via Matematica - Forum ScienzeMatematiche.it on 2/8/10
di salvo.tringali il dom 07 feb 2010, 17:24 in Analisi Matematica


La questione che intendo proporre è il prolungamento ideale di un problema posto solo ieri dal buon KB, riguardo agli zeri di una funzione olomorfa (qui). Ne abbiamo discusso un po' sulla chat del forum, ed è emerso chiaramente che la proprietà topologicamente rilevante è la connessione (per via del teorema di identità delle funzioni analitiche). Da cui...

Self-posed & solved (surely well-known). Dopo aver provato che la produttoria $\prod_{n=1}^\infty (1 + nz^n)$ converge nell'insieme $\Omega := \{z \in \mathbb{C}: |z| < 1\}$, mostrare che la funzione $\Omega \to \mathbb{C}: z \mapsto \prod_{n=1}^\infty (1 + nz^n)$ è olomorfa, e che l'insieme $Z(f)$ dei suoi zeri è (limitato e) infinito.

via Combinatorics and more by Gil Kalai on 2/2/10

 Click on the picture!

More riddles on this site:

Two math riddles

Greg’s dinosaur riddle

A riddle

The mystery beeping riddle

What can the scond prize possibly be

via Mathematics and Computation by Andrej Bauer on 1/7/10

Already a while ago videolectures.net published this tutorial on Computer Verified Exact Analysis by Bas Spitters and Russell O’Connor from Computability and Complexity in Analysis 2009. I forgot to advertise it, so I am doing this now. It is about an implementation of exact real arithmetic whose correctness has been verified in Coq. Russell also gave a quick tutorial on Coq.

via Mathematics and Computation by Andrej Bauer on 1/6/10

With Davorin Lešnik.

Abstract: We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Russian Constructivism the match between synthetic and metric topology breaks down, as even a very simple complete totally bounded space fails to be compact, and its topology is strictly finer than the metric topology. In contrast, in Brouwer’s intuitionism synthetic and metric notions of topology and compactness agree.

Download paper: csms_in_synthtop.pdf

via Combinatorics and more by Gil Kalai on 1/31/10

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? 

I’d love to learn about further basic or central examples and I think such examples serve as good invitations to various areas.

I asked this question over mathoverflow and it yielded around 100 examples. They are not equally fundamental and they are not equally suitable to be regarded as “examples,” but overall it is a very good list.  If you see some important example missing please, please add it.  Here are the examples classified to areas. (Of course, sometimes, the same example may fit several areas.)

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), 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 (*1854?), Chebyshev polynomials (?1854) punctured open set in C^n</a> (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, 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_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,

Number Theory:

 

Conics and pythagorean triples (ancient), Fermat equation (1637), eliptic curves, Fermat hypersurfaces,

Probability:

Normal distribution (1733), Brownian motion (1827), The Gaussian Orthogonal Ensemble, the Gaussian Unitary Ensemble, and the Gaussian Symplectic Ensemble, 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 (19th century), Fano plane (early 20th century ??), cyclic polytopes (1902), Delaunay triangulation (1934) Leech lattice (1965), Penrose tiling (1974), the 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?), 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)

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(?)), Fibonacci sequence (12th century; probably ancient),  Catalan numbers (mid 19th century), 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 dillema (1950), second price auction (1961)

The (partial) links are to the answers over MO which often have more information and external links (additions are moset welcome). I tried to find dates for the various examples and this was not easy. Corrections and additions are welcome! A date with a * like  *1970 refers to a date when this example had become important in view of a certain development. (E.g  3SAT (*1970) refers to the discovery of P,NP, and NP-completeness.)