most recent 30 from http://mathoverflow.net 2013-05-22T20:03:29Z http://mathoverflow.net/feeds/question/4121 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/4121/spec-z-analogue-of-thurston-program Ilya Nikokoshev 2009-11-04T17:25:13Z 2009-11-16T10:42:08Z

<p>It's been known for a while that primes in number fields can be thought of, from an algebraic point of view, to be similar to knots in 3-manifolds. A good reference (thanks to <a href="http://mathoverflow.net/questions/4075/questions-about-analogy-between-spec-z-and-3-manifolds" rel="nofollow">this question</a>) would be an article by Morishita, <a href="http://arxiv.org/abs/0904.3399" rel="nofollow">0904.3399</a>. </p> <p>There are therefore many good analogues of operations, such as covers, or objects, like zeta-functions, that are defined purely algebraically. For example, a <em>linking number</em> of two knots has an easy algebraic definition as the image of one knot in the homology of the complement to the other which is analogous to <em>residue symbol</em> in number theory.</p> <p>However, the operations of taking connected sum and cutting/gluing along a subsurface don't appear immediately to have an analogue in number fields. If you know how to make sense of "gluing" two schemes <img src="http://latex.mathoverflow.net/png?%5Cmathrm%7BSpec%7D%5C%2CO%5FK" alt="\mathrm{Spec}\,O K" title="" /> and <img src="http://latex.mathoverflow.net/png?%5Cmathrm%7BSpec%7D%5C%2CO%5FL" alt="\mathrm{Spec}\,O L" title="" /> along the "common element <img src="http://latex.mathoverflow.net/png?x%20%5Cin%20K%2C%20L" alt="x \in K, L" title="" />, by all means, please tell us!</p> <p>Either way, here's my question:</p> <blockquote> <p>What could be an analogue of the <a href="http://en.wikipedia.org/wiki/Geometrization%5Fconjecture" rel="nofollow">Thurston geometrization program</a> for number fields?</p> </blockquote> <p>(may be this analogue will not be using gluing-like operations after all?)</p>

http://mathoverflow.net/questions/4121/spec-z-analogue-of-thurston-program/4811#4811 Sam Nead 2009-11-10T04:33:45Z 2009-11-10T04:33:45Z

<p>McMullen's third lecture at the 2000 Washington DC AMS Colloquium is exactly addressing this question. See his slides at</p> <p><a href="http://www.math.harvard.edu/~ctm/expositions/home/text/talks/ams/dc00/html/index.html" rel="nofollow">http://www.math.harvard.edu/~ctm/expositions/home/text/talks/ams/dc00/html/index.html</a></p>

http://mathoverflow.net/questions/4121/spec-z-analogue-of-thurston-program/5586#5586 Thomas Riepe 2009-11-14T22:23:51Z 2009-11-16T10:42:08Z

<p>I was surprised that there exist even arithmetic analogies to solitons (<a href="http://terrytao.wordpress.com/2008/02/19/why-are-solitons-stable/" rel="nofollow" title="Terry Tao's blog">more</a>) and <a href="http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1987__65__131_0" rel="nofollow" title="numdam">Laumon</a>'s arithmetic version of an idea of Witten made a new proof of Weil II possible. What else may have arithmetic versions? <a href="http://math.berkeley.edu/~lott/ricciflow/perelman.html" rel="nofollow" title="Perelman repository">Ricci flow</a>?</p> <p>On the Ricci flow-renormalization issue in the comments below, <a href="http://golem.ph.utexas.edu/category/2007/02/huisken_on_uniformization.html#c029255" rel="nofollow" title="n-cat cafe">Urs Schreibers answer</a>, an other expert yesterday: "The renormalization that is involved is not the same as in QFT, except for the fact that it can also be thought as realizing, in that geometric context, a subtraction of divergences that has the effect of keeping the flow solutions from blowing up. Whether there is in that context any role for algebraic structures of renormalization, such as Hopf algebras accounting for nested divergences, is a good question."</p>