## 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

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

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The p-adic numbers. Discovered only in 1897.

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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 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

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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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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Smale's horseshoe map in dynamical systems.

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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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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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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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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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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

1 a c

0 1 b

0 0 1

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

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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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The trefoil knot in knot theory.

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The associahedron or Stasheff polytope.

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

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

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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 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

the Fischer-Griess Monster.

see:

http://en.wikipedia.org/wiki/Monster_group

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Poincare dodecahedral sphere, the 1904 example of a homology sphere was fundamental for the discovery of the fundamental group, and have led to the statement of the Poincare conjecture.

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The Alexander polynomial in knot theory.

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Hyperbolic toral automorphisms (viz. the cat map and its generalizations) are the fundamental examples of Anosov diffeomorphisms, and their suspensions are the fundamental examples of Anosov flows. This is because they are "structurally stable", i.e. small perturbations preserve the Anosov property and any Anosov diffeomorphism on a torus is topologically conjugate to a hyperbolic toral automorphism. I am actually not even aware of any other concrete examples of Anosov dynamics other than those derived from geodesic flows on hyperbolic spaces.