# 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. –  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

The example that launched category theory: (co)homology, for example simplicial homology and Čech cohomology. The various maps linking (co)homology groups for different 'resolutions' of a topological space (by triangulation or open sets resp.) were I think the first examples of natural transformations. This necessitated defining functors, and hence defining categories.

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Dirichlet Function is a fundamental example in Calculus where Riemann integral does not work. It is also a function which is discontinuous everywhere. The function D(x) is defined as D(x) = 1, if x is a rational number; otherwise D(x) = 0.

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The Thompson groups, as wikipedia said, have a collection of unusual properties which have made them counterexamples to many general conjectures in group theory. For example it is a finite generated finite presented torsion free group with infinite cohomology dimension. For more information see: http://en.wikipedia.org/wiki/Thompson_groups

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Fermat hypersurfaces and their Zeta functions.

Udi de Shalit proposed: One example that comes to my mind is the Zeta functions of the Fermat hypersurfaces, which were studied by Weil and are known to have helped him in formulating the Weil conjectures, later on proved by Grothendieck and Deligne.

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Dirichlet's theorem is the first use of analysis to prove a number theoretic result which does necessarily seem analytic. His proof leads to a lot of ideas about distributions of primes, many of which used analysis. It even leads to an analytic proof of one of the inequalities in class field theory, a result which can also be proved using a good deal of cohomology and which is therefore not exclusively analytic. (I am not counting the prime number theorem, since that is an asymptotic result and thus reeks of analysis as soon as it is conjectured. It was also proven later.)

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The (complex analytic) proof of the Prime Number Theorem is the first major use of complex analysis to prove results about asymptotic behavior of prime numbers, which, at first glance, do not at all seem to be tied to complex numbers. This has led to an enormous amount of mathematics.

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The field extension $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is the most basic example of an algebraic extension which is not Galois. In particular, one notices that it has no non-trivial automorphisms, and that this is related to the fact that not enough of the roots of $x^3-2$ are in this field. This leads to the key concept of Galois extensions and the relation between automorphisms and roots. It also led Galois to develop the concept of normal subgroup.

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Linear Algebra: In linear algebra the symmetric group $S_n=Bij(\{1,,n\})$ is a classical and important example if you learn basics in group theory.

Operator-algebra/ functional-analysis: Fundamental examples of $C^*$-algebras are $C(X)$ ( continuous functions on a compact Haudorff space X), $C_0(X)$ ( continuous functions vanishing at infinity on a localcompact Haudorff space X), $B(H)$ (bounded linear maps on a Hilbert space H). By Gelfand-Naimark you know a lot of abstract $C^*$algebras if you know these concrete examples.

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The ring Z, or the ring Z/6Z are examples of very many types of rings.

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The parallel-or functional is a fundamental example in denotational semantics.

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The most fundamental equation is $x^2+1=0$.

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