Comments on: 254A, Notes 2: Building Lie structure from representations and metrics

By: The closed graph theorem in various categories « What’s new

The closed graph theorem in various categories « What’s new — Wed, 21 Nov 2012 02:31:05 +0000

[...] There is also a number of closed graph theorems for topological groups, of which the following is typical (see Exercise 3 of these previous blog notes): [...]

By: 254A, Notes 8: The microstructure of approximate groups « What’s new

254A, Notes 8: The microstructure of approximate groups « What’s new — Thu, 01 Dec 2011 15:58:13 +0000

[...] was a key difficulty in the theory surrounding Hilbert’s fifth problem that was discussed in previous notes. A key tool in being able to resolve such structure was to build left-invariant metrics (or [...]

By: 254A, Notes 7: Models of ultra approximate groups « What’s new

254A, Notes 7: Models of ultra approximate groups « What’s new — Fri, 28 Oct 2011 04:37:20 +0000

[...] create a metric structure on strong approximate groups analogous to the Gleason metrics studied in previous notes, which can in turn be exploited (together with an induction on dimension argument) to fully [...]

By: Ben Hayes

Ben Hayes — Mon, 17 Oct 2011 00:43:11 +0000

Ah, when I go through the estimates I see the contradiction now. Thanks.

By: Terence Tao

Terence Tao — Sun, 16 Oct 2011 22:51:04 +0000

If exceeds 1/C, then one works instead with the largest positive integer m for which ; this will be an integer between 1 and n, and for small enough one can show that one must have to avoid a contradiction.

(More generally, one often encounters “almost circular reasoning” issues in these sorts of problems, in which one almost has to assume the result one is trying to prove as a hypothesis. Often, one has to deal with the apparent circularity by an induction argument, a bootstrap argument, a continuity argument, an infinite descent argument, or passing to the “first counterexample” or “last example”. The above is an instance of the latter strategy.)

By: Ben Hayes

Ben Hayes — Sun, 16 Oct 2011 18:26:40 +0000

In Lemma 17 I don’t quite see how you apply the escape property. To apply the escape property, you need to already know that whereas it appears that at this stage you only know

By: pavel zorin

pavel zorin — Sun, 09 Oct 2011 15:03:27 +0000

Dear Prof. Tao,

In the proof of theorem 14 one can use the continuity of scalar multiplication at (0,0) to make U bounded. This is easier and also shows that the continuity of addition is not needed in this step.

best regards,
pavel

[Good point; I've made the change. -T.]

By: 254A, Notes 5: The structure of locally compact groups, and Hilbert’s fifth problem « What’s new

Sat, 08 Oct 2011 20:58:03 +0000

[...] to the quotient of by a compact normal subgroup . By Cartan’s theorem (Theorem 2 of Notes 2), is also a Lie group. Among other things, this implies that the quotient homomorphism from the [...]

By: 254A, Notes 4: Building metrics on groups, and the Gleason-Yamabe theorem « What’s new

Tue, 04 Oct 2011 20:58:42 +0000

[...] Gleason-Yamabe theorem, we will use three major tools developed in previous notes. The first (from Notes 2) is a criterion for Lie structure in terms of a special type of metric, which we will call a [...]

By: 254A, Notes 3: Haar measure and the Peter-Weyl theorem « What’s new

254A, Notes 3: Haar measure and the Peter-Weyl theorem « What’s new — Tue, 27 Sep 2011 23:30:00 +0000

[...] the last few notes, we have been steadily reducing the amount of regularity needed on a topological group in order to [...]