I may be permitted to add another complementary example or two, the simplex and cube are for many purposes the least "Euclidean ball-like" convex bodies, so they are useful for understanding the limitations of the Euclidean ball as a prototype for arbitrary bodies.

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