Talk:Pontryagin duality

Just a note here with a copy of a discussion on User talk:Charles Matthews outlining plans for three pages: Pontryagin duality, dual group and Harmonic analysis. This is just in case anyone else wants to do anything to these pages.

[march, 10, 2009] It would be nice to add the dual of the set of p-adic numbers in the section Examples.

Hi Charles,

Pontryagin duality looks good, but I wondered why you created this instead of adding to and/or rewriting dual group. Both are about the same thing at different levels of sophistication. I threw dual group together the other day to expand an ugly stub -- essentially the first few lines describing characters. I have no problem with dual group being improved -- such is the way of the wiki. It suffers from the its history and while trying to be consistent with related pages. The best solution may be a merged page under Pontryagin duality (a more accurate title) with dual group redirecting.

If a merge was to happen, dual group suffers from not having any reasons for wanting duality (which yours covers from a couple of angles) and Pontryagin duality suffers from not actually saying what the dual group is (in terms that an undergrad maths student can handle). Your account of the development of the theory is also better. Maybe a solution is to import and tidy up some of the material on the concrete construction of the dual group (characters, topology) from dual group and make dual group a redirect? Possibly keep the classical examples as well, as they tie it to the Fourier transform/series articles. My attempt at referencing was deliberately lazy, so could be improved.

I have also since noticed some overlap with my version of dual group with the middle chunk of Harmonic analysis. I think it doesn't really belong there, as harmonic analysis on LCA groups is only a part of HA, whereas it reads as though it is the most important idea. My feeling is that HA (also served by a redirect from Fourier analysis) should really be a high-level page with a general flavour of what harmonic analysis is about, some history and many links to the disparate fields that come under or are related to HA - from the abstract to the applications (classical things like PDE solutions through to wavelets and whatever else). Do you know of anyone who has an interest in (or oversees/coordinates) this higher-level organisation of WP? The Wikipedia:WikiProject Mathematics page only deals with things at the article level.

Cheers, AndrewKepert 01:21, 18 Nov 2003 (UTC)

Generally, I agree with your assessment. I wrote the PD page quickly once I'd actually found dual group (I've thought subsequently of adding a bit more, on discrete and compact R-modules, and End(G) = opposite ring of End (G^)). It could indeed be good to merge the two under PD as main title, putting the dual group material first in the article (and moving across from HA anything that is really about LCA in general as you suggest).

There is no 'they' who sorts out policy-level stuff. We're on our own here ...

Charles Matthews 10:12, 18 Nov 2003 (UTC)

Thanks Charles - yes I know Wiki is anarchic. I also thought that informing somebody(s) who might consider the higher-level organisation as their patch might bring in a few tips and avoid subsequent angst when somebody reverted the lot!

I will go ahead with the merge as discussed (not today - too busy) and strip the corresponding section out of Harmonic analysis. I'll also tidy up notation - I prefer &gamma ∈ Γ = G^ to φ ∈ G' myself, but will change additive notation in the group to multiplicative. I think the merge will work well, given the complementary POV of the two pages. Then I will start thinking about how HA should be structured and fleshing it out. Hopefully whatever I do will offend sufficiently many people, thus attracting interest and contributions.

Cheers AndrewKepert 01:00, 19 Nov 2003 (UTC)

I've done the big move from dual group, and added some headers. This leaves detailed editing of the page.

Charles Matthews 08:47, 19 Nov 2003 (UTC)

Now moved the LCA stuff from harmonic analysis, and spliced it in. Leaves the page structure OK, I think - but needing a thorough edit on notation, excess links and so on.

Charles Matthews 09:30, 19 Nov 2003 (UTC)

What is it good for?

Can someone give some examples that show where Pontryagin duality can be used to prove something or understand something better? —The preceding unsigned comment was added by 217.189.228.213 (talk) 08:49, 3 May 2007 (UTC).