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45
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edited Oct 11 2011 at 13:06 |
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44
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edited Jul 7 2011 at 14:59 |
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43
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edited Apr 18 2011 at 13:18 |
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42
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edited Feb 21 2011 at 20:48 |
Real and Complex Analysis: Harmonic series (14th Cen.) {and Riemann zeta function (1859), 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 ?) 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, 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)
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41
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edited Feb 14 2011 at 14:00 |
Social Science: Prisoner dillema 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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40
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edited Feb 14 2011 at 7:19 |
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39
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edited Dec 14 2010 at 8:01 |
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38
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edited Dec 12 2010 at 14:23 |
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37
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edited Aug 30 2010 at 14:18 |
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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36
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edited Aug 30 2010 at 11:54 |
Topology: Spheres, Figure-eight knot (ancient), trefoil knot (ancient?) (Borromean rings (ancient?)), the torus (ancient?), 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), Discrete metric spaces, Sorgenfrey line, Complex projective space, the cotangent bundle (?), The Grassmannian variety,homotopy group of spheres (*1951), Milnor exotic spheres (1965)
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35
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edited Jun 22 2010 at 9:21 |
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34
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edited Jun 22 2010 at 6:14 |
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33
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edited May 19 2010 at 10:45 |
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), Dynamics: Logistic map (1845?), Smale's horseshoe map(1960). Mandelbrot set (1978/80) (Julia set), cat map, (Anosov diffeomorphism) Topology: Spheres, Figure-eight knot (ancient), trefoil knot (ancient?) (Borromean rings (ancient?)), the torus (ancient?), Cantor set (1883), 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), 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),
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32
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edited May 18 2010 at 14:30 |
Combinatorics: tic-tac-toe (ancient Egypt(?)) (The game of nim (ancient China(?))), Binomial 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)
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31
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edited May 18 2010 at 13:09 |
In order to make it a more useful source, I list all the answers in categories, and added (for most) a date and (for 1/32/5) 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: Alef_omega $\aleph_\omega$ (~1890), Russell's paradox (1901), Topology: Spheres, Figure-eight knot]7 (ancient), trefoil knot (ancient?) (Borromean rings (ancient?)), the torus (ancient?), Cantor set (1883), 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), Discrete metric spaces, Complex projective space, the cotangent bundle (?), The Grassmannian variety,homotopy group of spheres (*1951), Milnor exotic spheres (1965)
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30
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edited May 13 2010 at 17:19 |
Combinatorics: tic-tac-toe (ancient Egypt(?)) (The game of nim (ancient China(?))), Binomial 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)
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29
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edited May 9 2010 at 6:08 |
Functional analysis: Unilateral shift, The spaces $\ell_p$, $L_p$ and $C(k), C(k)$, Tsirelson spaces (1974), Cuntz algebra,
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28
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edited May 7 2010 at 6:39 |
Functional analysis: Unilateral shift, The spaces $\ell_p$, $L_p$ and $C(k), Tsirelson spaces (1974), Cuntz algebra, Algebra: Z and Z/6Z (Middle Ages?), symmetric and alternating groups (*1832), Gaussian integers ($Z[\sqrt -1]$) (1832), $Z[\sqrt(-5)]$,su_3 $su_3$ (su_2)$su_2)$, 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, 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), 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), 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) Social Science: Prisoner dillema (1950), 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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27
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edited Apr 27 2010 at 19:32 |
Algebra: Z and Z/6Z (Middle Ages?), symmetric and alternating groups (*1832), Gaussian integers ($Z[\sqrt -1]$) (1832), $Z[sqrt(-5)],Z[\sqrt(-5)]$,su_3 (su_2), full matrix ring over a ring, SU(2), quaternions (1843), p-adic numbers (1897), Young tableaux (1900) and Schur polynomials, Hopf algebras (1941) Fischer-Griess monster (1973), Heisenberg group, ADE-classification (and Dynkin diagrams), Prufer p-groups,
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26
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edited Mar 2 2010 at 20:37 |
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25
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edited Jan 14 2010 at 8:31 |
Probability: Normal distribution (1733), Brownian motion (1827), The Gaussian Orthogonal Ensemble, the Gaussian Unitary Ensemble, and the Gaussian Symplectic Ensemble, SLE (1999),
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24
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edited Jan 13 2010 at 17:24 |
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23
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edited Jan 13 2010 at 13:34 |
Algebra: Z and Z/6Z (Middle Ages?), symmetric and alternating groups (*1832), Gaussian integers ($Z[\sqrt -1]$) (1832), $Z[sqrt(-5)],,su_3 (su_2) Z[sqrt(-5)], full matrix ring over a ring, SU(2), quaternions (1843), p-adic numbers (1897), Young tableaux (1900) and Schur polynomials, Hopf algebras (1941) Fischer-Griess monster (1973), Heisenberg group, ADE-classification (and Dynkin diagrams), Prufer p-groups, Topology: Spheres, Figure-eight knot]7 (ancient), trefoil knot (ancient?) (Borromean rings (ancient?)), the torus (ancient?), Cantor set (1883), Poincare dodecahedral sphere (1904), Alexander polynomial (1923), Hopf fibration (1931), The standard embedding of the torus in R^3 (*1934 in Morse theory), Discrete metric spaces, Complex projective space, the cotangent bundle (?), The Grassmannian variety,homotopy group of spheres (*1951), Milnor exotic spheres (1965)
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22
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edited Jan 10 2010 at 20:18 |
In order to make it a more useful source, I list all the answers in categories, and added (so far to 1/3 of themfor most) a date and (for 1/3) 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! Geometry: Platonic solids (ancient), the Euclidean ball (ancient), The configuration of 27 lines on a cubic surface, 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]7 (ancient), the torus, The standard embedding of trefoil knot (ancient?) (Borromean rings (ancient?)), the torus in R^3, ,(ancient?), Cantor set (1883), Poincare dodecahedral sphere (1904), Alexander polynomial (1923), Hopf fibration (1931), The standard embedding of the torus in R^3 (*1934 in Morse theory), Complex projective space, trefoil knot, the cotangent bundle (?), The Grassmannian variety,homotopy group of spheres, (*1951), Milnor exotic spheres (1965)
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21
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edited Jan 8 2010 at 10:10 |
Graph theory: 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(?))), Catalan numbers (mid 19th century), (Fibonacci sequence)sequence (12th century; probably ancient), Kirkman's schoolgirl problem (1850) tic-tac-toe (and nim), 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 dillema (1950), second price auction (1961)
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20
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edited Jan 5 2010 at 10:22 |
Real and Complex Analysis: Harmonic series (14th Cen.) and Riemann zeta function (1859), li(x), The elliptic integral that launched Riemann surfaces (*1854?), Chebyshev polynomials (?1854) punctured open set in C^n (Hartog's theorem?theorem *1906 ?) Partial differential equations: KdV equations (1877), Laplace equation , (1773), the heat equation, wave equation, Navier-Stokes equation ,(1822),KdV equations (1877), Graph theory: Petersen Graph (1886), two edge-colorings of K_6, K_33 and K_5 (Kuratowski's theorem 1930), Tutte graph , (1946), Margulis's expanders (1973) and Ramanujan graphs , (1986),
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19
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edited Jan 5 2010 at 10:05 |
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.)
In order to make it a more useful source, I list all the answers in categories, and added (so far to 1/3 of them) a date and 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: Alef_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), Feynman path integral (1948),
Real and Complex Analysis: Harmonic series(14th series (14th Cen.) and Riemann zeta function (1859), li(x), The elliptic integral that launched Riemann surfaces (?), *1854?), Chebyshev polynomials (?1854) punctured open set in C^n (?)Hartog's theorem?)
Partial differential equations: KdV equations (1877), Laplace equation, the heat equation, wave equation, Navier-Stokes equation,
Functional analysis: Unilateral shift, Tsirelson spaces (1974), Cuntz algebra,
Algebra: Z and Z/6Z (Middle Ages?), su_3 (su_2) Z[sqrt(-5)], full matrix ring over a ring, SU(2), quaternions (1843), p-adic numbers (1897), Young tableaux (1900) and Schur polynomials, 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), eliptic curves, Fermat hypersurfaces,
Probability: Normal distribution (1733), Brownian motion (1827), SLE (1999),
Dynamics: Logistic map (1845?), 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, construction of regular heptadecagon (*1796), Hyperbolic geometry (1830), Reuleaux triangle, Fano plane, 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]7 (ancient), the torus, The standard embedding of the torus in R^3, ,Cantor set (1883), Poincare dodecahedral sphere (1904), Alexander polynomial (1923), Hopf fibration (1931), Complex projective space, trefoil knot, the cotangent bundle (?),homotopy group of spheres, Milnor exotic spheres
Graph theory: Petersen Graph (1886), two edge-colorings of K_6, K_33 and K_5, Tutte graph, Margulis's expanders and Ramanujan graphs,
Combinatorics: Catalan numbers, (Fibonacci sequence), Kirkman's schoolgirl problem (1850) tic-tac-toe (and nim), surreal numbers, alternating sign matrices
Algorithms and Computer Science: Newton Raphson method, Turing machine (1937), RSA, universal quantum computer (1985)
Social Science: Prisoner dillema (1950), second price auction
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18
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edited Dec 25 2009 at 9:07 |
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.)
In order to make it a more useful source, I list all the answers in categories, and added (so far to 1/3 of them) a date and a link to the answer. (~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: Alef_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), Feynman path integral (1948),
Real and Complex Analysis: Harmonic series(14th Cen.) and Riemann zeta function (1859), li(x), The elliptic integral that launched Riemann surfaces (?), Chebyshev polynomials (?1854) punctured open set in C^n (?)
Partial differential equations: KdV equations (1877), Laplace equation, the heat equation, wave equation, Navier-Stokes equation,
Functional analysis: Unilateral shift, Tsirelson spaces (1974), Cuntz algebra,
Algebra: Z and Z/6Z (Middle Ages?), su_3 (su_2) Z[sqrt(-5)], full matrix ring over a ring, SU(2), quaternions (1843), p-adic numbers (1897), Young tableaux (1900) and Schur polynomials, 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), eliptic curves, Fermat hypersurfaces,
Probability: Normal distribution (1733), Brownian motion (1827), SLE (1999),
Dynamics: Logistic map (1845?), 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, construction of regular heptadecagon (*1796), Hyperbolic geometry (1830), Reuleaux triangle, Fano plane, 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]7 (ancient), the torus, The standard embedding of the torus in R^3, ,Cantor set (1883), Poincare dodecahedral sphere (1904), Alexander polynomial (1923), Hopf fibration (1931), Complex projective space, trefoil knot, the cotangent bundle (?),homotopy group of spheres, Milnor exotic spheres
Graph theory: Petersen Graph (1886), two edge-colorings of K_6, K_33 and K_5, Tutte graph, Margulis's expanders and Ramanujan graphs,
Combinatorics: Catalan numbers, (Fibonacci sequence), Kirkman's schoolgirl problem (1850) tic-tac-toe (and nim), surreal numbers, alternating sign matrices
Algorithms and Computer Science: Newton Raphson method, Turing machine (1937), RSA, universal quantum computer (1985)
Social Science: Prisoner dillema (1950), second price auction
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17
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edited Dec 18 2009 at 13:51 |
Algebra: Z and Z/6Z (Middle Ages?), su_3 (su_2) Z[sqrt(-5)], full matrix ring over a ring, SU(2), quaternions (1843), p-adic numbers , quaternions, (1897), Young tableaux (1900) and Schur polynomials, Hopf algebras (1941) Fischer-Griess monster , (1973), Heisenberg group, ADE-classification (and Dynkin diagrams), Prufer p-groups, Geometry: Platonic solids (ancient), the Euclidean ball(ancient), ball (ancient), The configuration of 27 lines on a cubic surface, construction of regular heptadecagon (*1796), Hyperbolic geometry (1830), Reuleaux triangle, Fano plane, 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]7 (ancient) ,Cantor set, ancient), the torus, The standard embedding of the torus in R^3, ,Cantor set (1883), Poincare dodecahedral sphere , (1904), Alexander polynomial (1923), Hopf fibration (1931), Complex projective space, trefoil knot, the cotangent bundle (?),Milnor exotic spheres, Alexander polynomial,homotopy group of spheres, Milnor exotic spheres Graph theory: Petersen Graph, (1886), two edge-colorings of K_6, K_33 and K_5, Tutte graph, Margulis's expanders and Ramanujan graphs, Combinatorics: Catalan numbers, (Fibonacci sequence), Kirkman's schoolgirl problem (1850) tic-tac-toe (and nim), surreal numbers, alternating sign matrices
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16
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edited Dec 15 2009 at 2:43 |
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15
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edited Dec 14 2009 at 16:43 |
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.)
In order to make it a more useful source, I list all the answers in categories, and added (so far to 1/3 of them) a date and a link to the answer. (~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: Alef_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), The harmonic oscillator,(?) Dirac's delta function (1927), Feynman path integral (1948),
Real and Complex Analysis: Harmonic series(14th Cen.) and Riemann zeta function (1859), li(x), The elliptic integral that launched Riemann surfaces (?), Chebyshev polynomials (?1854) punctured open set in C^n (?)
Partial differential equations: KdV equations (1877), Laplace equation, the heat equation, wave equation, Navier-Stokes equation,
Functional analysis: Unilateral shift, Tsirelson spaces (1974), Cuntz algebra,
Algebra: Z and Z/6Z (Middle Ages?), su_3 (su_2) Z[sqrt(-5)], full matrix ring over a ring, SU(2), p-adic numbers, quaternions, Young tableaux and Schur polynomials, Fischer-Griess monster, Heisenberg group, ADE-classification, Prufer p-groups,
Number Theory: conics and pythagorean triples (ancient), Fermat equation (1637), eliptic curves, Fermat hypersurfaces,
Probability: Normal distribution (1733), Brownian motion (1827), SLE (1999),
Dynamics: Logistic map (1845?), Mandelbrot set (1978/80) (Julia setset), cat map, (Anosov diffeomorphism)
Geometry: Platonic solids (ancient), the Euclidean ball(ancient), The configuration of 27 lines on a cubic surface, construction of regular heptadecagon (*1796), Hyperbolic geometry (1830), Reuleaux triangle, Fano plane, cyclic polytopes (1902), Leech lattice (1965), Penrose tiling (1974), noncommutative torus, cone of positive semidefinite matrices, the associahedron (1961)
Topology: Spheres, Figure-eight knot]7 (ancient) ,Cantor set, the torus, The standard embedding of the torus in R^3, Poincare dodecahedral sphere, Hopf fibration (1931), Complex projective space, trefoil knot, the cotangent bundle (?), Milnor exotic spheres, Alexander polynomial,homotopy group of spheres
Graph theory: Petersen Graph, two edge-colorings of K_6, K_33 and K_5, Tutte graph, Margulis's expanders and Ramanujan graphs,
Combinatorics: Catalan numbers, (Fibonacci sequence), tic-tac-toe (and nim), surreal numbers, alternating sign matrices
Algorithms and Computer Science: Newton Raphson method, RSA
Social Science: Prisoner dillema (1950), second price auction
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14
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edited Dec 14 2009 at 14:35 |
In order to make it a more useful source, I list all the answers in categories, and added (so far to 1/3 of them) a date and a link to the answer. (~year means approximate year, *year means a year when an older example becomes central in view of some discovery.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! Halting problem (1936), Goedel constructible universe L (1938), McKinsey formula in modal logic (~1941), 3SAT ( *1970),The *1970), The theory of Algebraically closed fields ( ACF),ACF) (?),Physics: Brachistochrone problem (1696), The harmonic oscillator,(?) Dirac's delta function (1927), Feynman path integral (1948), Real and Complex Analysis: Harmonic series(14th Cen.) and Riemann zeta function (1859), li(x), The elliptic integral that launched Riemann surfaces, (?), Chebyshev polynomials (?1854) punctured open set in C^n (?) Algebra: Z and Z/6Z (Middle Ages?), su_3 (su_2) Z[sqrt(-5)], full matrix ring over a ring, SU(2), p-adic numbers, quaternions, Young tableaux and Schur polynomials, Fischer-Griess monster, Heisenberg group, ADE-classification, Prufer p-groups,
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13
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edited Dec 13 2009 at 17:43 |
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.)
In order to make it a more useful source, I list all the answers in categories, and added (so far to 1/5 1/3 of them) a date and a link to the answer. (~year means approximate year, *year means a year when an older example becomes central in view of some discovery.) Of course, if you see some important example missing, add it!
Logic and foundations: Alef_omega (~1890), Russell's paradox (1901),
Halting problem (1936), Goedel constructible universe L (1938), McKinsey formula in modal logic (~1941), 3SAT (*1970),
The *1970),The theory of Algebraically closed fields (ACF),
Physics: Brachistochrone problem (1696), The harmonic oscillator, Dirac's delta function (1927), Feynman path integral (1948),
Real and Complex Analysis: Harmonic series(14th Cen.) and Riemann zeta function (1859), li(x), The integral that launched Riemann surfaces, punctured open set in C^n
Partial differential equations: KdV equations (1877), Laplace equation, the heat equation, wave equation, Navier-Stokes equation,
Functional analysis: Unilateral shift, Tsirelson spaces (1974), Cuntz algebra,
Algebra: Z and Z/6Z, su_3 (su_2) Z[sqrt(-5)], full matrix ring over a ring, SU(2)
SU(2), p-adic numbers, quaternions, Young tableaux and Schur polynomials, Fischer-Griess monster, Heisenberg group, ADE-classification, Prufer p-groups,
Number Theory: conics and pythagorean triples (ancient),Fermat equation (1637), eliptic curves, Fermat hypersurfaces,
Probability: Normal distribution (1733), Brownian motion (1827), SLE (1999),
Dynamics: Logistic map (1845?), Mandelbrot set (1978/80) (Julia set)
Geometry: Platonic solids (ancient), the Euclidean ball(ancient), The configuration of 27 lines on a cubic surface, construction of regular heptadecagon (*1796), Hyperbolic geometry (1830), Reuleaux triangle, Fano plane, cyclic polytopes (1902), Leech lattice (1965), Penrose tiling (1974), noncommutative torus, cone of positive semidefinite matrices, the associahedron (1961)
Topology: Spheres, Figure-eight knot]7 (ancient) ,Cantor set, the torus, The standard embedding of the torus in R^3, Poincare dodecahedral sphere, Hopf fibration (1931), Complex projective space, trefoil knot, the cotangent bundle (?), Milnor exotic spheres, Alexander polynomial,homotopy group of spheres
Graph theory: Petersen Graph, two edge-colorings of K_6, K_33 and K_5, Tutte graph, Margulis's expanders and Ramanujan graphs,
Combinatorics: Catalan numbers, (Fibonacci sequence), tic-tac-toe (and nim), surreal numbers, alternating sign matrices
Algorithms and Computer Science: Newton Raphson method, RSA
Social Science: Prisoner dillema (1950), second price auction
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edited Dec 13 2009 at 11:38 |
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.)
In order to make it a more useful source, I list all the answers in categories, and added (so far to 1/5 of them) a date and a link to the answer. (~year means approximate year, *year means a year when an older example becomes central in view of some discovery.) Of course, if you see some important example missing, add it!
Logic and foundations: Alef_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), The harmonic oscillator, Dirac's delta function (1927), Feynman path integral (1948),
Real and Complex Analysis: Harmonic series(14th Cen.) and Riemann zeta function (1859), li(x), The integral that launched Riemann surfaces, punctured open set in C^n
Partial differential equations: KdV equations (1877), Laplace equation, the heat equation, wave equation, Navier-Stokes equation,
Functional analysis: Unilateral shift, Tsirelson spaces (1974), Cuntz algebra,
Algebra: Z and Z/6Z, su_3 (su_2) Z[sqrt(-5)], full matrix ring over a ring, SU(2)
p-adic numbers, quaternions, Young tableaux and Schur polynomials, Fischer-Griess monster,
Heisenberg group, ADE-classification, Prufer p-groups,
Number Theory: conics and pythagorean triples (ancient),Fermat equation (1637), eliptic curves, Fermat hypersurfaces,
Probability: Normal distribution (1733), Brownian motion (1827), SLE (1999),
Dynamics: Logistic map (1845?), Mandelbrot set (1978/80) (Julia set)
Geometry: Platonic solids (ancient), the Euclidean ball(ancient), The configuration of 27 lines on a cubic surface, construction of regular heptadecagon (*1796), Hyperbolic geometry (1830), Reuleaux triangle, Fano plane, cyclic polytopes (1902), Leech lattice (1965), Penrose tiling (1974), noncommutative torus, cone of positive semidefinite matrices, the associahedron (1961)
Topology: Spheres, Figure-eight knot]7 (ancient) ,Cantor set, the torus, The standard embedding of the torus in R^3, Poincare dodecahedral sphere, Hopf fibration (1931), Complex projective space, trefoil knot, the cotangent bundle (?), Milnor exotic spheres, Alexander polynomial,
Graph theory: Petersen Graph, two edge-colorings of K_6, K_33 and K_5, Tutte graph, Margulis's expanders and Ramanujan graphs,
Combinatorics: Catalan numbers, (Fibonacci sequence), tic-tac-toe (and nim), surreal numbers, alternating sign matrices
Algorithms and Computer Science: Newton Raphson method, RSA
Social Science: Prisoner dillema (1950), second price auction
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edited Dec 13 2009 at 11:32 |
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.)
In order to make it a more useful source, I list all the answers in categories, and added (so far to 1/5 of them) a date and a link to the answer. (~year means approximate year, *year means a year when an older example becomes central in view of some discovery.) Of course, if you see some important example missing, add it!
Logic and foundations: Alef_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), The harmonic oscillator, Dirac's delta function (1927), Feynman path integral (1948),
Real and Complex Analysis: Harmonic series(14th Cen.) and Riemann zeta function (1859), li(x), The integral that launched Riemann surfaces, punctured open set in C^n
Partial differential equations: KdV equations (1877), Laplace equation, the heat equation, wave equation, Navier-Stokes equation,
Functional analysis: Unilateral shift, Tsirelson spaces (1974), Cuntz algebra,
Algebra: Z and Z/6Z, su_3 (su_2) Z[sqrt(-5)], full matrix ring over a ring, SU(2)
p-adic numbers, quaternions, Young tableaux and Schur polynomials, Fischer-Griess monster,
Heisenberg group, ADE-classification, Prufer p-groups,
Number Theory: Fermat equation, eliptic curves, conics and pythagorean triples (ancient),Fermat equation (1637), eliptic curves, Fermat hypersurfaces,
Probability: Normal distribution , (1733), Brownian motion (1827), SLE (1999),
Dynamics: Logistic map, (1845?), Mandelbrot set (1978/80) (Julia set)
Geometry: Platonic solids (ancient), Hyperbolic geometrythe Euclidean ball(ancient), The configuration of 27 lines on a cubic surface, construction of regular heptadecagon (*1796), Hyperbolic geometry (1830), Reuleaux triangle, Fano plane, cyclic polytopes (1902), Leech lattice , the Euclidean ball, (1965), Penrose tiling , cyclic polytopes, (1974), noncommutative torus, cone of positive semidefinite matrices, Reuleaux triangle, the associahedron (1961)
Topology: Spheres, Figure-eight knot]7 (ancient) ,Cantor set, the torus, The standard embedding of the torus in R^3, Poincare dodecahedral sphere, Hopf fibration (1931), Complex projective space, trefoil knot, the cotangent bundle (?), Milnor exotic spheres, Alexander polynomial,
Graph theory: Petersen Graph, two edge-colorings of K_6, K_33 and K_5, Tutte graph, Margulis's expanders and Ramanujan graphs,
Combinatorics: Catalan numbers, (Fibonacci sequence), tic-tac-toe (and nim), alternating sign matrices
Algorithms and Computer Science: Newton Raphson method, RSA
Social Science: Prisoner dillema, (1950), second price auction
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edited Dec 13 2009 at 9:02 |
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.)
In order to make it a more useful source, I list all the answers in categories, and added (so far to 1/5 of them) a date and a link to the answer. (~year means approximate year, *year means a year when an older example becomes central in view of some discovery.) Of course, if you see some important example missing, add it!
Logic and foundations: Alef_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), The harmonic oscillator, Dirac's delta function (1927), Feynman path integral (1948),
Real and Complex Analysis: Harmonic series(14th Cen.) and Riemann zeta function (1859), li(x), The integral that launched Riemann surfaces, punctured open set in C^n
Partial differential equations: KdV equations (1877), Laplace equation, the heat equation, wave equation, Navier-Stokes equation,
Functional analysis: Unilateral shift, Tsirelson spaces , (1974), Cuntz algebra,
Algebra: Z and Z/6Z, su_3 (su_2) Z[sqrt(-5)], full matrix ring over a ring, SU(2)
p-adic numbers, quaternions, Young tableaux and Schur polynomials, Fischer-Griess monster,
Heisenberg group, ADE-classification, Prufer p-groups,
Number Theory: Fermat equation, eliptic curves, conics and pythagorean triples, Fermat hypersurfaces,
Probability: Normal distribution, Brownian motion, (1827), SLE ,(1999),
Dynamics: Logistic map, Mandelbrot set (Julia set)
Geometry: Platonic solids , (ancient), Hyperbolic geometry, The configuration of 27 lines on a cubic surface, construction of regular heptadecagon, Fano plane, Leech lattice, the Euclidean ball, Penrose tiling, cyclic polytopes, noncommutative torus, cone of positive semidefinite matrices, Reuleaux triangle, the associahedron
Topology: Spheres, Figure-eight knot]7 (ancient) ,Cantor set, the torus, The standard embedding of the torus in R^3, Poincare dodecahedral sphere, Hopf fibration (1931), Complex projective space, trefoil knot, the cotangent bundle (?), Milnor exotic spheres, Alexander polynomial,
Graph theory: Petersen Graph, two edge-colorings of K_6, K_33 and K_5, Tutte graph, Margulis's expanders and Ramanujan graphs,
Combinatorics: Catalan numbers, (Fibonacci sequence), tic-tac-toe (and nim), alternating sign matrices
Algorithms and Computer Science: Newton Raphson method, RSA
Social Science: Prisoner dillema, second price auction
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edited Dec 13 2009 at 8:11 |
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.)
In order to make it a more useful source, I list all the answers in categories, and added (so far to 1/7 1/5 of them) a date and a link to the answer. (~year means approximate year, *year means a year when an older example becomes central in view of some discovery.) Of course, if you see some important example missing, add it!
Logic and foundations: Alef_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), The harmonic oscillator, Dirac's delta function (1927), Feynman path integral (1948),
Real and Complex Analysis: Harmonic series(14th Cen.) and Riemann zeta function (1859), li(x), The integral that launched Riemann surfaces, punctured open set in C^n
Partial differential equations: KdV equations, (1877), Laplace equation, the heat equation, wave equation, Navier-Stokes equation,
Functional analysis: Unilateral shift, Tsirelson spaces, Cuntz algebra,
Algebra: Z and Z/6Z, su_3 (su_2) Z[sqrt(-5)], full matrix ring over a ring, SU(2)
p-adic numbers, quaternions, Young tableaux and Schur polynomials, Fischer-Griess monster,
Heisenberg group, ADE-classification, Prufer p-groups,
Number Theory: Fermat equation, eliptic curves, conics and pythagorean triples, Fermat hypersurfaces,
Probability: Normal distribution, Brownian motion, SLE,
Dynamics: Logistic map, Mandelbrot set (Julia set)
Geometry: Platonic solids, Hyperbolic geometry, The configuration of 27 lines on a cubic surface, construction of regular heptadecagon, Fano plane, Leech lattice, the Euclidean ball, Penrose tiling, cyclic polytopes, noncommutative torus, cone of positive semidefinite matrices, Reuleaux triangle, the associahedron
Topology: Spheres, Figure-eight knot]7 (ancient) ,Cantor set, the torus, The standard embedding of the torus in R^3, Poincare dodecahedral sphere, Hopf fibration, (1931), Complex projective space,Figure-eight knotspace, trefoil knot, the cotangent bundle, Milnot (?), Milnor exotic spheres, Alexander polynomial,
Graph theory: Petersen Graph, two edge-colorings of K_6, K_33 and K_5, Tutte graph, Margulis's expanders and Ramanujan graphs,
Combinatorics: Catalan numbers, (Fibonacci sequence), tic-tac-toe (and nim), alternating sign matrices
Algorithms and Computer Science: Newton Raphson method, RSA
Social Science: Prisoner dillema, second price auction
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edited Dec 13 2009 at 7:56 |
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.)
In order to make it a more useful source, I list all the answers in categories, and added (so far to 1/7 of them) a date and a link to the answer. (~year means approximate year, *year means a year when an older example becomes central in view of some discovery.) Of course, if you see some important example missing, add it!
Logic and foundations: Alef_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), The harmonic oscillator, Dirac's delta function (1927), Feynman path integral (1948),
Real and Complex Analysis: Harmonic series(14th Cen.) and Riemann zeta function (1859), li(x), The integral that launched Riemann surfaces, punctured open set in C^n
Partial differential equations: KdV equations, Laplace equation, the heat equation, wave equation, Navier-Stokes equation,
Functional analysis: Unilateral shift, Tsirelson spaces, Cuntz algebra,
Algebra: Z and Z/6Z, su_3 (su_2) Z[sqrt(-5)], full matrix ring over a ring, SU(2)
p-adic numbers, quaternions, Young tableaux and Schur polynomials, Fischer-Griess monster,
Heisenberg group, ADE-classification, Prufer p-groups,
Number Theory: Fermat equation, eliptic curves, conics and pythagorean triples, Fermat hypersurfaces,
Probability: Normal distribution, Brownian motion, SLE,
Dynamics: Logistic map, Mandelbrot set (Julia set)
Geometry: Platonic solids, Hyperbolic geometry, The configuration of 27 lines on a cubic surface, construction of regular heptadecagon, Fano plane, Leech lattice, the Euclidean ball, Penrose tiling, cyclic polytopes, noncommutative torus, cone of positive semidefinite matrices, Reuleaux triangle, the associahedron
Topology: Spheres, Cantor set, the torus, The standard embedding of the torus in R^3, Poincare dodecahedral sphere, Hopf fibration, Complex projective space,Figure-eight knot, trefoil knot, cotangent bundle, Milnot exotic spheres, Alexander polynomial,
Graph theory: Petersen Graph, Fano plane, two edge-colorings of K_6, K_33 and K_5, Tutte graph, Margulis's expanders and Ramanujan graphs,
Combinatorics: Catalan numbers, (Fibonacci sequence), tic-tac-toe (and nim), alternating sign matrices
Algorithms and Computer Science: Newton Raphson method, RSA
Social Science: Prisoner dillema, second price auction
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edited Dec 11 2009 at 8:43 |
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.)
In order to make it a more useful source, I list all the answers in categories, and added (so far to 1/7 of them) a date and a link to the answer. (~year means approximate year, *year means a year when an older example becomes central in view of some discovery.) Of course, if you see some important example missing, add it!
Logic and foundations: Alef_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), The harmonic oscillator, Dirac's delta function (1927), Feynman path integral (1948),
Real and Complex Analysis: Harmonic series(14th Cen.) and Riemann zeta function (1859), li(x), The integral that launched Riemann surfaces, punctured open set in C^n
Partial differential equations: KdV equations, Laplace equation, the heat equation, wave equation, Navier-Stokes equation,
Functional analysis: Unilateral shift, Tsirelson spaces, Cuntz algebra,
Algebra: Z and Z/6Z, su_3 (su_2) Z[sqrt(-5)], full matrix ring over a ring, SU(2)
p-adic numbers, quaternions, Young tableaux and Schur polynomials, Fischer-Griess monster,
Heisenberg group, ADE-classification, Prufer p-groups,
Number Theory: Fermat equation, eliptic curves, conics and pythagorean triples, Fermat hypersurfaces,
Probability: Normal distribution, normal distributionBrownian motion, SLE,
Dynamics: Logistic map, Mandelbrot set (Julia set)
Geometry: Platonic solids, Hyperbolic geometry, The configuration of 27 lines on a cubic surface, construction of regular heptadecagon, Leech lattice, the Euclidean ball, Penrose tiling, cyclic polytopes, noncommutative torus, cone of positive semidefinite matrices, Reuleaux triangle, the associahedron
Topology: Spheres, Cantor set, the torus, The standard embedding of the torus in R^3, Poincare dodecahedral sphere, Hopf fibration, Complex projective space,Figure-eight knot, trefoil knot, cotangent bundle, Milnot exotic spheres, Alexander polynomial,
Graph theory: Petersen Graph, Fano plane, two edge-colorings of K_6, K_33 and K_5, Tutte graph, Margulis's expanders and Ramanujan graphs,
Combinatorics: Catalan numbers, (Fibonacci sequence), tic-tac-toe (and nim), alternating sign matrices
Algorithms and Computer Science: Newton Raphson method, RSA
Social Science: Prisoner dillema, second price auction
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edited Dec 11 2009 at 8:28 |
In order to make it a more useful source, I list all the answers in categories, and added (so far to 1/7 of them) a date and a link to the answer. (~year means approximate year, *year means a year when an older example becomes central in view of some discovery.) Of course, if you see some important example missing, add it! Logic and foundations: Alef_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), The harmonic oscillator, Dirac's delta function (1927), Feynman path integral (1948), Real and Complex Analysis: Harmonic series(14th Cen.) and Riemann zeta function (1859), li(x), The integral that launched Riemann surfaces, punctured open set in C^n Partial differential equations: KdV equations, Laplace equation, the heat equation, wave equation, Navier-Stokes equation, Functional analysis: Unilateral shift, Tsirelson spaces, Cuntz algebra, Algebra: Z and Z/6Z, su_3 (su_2) Z[sqrt(-5)], full matrix ring over a ring, SU(2)p-adic numbers, quaternions, Young tableaux and Schur polynomials, Fischer-Griess monster,Heisenberg group, ADE-classification, Prufer p-groups, Number Theory: Fermat equation, eliptic curves, conics and pythagorean triples, Fermat hypersurfaces, Probability: Normal distribution, normal distribution, SLE, Dynamics: Logistic map, Mandelbrot set (Julia set) Geometry: Platonic solids, Hyperbolic geometry, The configuration of 27 lines on a cubic surface, construction of regular heptadecagon, Leech lattice, the Euclidean ball, Penrose tiling, cyclic polytopes, noncommutative torus, cone of positive semidefinite matrices, Reuleaux triangle, the associahedron Topology: Spheres, Cantor set, the torus, The standard embedding of the torus in R^3, Poincare dodecahedral sphere, Hopf fibration, Complex projective space,Figure-eight knot, trefoil knot, cotangent bundle, Milnot exotic spheres, Alexander polynomial, Graph theory: Petersen Graph, Fano plane, two edge-colorings of K_6, K_33 and K_5, Tutte graph, Margulis's expanders and Ramanujan graphs, Combinatorics: Catalan numbers, (Fibonacci sequence), tic-tac-toe (and nim), alternating sign matrices Algorithms and Computer Science: Newton Raphson method, RSA Social Science: Prisoner dillema, second price auction
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edited Nov 27 2009 at 12:17 |
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 is was offered.)
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edited Nov 22 2009 at 22:12 |
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 is offered.)
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edited Nov 21 2009 at 17:08 |
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 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 is offered.)
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edited Nov 12 2009 at 9:37 |
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Post Made Community Wiki by Scott Morrison♦
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occurred Nov 11 2009 at 19:33
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asked Nov 11 2009 at 7:52 |
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 per post, please)
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