Organizing principles of mathematics - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T19:09:11Z http://mathoverflow.net/feeds/question/48692 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48692/organizing-principles-of-mathematics Organizing principles of mathematics Nikita Kalinin 2010-12-08T23:41:38Z 2010-12-23T05:22:14Z <p>In his famous paper <a href="http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf" rel="nofollow">"The two cultures of Mathematics"</a> T. Gowers gives examples of organizing principles in combinatorics.</p> <p>(i) Obviously if events \$E1, \cdots,E_n\$ are independent and have non-zero probability, then with non-zero probability they all happen at once. In fact, this can be usefully true even if there is a very limited dependence. [EL,J]</p> <p>(ii) All graphs are basically made out of a few random-like pieces, and we know how those behave. [Sze]</p> <p>(iii) If one is counting solutions, inside a given set, to a linear equation, then it is enough, and usually easier, to estimate Fourier coefficients of the characteristic function of the set.</p> <p>(iv) Many of the properties associated with random graphs are equivalent, and can therefore be taken as sensible definitions of pseudo-random graphs. [CGW,T]</p> <p>(v) Sometimes, the set of all eventually zero sequences of zeros and ones is a good model for separable Banach spaces, or at least allows one to generate interesting hypotheses.</p> <p>(vi) Concentration of Measure</p> <p>More examples (by Tao and other) you can see at <a href="http://ncatlab.org/davidcorfield/show/Two+Cultures" rel="nofollow">http://ncatlab.org/davidcorfield/show/Two+Cultures</a></p> <p>Do you know another examples in various areas? I mean, for example, globalization techniques in topology (structure functor in Hirsh, <em>Differential Topology</em>, \$\S 2.11\$ and Mayer–Vietoris sequence, in Bott &amp; Tu, <em>Differential Forms in Algebraic Topology</em> \$\S 5\$).</p> <p>So, many proofs look like "prove the local version of theorem and globalize".</p> <p>Do you know such principles? It should be more specific than undergraduate course but it should be common used in your branch and be situated in "common wisdom" of mathematics.</p> http://mathoverflow.net/questions/48692/organizing-principles-of-mathematics/48715#48715 Answer by anon for Organizing principles of mathematics anon 2010-12-09T05:18:17Z 2010-12-09T05:18:17Z <p>The Choquet theory in convex analysis / functional analysis / whatever you want to call it. An element of a convex set should be some kind of "average" of extreme points. This has the status of a theorem for compact sets in normed linear spaces but is a useful guiding principle for not-necessarily-compact sets in not-necessarily-normed linear spaces. Chapter 14 in Lax's Functional Analysis book gives good examples of the wide array of applications of the same simple idea.</p>