Comments on: The structure of approximate groups
http://terrytao.wordpress.com/2011/10/24/the-structure-of-approximate-groups/
Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence TaoSat, 01 Nov 2014 03:27:12 +0000hourly1http://wordpress.com/By: Additive limits | What's new
http://terrytao.wordpress.com/2011/10/24/the-structure-of-approximate-groups/#comment-429518
Sun, 12 Oct 2014 19:57:33 +0000http://terrytao.wordpress.com/?p=5403#comment-429518[…] is likely that this theorem can be extended to the noncommutative setting, using the noncommutative Freiman theorem of Emmanuel Breuillard, Ben Green, and myself, but I have not attempted to do so here; in a separate direction, there should be extensions that […]
]]>By: Carry propagation-free number systems and a kind of approximate groups (I) | chorasimilarity
http://terrytao.wordpress.com/2011/10/24/the-structure-of-approximate-groups/#comment-225242
Sun, 21 Apr 2013 08:13:03 +0000http://terrytao.wordpress.com/?p=5403#comment-225242[...] (that’s what qualifies as a kind of an approximate group). [...]
]]>By: Small doubling in groups « What’s new
http://terrytao.wordpress.com/2011/10/24/the-structure-of-approximate-groups/#comment-215443
Fri, 01 Feb 2013 18:19:52 +0000http://terrytao.wordpress.com/?p=5403#comment-215443[...] have Lie structure; this connection was first observed and exploited by Hrushovski, and then used by Breuillard, Green, and myself to obtain the analogue of Freiman’s theorem for an arbitrary nonabelian [...]
]]>By: Approximate groupoids again « chorasimilarity
http://terrytao.wordpress.com/2011/10/24/the-structure-of-approximate-groups/#comment-212074
Fri, 04 Jan 2013 15:05:54 +0000http://terrytao.wordpress.com/?p=5403#comment-212074[...] Here is the path I would like to pursue further. The notion of approximate groupoid (see here for the definition) is not complete, because it is flattened, i.e. the set of arrows should be seen as a set of variables. What I think is that the correct notion of approximate groupoid is a polynomial functor over groupoids (precisely a specific family of such functors). The category Grpd is cartesian closed, so it has an associated model of (typed) lambda calculus. By using this observation I could apply emergent algebra techniques (under the form of my graphic lambda calculus, which was developed with — and partially funded by – this application in mind) to approximate groupoids and hope to obtain streamlined proofs of Breuillard-Green-Tao type results. [...]
]]>By: A geometric viewpoint on computation? | chorasimilarity
http://terrytao.wordpress.com/2011/10/24/the-structure-of-approximate-groups/#comment-143717
Sun, 20 May 2012 11:13:55 +0000http://terrytao.wordpress.com/?p=5403#comment-143717[...] the profound resemblance between geometrical results of Gromov on groups with polynomial growth and combinatorial results of Breuillard, Gree, Tao on approximate groups? In both cases a nilpotent structure emerges from considering larger and larger scales. The word [...]
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http://terrytao.wordpress.com/2011/10/24/the-structure-of-approximate-groups/#comment-137326
Mon, 09 Apr 2012 11:57:39 +0000http://terrytao.wordpress.com/?p=5403#comment-137326Thank you for the answer, very interesting! A Gleason metric is a CC metric if and only if the nilpotent group is abelian, because of the commutator estimate.
]]>By: 254A, addendum: Some notes on nilprogressions « What’s new
http://terrytao.wordpress.com/2011/10/24/the-structure-of-approximate-groups/#comment-133749
Sun, 18 Mar 2012 05:12:38 +0000http://terrytao.wordpress.com/?p=5403#comment-133749[...] groups, and use it to prove the above proposition, which turns out to be a bit tricky. (In my paper with Breuillard and Green, we avoid the need for this proposition by restricting attention to a special type of [...]
]]>By: A nilpotent Freiman dimension lemma | t1u
http://terrytao.wordpress.com/2011/10/24/the-structure-of-approximate-groups/#comment-117595
Thu, 22 Dec 2011 00:50:03 +0000http://terrytao.wordpress.com/?p=5403#comment-117595[...] remark that our previous paper established a similar result, in which the dimension bound was improved to , but at the cost of [...]
]]>By: A nilpotent Freiman dimension lemma « What’s new
http://terrytao.wordpress.com/2011/10/24/the-structure-of-approximate-groups/#comment-117588
Thu, 22 Dec 2011 00:24:08 +0000http://terrytao.wordpress.com/?p=5403#comment-117588[...] remark that our previous paper established a similar result, in which the dimension bound was improved to , but at the cost of [...]
]]>By: Approximate algebraic structures, emergent algebras | chorasimilarity
http://terrytao.wordpress.com/2011/10/24/the-structure-of-approximate-groups/#comment-114044
Fri, 09 Dec 2011 15:52:47 +0000http://terrytao.wordpress.com/?p=5403#comment-114044[...] algebra can also be seen as an approximate algebraic structure! But in a different sense than approximate groups. The operations themselves are approximately associative, for [...]
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