Fundamental Examples

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.)

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) eliptic curves, transendental 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 Desrague 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 (mid 19th 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)

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I think that this should be community wiki. – Andrew Stacey Nov 11 2009 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. – Andrew Stacey Nov 11 2009 at 9:50
I've hit this with the wiki hammer. – Scott Morrison Nov 11 2009 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 2009 at 8:03
Why does this question have a bounty anyway? – Kevin Lin Nov 21 2009 at 17:33

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 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. – Andrew Stacey Nov 11 2009 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 *un*representative. – HW Nov 11 2009 at 17:36

Laplace's equation is the fundamental example of a PDE.

If I could broaden the question to allow a triumvirate of examples, I'd say Laplace's equation, the heat equation, and the wave equation are the canonical examples of PDEs, representing elliptic, parabolic, and hyperbolic equations respectively.

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

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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 2009 at 10:28

I do think Milnor's exotic sphere distinguish differential topology from general topology, but i don't know if this is an example of the kind you want.

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It has often been said that if you understand $su_3$ you understand all simple Lie algebras, so that should make it the fundamental example. (Personally I think that it suffices to understand $su_2$!)

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On the same line, SU(2) (compared to U(1), which is abelian) already shows a lot of features of the compact non abelian groups (Peter-Weil theorem, irreducible representations of (every) dimension greater than 1, ...). – Gian Maria Dall'Ara Nov 11 2009 at 9:11
I guess you could say that the heat equation "shaped" functional analysis since so many of the early tools were developed to study just that equation (more so if you say fourier analysis). Similarly (as has been noted) the spheres are currently shaping algebraic topology since so many of the tools are developed to get at the stable homotopy of spheres. I didn't put my comment on your "harmonic oscillator" example because that has played a role in shaping quantum mechanics. Part of it is purely timing: good examples have a chance to shape a subject if they are encountered in its infancy. – Andrew Stacey Nov 11 2009 at 10:33

The cotangent bundle is the fundamental example of symplectic manifold/phase space.

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

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There's always the venerable normal distribution (for probability theory).

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