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

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Of course, for this question and many others there is no meaning to "correct answer". In fact I liked all the answers to the question and I hope more answers will come along. Jose was the most valuable partner to this endeavor and he contributed several great answers both before and after the boundy was announced. – Gil Kalai Nov 26 2009 at 12:39
@Gil, now that the bounty has been delivered you might "unaccept" this answer. – Scott Morrison Nov 30 2009 at 16:20
Is it possible? Is it moral? – Gil Kalai Dec 1 2009 at 14:53

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

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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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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 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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The irrational rotation on the torus $\mathbb{T^2}$: $T_t (z,w) = (e^{2 \pi i \alpha }z, e^{2 \pi i \beta} w)$, where $\alpha, \beta \in \mathbb{R}, \frac{\alpha}{\beta} \notin \mathbb{Q}$. It is perhaps the first nontrivial example of an ergodic dynamical system. By considering its orbits $t \mapsto T_t (z,w)$, one gets examples of one-to-one immersions (with dense image) which are not embeddings, immersed submanifolds which are not closed/regular submanifolds, Lie subgroups which are not closed subgroups, non-Hausdorff quotient space and similar phenomena.

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The Lorenz system of ordinary differential equations: $$\dot x=\sigma(y-x)$$ $$\dot y = rx-y-xz$$ $$\dot z = xy-bz$$ ($\sigma$, $r$, $b$ are parameters) is a good example in dynamical system. It is an example of a deterministic system displaying chaotic behaviour. Also the Lorenz attractor. Date: 1963.

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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 Harper has an entire chapter of his book devoted to this group.

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Motivated by Amit Kumar Gupta's answer about the continuum hypothesis, let me add an example that is less natural but has inspired an amazing amount of set theory, namely Suslin's Hypothesis. This conjecture, proposed in 1920 and now known to be independent of ZFC, says that the real line with its usual ordering relation is characterized up to isomorphism by the following properties:

• dense linear order without endpoints

• Dedekind-complete

• No uncountable family of pairwise disjoint open intervals.

The point of the conjecture is that it was proved much earlier by Cantor that one gets a characterization of $\mathbb R$ if one puts in place of the last property the stronger statement that there is a countable dense set. So Suslin is simply asking whether one can weaken this separability assumption to the third property in the list above (often called the "countable chain condition"). I can't claim that this question is anywhere near as natural as the continuum hypothesis, but what makes it important (in my opinion) is its impact on the development of set theory. The fact that Suslin's hypothesis is false in Gödel's constructible universe $L$ was one of the first applications (and probably a major motivation, though I don't actually know that) for Jensen's theory of the fine structure of $L$, a theory that has grown tremendously as a component of the inner model program in contemporary set theory. The fact that Suslin's hypothesis is consistent with ZFC was the initial application and the motivation for the theory of iterated forcing, now a central tool in set theory. It also provided the occasion for the invention of Martin's axiom. That axiom and the combinatorial principles isolated by Jensen from the fine structure of $L$ have become standard tools for proving independence results without explicitly referring to forcing or to $L$.

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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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Schwarzschild metric as a prototype of black hole was a fundamental example in the development of General Relativity (for instance, it is often referred to when "defending" the ADM mass as a natural concept of mass in General Relativity).

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The free group factors $L(\mathbb{F}_{n})$, which are the closer in the weak operator topology of the left regular representation of the free group $\mathbb{F}_n$, are fundamental examples in von Neumann algebras. The isomorphism question is the root of the so important Free Probability theory of Voiculescu.

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The hyperfinite $\mathrm{II}_1$ factor $\mathcal{R}$ and its ultrapower $\mathcal{R}^{\omega}$ are fundamental examples in von Neumann algebras and Connes' embedding problem.

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The semicircular law and the Marchenko-Pastur distribution are fundamental examples of probability distributions in random matrix theory.

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to understand curves, first study the abel map. and then the torelli map. [perhaps I should expand this rather succinct answer.]

8320 Spring 2010, day one Introduction to Riemann Surfaces

We will describe how Riemann used topology and complex analysis to study algebraic curves over the complex numbers. [The main tools and results have analogs in arithmetic, which I hope are more easily understood after seeing the original versions.]

The idea is that an algebraic curve C, say in the plane, is the image by a holomorphic map, of an abstract complex manifold, the Riemann surface X of the curve, where X has an intrinsic complex structure independent of its representation in the plane. Then Riemann's approach to classifying all complex curves, is to classify such Riemann surfaces, and then for each such surface to classify all maps from it to projective space. Briefly, the Torelli maps classifies complex surfaces, and the abel maps classify all projective models of a given complex surface.

More precisely, we will construct two fundamental functors of an algebraic curve:

i) the Riemann surface X, and

ii) the Jacobian variety J(X), and

natural transformations X^(d)--->J(X), the Abel maps, from the “symmetric powers” X^(d) of X, to J(X).

The Riemann surface X

The first construction is the Riemann surface of a plane curve: {irreducible plane curves C: f(x,y)=0} ---> {compact Riemann surfaces X}.

The first step is to compactify the affine curve C: f(x,y) =0 in A^2, the affine complex plane, by taking its closure in the complex projective plane P^2. Then one separates intersection points of C to obtain a smooth compact surface X. X inherits a complex structure from the coordinate functions of the plane. If f is an irreducible polynomial, X will be connected. Then X will have a topological genus g, and a complex structure, and will be equipped with a holomorphic map ƒ:X--->C of degree one, i.e. ƒ will be an isomorphism except over points where the curve C is not smooth, e.g. where C crosses itself or has a pinch.

This analytic version X of the curve C retains algebraic information about C, e.g. the field M(X) of meromorphic functions on X is isomorphic to the field Rat(C) of rational functions on C, the quotient field k[x,y]/(f), where k = complex number field. It turns out that two curves have isomorphic Riemann surfaces if and only if their fields of rational functions are isomorphic, if and only if the curves are equivalent under maps defined by mutually inverse pairs of rational functions.

Since the map X--->C is determined by the functions (x,y) on X, which generate the field Rat(C), classifying algebraic curves up to “birational equivalence” becomes the question of classifying these function fields, and classifying pairs of generators for each field, but Riemann’s approach to this algebraic problem will be topological/analytic.

We already can deduce that two curves cannot be birationally equivalent unless their Riemann surfaces have the same genus. This solves the problem that interested the Bernoullis as to why most integrals of form dx/sqrt(cubic in x) cannot be “rationalized” by rational substitutions. I.e. only curves of genus zero can be so rationalized and y^2 = (cubic in x) usually has positive genus.

The symmetric powers X^(d)

To recover C, we seek to encode the map ƒ:X--->C, i.e. ƒ:X--->P^2, by intrinsic geometric data on X. If the polynomial f defining C has degree d, then each line L in the plane P^2 meets C in d points, counted properly. Thus we get an unordered d tuple of points L.C, possibly with repetitions, on C, hence when pulled back via ƒ, we get such a d tuple called a positive “divisor” D = ƒ^(-1)(L) of degree d on X. (D = n1p1+...nk pk, where nj are positive integers, n1+...nk = d.)

Since lines L in the plane move in a linear space dual to the plane, and (if d ≥ 2) each line is spanned by the points where it meets C, we get an injection P^2*--->{unordered d tuples of points of X}, taking L to ƒ^(-1)(L).

If X^d is the d - fold Cartesian product of X, and Sym(d) is the symmetric group of permutations of d objects, and we define X^(d) = X^d/Sym(d) = the “symmetric product” of X, d times, then the symmetric product X^(d) parametrizes unordered d tuples, and inherits a complex structure as well.

Thus the map ƒ:X--->C yields a holomorphic injection P^2*--->Π of the projective plane into X^(d). I.e. the map ƒ determines a complex subvariety of X^(d) isomorphic to a linear space Π ≈ P^2*. Now conversely, this “linear system” Π of divisors of degree d on X determines the map ƒ back again as follows:

Define ƒ:X--->Π* = P^2** =P^2, by setting ƒ(p) = the line in Π consisting of those divisors D that contain p. Then this determines the point ƒ(p) on C in P^2, because a point in the plane is determined by the lines through that point. [draw picture] Thus the problem becomes one of determining when the product X^(d) contains a holomorphic copy of P^2, or copies of P^n for models of X in other projective spaces.

The Jacobian variety J(X) and the Abel map X^(d)--->J(X).

For this problem, Riemann introduced a second functor the “Jacobian” variety J(X) = k^g/lattice, where k^g complex g -dimensional space. J(X) is a compact g dimensional complex group, and there is a natural holomorphic map Abel:X^(d)--->J(X), defined by integrating a basis of the holomorphic differential forms on X over paths in X.

Abel collapses each linear system Π ≈ P^n* to a point by the maximum principle, since the coordinate functions of k^g have a local maximum on the compact simply connected variety Π. Conversely, each fiber of the Abel map is a linear system in X^(d). Existence of linear systems Π on X: the Riemann - Roch theorem.

By dimension theory of holomorphic maps, every fiber of the abel map X^(d)--->J(X) has dimension ≥ d-g. Hence every positive divisor D of degree d on X is contained in a maximal linear system |D| , where dim|D| ≥ d-g. This is called Riemann’s inequality, or the “weak” Riemann Roch theorem.

The Roch part analyzes the relation between D and the divisor of a differential form to compute dim|D| more precisely. Note if D is the divisor cut by one line in the plane of C, and E is cut by another line, then E belongs to |D|, and the difference E-D is the divisor of the meromorphic function defined by the quotient of the linear equations for the two lines.

If D is a not necessarily positive divisor, we define |D| to consist of those positive divisors E such that E-D is the divisor of a meromorphic function on X. If there are no such positive divisors, |D| is empty and has “dimension” equal to -1. Then if K is the divisor of zeroes of a holomorphic differential form on X, the full Riemann Roch theorem says:

dim|D| = d-g +1+dim|K-D|, where the right side = d-g when d > deg(K).

This sketch describes the abel maps and their relation to the RRT. The assignment X-->J(X) is the Torelli map, and classifies X by the numerical data in the lattice defining J(X), i.e. periods of integrals of the first kind on X. This assignment gives birth to the whole subject of "moduli" as numerical invariants of complex or geometric structure.a

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I think polynomials are one of the greatest inventions of humankind. Not only are they extremely flexible and come up in so many domains of math, but they've lead to interesting breakthroughs. For example, trying to find a closed formed solution to the quintic polynomial lead Galois to develop groups, right?

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I think no one's pointed Lorenz equations.

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The pseudo-arc in continuum theory.

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The Möbius strip or Möbius band (a surface with only one side and only one boundary component).

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To make this question and the various examples a more useful source this is a designated answer to point out connections between the various examples we collected. please indicate only strong, definite, nontrivial, and clear connections.

1) The Petersen graph is obtained by identifying antipodal vertices and edges in the graph of the dodecahedron - one of the five platonic solids. Such an identification gives a polyhedral complex realizing the real projective plane. Applying this operation to the icosahedron leads to a 6-vertex triangulation of the real projective plane.

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The cyclotomic field $\mathbb{Q}(\zeta_p)/\mathbb{Q}$ is the most basic example of a field extension in which splitting of primes depends on an obvious congruence condition. Specifically, if $\ell$ is another prime, then the Frobenius of $\ell$ is $\ell \mod p \in (\mathbb{Z}/p)^\times = \mathrm{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q})$. In particular, $\ell$ splits in the field iff its Frobenius is trivial, and this is true iff $\ell \equiv 1 \mod p$. We can then relate other congruences to splitting in subfields of $\mathbb{Q}(\zeta_p)$, etc. The theorems of global class field theory show that this basic concept holds in a very general case, although the general case is much harder to prove. This basic example, does, however, motivate the ideas in class field theory, which have greatly influenced modern number theory and related areas. (As an added note, the fact that the Artin reciprocity law is true for cyclotomic fields is actually a key ingredient in the proof for general abelian extensions!)

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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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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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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 rational parametrization of the locus of the equation $X^2+Y^2=1$ by $(\frac{t^2-1}{t^2+1},\frac{2t}{t^2+1})$. It can be viewed geometrically by taking a line that intersects the unit circle at one rational point and then considering all possible (rational) slopes of the line (including infinity), which are in correspondence with (rational) points of the circle. This is the most basic example of using a geometric idea to find solutions to a diophantine equation, and it leads to very deep mathematics.

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The gamma function is a fundamental example of an interesting function defined only on the integers which has an analytic (meromorphic) continuation to the whole complex plane. This ability to extend an interesting, seemingly discrete function to a complex differentiable function motivates a lot of later material.

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

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Smale's horseshoe map in dynamical systems.

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The Sorgenfrey line is an example that has motivated a lot of research in general topology, mostly generalized metric properties and ordered space theory. It's an example of a hereditary normal space with non-normal square, it is separable, Lindelöf, first countable, but not second countable; a generalized ordered space that is not orderable, and many more.

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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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