Number of ways to construct mathematical objects - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:55:02Z http://mathoverflow.net/feeds/question/39783 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39783/number-of-ways-to-construct-mathematical-objects Number of ways to construct mathematical objects Thierry Zell 2010-09-23T18:35:28Z 2010-09-25T07:36:52Z <p>This question stems from this other one mentioning <a href="http://mathoverflow.net/questions/28948/gromovs-list-of-7-constructions-in-differential-topology" rel="nofollow">7 ways of constructing smooth manifolds</a>. I quote:</p> <blockquote> <p>At the 2010 Clay Research Conference, Gromov explained that we know of only 7 different methods for constructing smooth manifolds.[...]</p> <ul> <li> Algebraic geometry (affine and projective varieties, ...) <li> Lie groups (homogeneous spaces, ...) <li> General position arguments (Morse theory, Pontryagin-Thom construction, ...) <li> Solutions to PDE (Moduli spaces in gauge theory, Floer theory, ...) <li> Surgery (Cut and paste techniques, ...) <li> Markov processes <li> [and also bundles seems to be the consensus in the answers to the cited question] </ul> </blockquote> <p>Note that all of these methods are actually areas of mathematics in their own right (five of the six listed in that question involve trailing dots), so that got me thinking that 7 methods is actually a sign of a rich subject and a fairly ubiquitous concept. I have trouble comparing this to anything else: e.g., would you say that there are fewer ways to build a group? I don't know nearly enough about groups to answer that one. So here's my question:</p> <p><strong>Do you know of an abstract math construct that can be built in truly more than seven ways?</strong></p> <p>I realize that this is somewhat in the eye of the beholder (hence the soft question tag) since it may not be obvious where to draw the line between methods. But for the comparison to make sense, you need to consider broad categories like Gromov does.</p> http://mathoverflow.net/questions/39783/number-of-ways-to-construct-mathematical-objects/39784#39784 Answer by dfranke for Number of ways to construct mathematical objects dfranke 2010-09-23T18:58:55Z 2010-09-23T20:45:31Z <p>How about models of computation?</p> <ul> <li>Turing machines</li> <li>Register machines</li> <li>Cellular automata</li> <li>$\mu$-recursive functions</li> <li>The untyped $\lambda$-calculus</li> <li>Unrestricted grammars</li> <li>Term-rewriting systems</li> <li>Recursively-enumerable subsets of $\mathbb{N}$</li> <li>$\ldots$</li> </ul> <p>To this list we can add finite presentations of semigroups and groups, diophantine sets, and many other mathematical structures that realize universal computation.</p> http://mathoverflow.net/questions/39783/number-of-ways-to-construct-mathematical-objects/39918#39918 Answer by Mark Sapir for Number of ways to construct mathematical objects Mark Sapir 2010-09-25T00:02:42Z 2010-09-25T07:36:52Z <p>There are lots of different ways to construct a finitely generated (discrete) group. For example (in no particular order):</p> <ol> <li><p>Fundamental groups of topological spaces;</p></li> <li><p>Groups of symmetries of mathematical objects (including fields, manifolds, simplicial, real and $\Lambda$-trees, etc.);</p></li> <li><p>Groups given by presentations satisfying various small cancelation conditions (from Tartakovsky to Olshanskii to Gromov);</p></li> <li><p>Lattices in Lie groups;</p></li> <li><p>Wreath products of various kinds;</p></li> <li><p>Direct limits of sequences of groups and their homomorphisms (including various "monsters", etc.);</p></li> <li><p>Free constructions (HNN extensions and amalgamated products);</p></li> <li><p>Groups simulating various computing devices (there are several different constructions here);</p></li> <li><p>Groups acting on locally finite rooted trees (including Grigorchuk groups and iterated monodromy groups of Nekrashevych);</p></li> </ol> <p>10 Automatic groups,</p> <p>..................</p>