Comments on: 254A, Notes 6: Ultraproducts as a bridge between hard analysis and soft analysis
http://terrytao.wordpress.com/2011/10/15/254a-notes-6-ultraproducts-as-a-bridge-between-hard-analysis-and-soft-analysis/
Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence TaoFri, 29 Aug 2014 22:21:55 +0000hourly1http://wordpress.com/By: Ultraproducts as a Bridge Between Discrete and Continuous Analysis | What's new
http://terrytao.wordpress.com/2011/10/15/254a-notes-6-ultraproducts-as-a-bridge-between-hard-analysis-and-soft-analysis/#comment-255857
Sun, 08 Dec 2013 00:05:44 +0000http://terrytao.wordpress.com/?p=5380#comment-255857[…] analysis“. Some of the material here is drawn from previous blog posts, notably “Ultraproducts as a bridge between hard analysis and soft analysis” and “Ultralimit analysis and quantitative algebraic geometry“‘. The text […]
]]>By: Rectification and the Lefschetz principle | What's new
http://terrytao.wordpress.com/2011/10/15/254a-notes-6-ultraproducts-as-a-bridge-between-hard-analysis-and-soft-analysis/#comment-219475
Thu, 14 Mar 2013 21:17:45 +0000http://terrytao.wordpress.com/?p=5380#comment-219475[...] can be used as a bridge to connect quantitative and qualitative results (as discussed in these previous blog posts), we will deduce Theorem 3 from the following (well-known) qualitative [...]
]]>By: Multiple recurrence in quasirandom groups « What’s new
http://terrytao.wordpress.com/2011/10/15/254a-notes-6-ultraproducts-as-a-bridge-between-hard-analysis-and-soft-analysis/#comment-197379
Thu, 29 Nov 2012 01:32:06 +0000http://terrytao.wordpress.com/?p=5380#comment-197379[...] this mixing theorem. Instead, we had to first use the machinery of ultraproducts (as discussed in this previous post) to convert the finitary statement about a quasirandom group to an infinitary statement about a [...]
]]>By: Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets « What’s new
http://terrytao.wordpress.com/2011/10/15/254a-notes-6-ultraproducts-as-a-bridge-between-hard-analysis-and-soft-analysis/#comment-192677
Wed, 14 Nov 2012 17:42:13 +0000http://terrytao.wordpress.com/?p=5380#comment-192677[...] into equivalent qualitative, infinitary problems in zero characteristic (in the spirit of this blog post). This allows one to use several tools from those fields as “black boxes”; not just the [...]
]]>By: Walsh’s ergodic theorem, metastability, and external Cauchy convergence « What’s new
http://terrytao.wordpress.com/2011/10/15/254a-notes-6-ultraproducts-as-a-bridge-between-hard-analysis-and-soft-analysis/#comment-186993
Sun, 28 Oct 2012 20:24:16 +0000http://terrytao.wordpress.com/?p=5380#comment-186993[...] will assume some familiarity with nonstandard analysis, as covered for instance in these previous blog [...]
]]>By: Definable subsets over (nonstandard) finite fields, and almost quantifier elimination « What’s new
http://terrytao.wordpress.com/2011/10/15/254a-notes-6-ultraproducts-as-a-bridge-between-hard-analysis-and-soft-analysis/#comment-169850
Wed, 12 Sep 2012 23:36:33 +0000http://terrytao.wordpress.com/?p=5380#comment-169850[...] this ultraproduct. (We will be using the notation of ultraproducts and nonstandard analysis from this previous blog post.) This approach is equivalent to the more finitary approach mentioned in the previous paragraph, at [...]
]]>By: A cheap version of nonstandard analysis « What’s new
http://terrytao.wordpress.com/2011/10/15/254a-notes-6-ultraproducts-as-a-bridge-between-hard-analysis-and-soft-analysis/#comment-136449
Tue, 03 Apr 2012 00:55:01 +0000http://terrytao.wordpress.com/?p=5380#comment-136449[...] an ultrapower of with respect to some non-principal ultrafilter over the natural numbers; see e.g. this blog post for details. Once one is comfortable with ultrafilters and ultrapowers, this becomes quite a simple [...]
]]>By: 254B, Notes 5: Product theorems, pivot arguments, and the Larsen-Pink non-concentration inequality « What’s new
http://terrytao.wordpress.com/2011/10/15/254a-notes-6-ultraproducts-as-a-bridge-between-hard-analysis-and-soft-analysis/#comment-126691
Sun, 05 Feb 2012 19:33:45 +0000http://terrytao.wordpress.com/?p=5380#comment-126691[...] use the tool of ultraproducts (which we will phrase in the language of nonstandard analysis). See this previous blog post for a discussion of ultraproducts and how they can be used to turn quantitative (or [...]
]]>By: Anonymous
http://terrytao.wordpress.com/2011/10/15/254a-notes-6-ultraproducts-as-a-bridge-between-hard-analysis-and-soft-analysis/#comment-104153
Sat, 12 Nov 2011 02:28:02 +0000http://terrytao.wordpress.com/?p=5380#comment-104153In definition 5, in the definition of a standard k-ary relation, the R_k should be replaced with X_k.

[Corrected, thanks - T.]

]]>By: Ben Hayes
http://terrytao.wordpress.com/2011/10/15/254a-notes-6-ultraproducts-as-a-bridge-between-hard-analysis-and-soft-analysis/#comment-104066
Fri, 11 Nov 2011 23:33:47 +0000http://terrytao.wordpress.com/?p=5380#comment-104066In Exercise 19, should $N$ be strictly non-standard? I think in the case $N=1,$ then the quotient group is just $Z.$