Fundamental Examples

Question

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.)

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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.

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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)

Answer 1

The harmonic oscillator is a fundamental example in both classical and quantum mechanics.

Answer 2

The integral $\int \frac{dx}{\sqrt{x(x-1)(x-\lambda)}}$ for $\lambda\neq 0,1$ essentially launched both complex analysis and algebraic geometry, via Riemann's discovery of the Riemann surface that is the natural domain of a function, leading to both analytic theory of Riemann surfaces and to the study of algebraic curves, leading to...complex analysis including in several variables and complex algebraic geometry, as we now know it.

Answer 3

Relevant to many areas (but mostly topology) is the Cantor set. It is an example of a set with properties too numerous to list here. To name a few: it is uncountable, compact, nowhere dense and it has Lebesgue measure 0.

Answer 4

The Petersen Graph in graph theory.

Answer 5

For a picture that launched a thousand papers, I'd nominate the bifurcation diagram of the logistic map.

(image via Wikipedia).

Answered by Martin M. W.

Answer 6

The Hopf fibration

Answer 7

The Fermat Equation xⁿ + yⁿ - zⁿ = 0.

This has truly been much more than an example in both algebra and number theory: it was one of the main motivations to develop the theory of unique factorization domains, Dedekind domains, class numbers, regular primes, etc. in the 19th century. In the late 20th century it provided a motivation for Wiles to work on modularity of elliptic curves.

In the 21st century, the equation c₁ x^a + c₂ y^b - c₃ z^c = 0 is similarly motivational for things like Q-curves, Galois representations, hypergeometric abelian varieties...

Answer 8

Brownian motion has a central role in the theory of stochastic processes

Answer 9

The Prisoner's Dilemma in Game Theory.

Answer 10

The Brachistochrone problem. Solved by both Newton/Bernoulli. It is considered to be the fundamental/first problem which led to the formulation of the Calculus of Variations.

Find the shape of the curve down which a bead sliding from rest and accelerated by gravity will slip (without friction) from one point to another in the least time.

Answer 11

In complex dynamics: The Mandelbrot set

Answer 12

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

Answer 13

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.

Answer 14

Russell's paradox

Showing that Frege's set theory leads to contradiction.

The example can be described as the set of all sets $A$ such that $A \notin A$. It is related in spirit to Cantor's proof that a power set has larger cardinality than a set and to the ancient "liar paradox".

Answer 15

Someone has already mentioned tori, but I think elliptic curves in algebraic geometry merit their own separate mention.

Answered by Kevin Lin

Answer 16

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

Answer 17

SAT (Boolean satisfiability problem) in complexity theory/theoretical computer science -- it's the canonical example of an NP-complete problem not just because it came first, but because it launched a thousand other research papers all on its own. (3SAT is maybe more canonical, but the more general form is better to generalize and study for its own sake.)

Answered by: Harrison Brown

Answer 18

Answered by James: The Platonic solids. They are fundamental, collectively and individually, to many areas of mathematics.

Answer 19

1The Fano plane in finite geometry

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

Answer 20

The ring $\mathbb{Z}[\sqrt{-5}]$ is a fundamental example of non-unique factorization in rings in algebraic integers. Perhaps of more historical relevance is the example of $p=37$ that shows that Lamé's "proof" of Fermat's Last Theorem fails. I'm not sure Kummer used $p=37$ though. In any case, examples of non-unique factorization in rings in algebraic integers lead to the whole theory of ideal numbers and later Dedekind domains and set the tone for algebraic number theory.

Answer 21

The Penrose tiling (image from Wikipedia). It's a fundamental example not just for aperiodic tilings more generally, but for Connes' work on noncommutative geometry.

Answer 22

The Catalan numbers are definitely a fundamental example in combinatorics.

Answered by Qiaochu Yuan

Answer 23

The quaternions

Probably the natural numbers, real numbers, and complex numbers are "too fundamental" to count here. But the field of quaternions discovered in the mid 19 century qualifies.

Answer 24

A lot of algebraic topology was developed with computing the higher homotopy groups of spheres in mind.

Answered by: David Lehavi

Answer 25

In geometry, group theory and other areas: The Leech Lattice:

Answer 26

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

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.

Answer 27

The KdV equation in integrable systems. It was through a numerical study of KdV that the word soliton was coined. This numerical study lead to much analytical work, including the development of Lax Pairs. (Answer by Aaron Hoffman)

Answer 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.

Answered by: Yuhao Huang

Answer 29

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 $su_2$!)

Answer 30

The Noncommutative Torus in non-commutative geometry. (Maybe it has just shaped the subject because it is about the only thing one can handle explicitly)