Still some way off being organised or motivated enough to follow through on the vague promises made in this earlier post. Have been a bit busy since the start of the year juggling work projects and the knock-on effects; currently need to teach myself about Hilbert algebras and semifinite traces, and — more embarrassingly — the basics of calculus on differentiable manifolds.
(The latter is something which I found completely opaque as an undergraduate; the course I took seemed to be very good for those who already had some intuition, but not so good for those of us who lacked it. However, it might be easier to revisit it now that the formal language and notation doesn’t scare me as much. We shall see…)
(Edit: links added to Encyclopaedia of Mathematics and Wikipedia.)
No, this isn’t a rant about bean-counting or spurious self-citation… but you should read both of those anyway.
Instead, this is me bemoaning one thing that has bugged me on and off when trying to learn more about certain topics – and by learning, I don’t mean in the leisurely sense of broadening one’s mathematical background, but rather as a necessary precursor to finding and exploring avenues of research.
So: to those who make use in textbooks or survey articles or even their own research articles of big results “due to Connes” or “due to Dixmier” – could one of you at least take the time to untangle how the proofs actually work? Not in detail, but just to point out that statements along the lines of
the von Neumann algebra of an almost connected group is Connes-amenable, and this is due to Connes in his Annals paper classifying injective factors
are actually misleading? One might get the impression that people cite these things without looking at the original papers to see where the proof is given… (Hint: if Pukanszky doesn’t get mentioned, then the account is either inadvertently or willfully cutting corners.)
May expand on this particular peeve of mine – and a more general bugbear of mine, that the incentives in mathematics have drifted away from understanding and refining what we’ve proved, to shouty advertising and Toiling At The Coalface – if I can get my own notes organised. So probably not, then…
Edited 21-12-09: On rereading this bit of grumbling, I see that I may have given a misleading impression, or at least sinned against some W3C specification, with that “blockquote” above. It isn’t a genuine quote from anything, but rather my own version of what several sources have seemed to imply. Moreover, the result claimed is both true and non-trivial — but I think I’ll save further discussion for a separate post.
Well, I’ve been on the verge of writing some new posts for the last week or so, but somehow the energy/focus keeps ebbing away.
As a vague attempt to hold myself to some commitments, some things I hope to post about, albeit probably not in that order:
- Updated links looking back on the many achievements and exacting standards of I. M. Gelfand (thanks to Philip Brooker in the comments for alerting me to some I hadn’t seen before);
- The obligatory post mentioning Math Overflow;
- Something simple on the Pelczynski decomposition, as much to refresh my own memory as anything else;
- A tridiagonalization trick and norms in the GOE;
- A non-slick proof that we can’t take square roots of the shift;
- Maybe, just maybe, if I am infused with enough zeal or caffeine, some posts on Hochschild (co)homology as learned patchily by a callow PhD student with an odd combination of prior background.
One of the greats of 20th century mathematics, and one who by several accounts was an inspiring figure to others.
Update 07-10-09: missing link reinstated. More information about Gelfand’s achievements and role can be found via the following blog posts and their comment threads:
- John Baez at the n-category cafe
- Vladimir Dotsenko at être moral, être sincère
- Ben Webster at the Secret Blogging Seminar
and no doubt elsewhere. (Unfortunately I can’t read or speak Russian, otherwise I would try to find some links from those who had experience of Gelfand’s seminar.)
Having let the Shavgulidze-Thompson project slide out of the intray and into the mountain of Unfinished Loose Ends, I feel I should compensate with something vaguely mathematical. Hence this post, which is a follow up to some comments I left on a post at the Secret Blogging Seminar.
More precisely, in response to Q2 on that post, I left some rather dim-witted and error-strewn comments, only to have light shed by this subsequent observation from Greg Kuperberg:
Proposition: Let A and B be two Hermitian matrices. Then the spectrum of A+iB lies in the rectangle formed by the first and last eigenvalues of A and B.
Once GK stated the correct result, I realised that it followed from some facts that I really should have known – or knew, but had momentarily forgotten. It seems that the argument I had in mind is slightly different, at least in presentation, from the proof GK had in mind, and so I thought I’d give it here. (His reasoning seems like it should be more robust, and extend more easily to the case of bounded operators on infinite-dimensional Hilbert space.)
Claim. Let 
Proof. Since 
Similarly, there is an orthonormal basis 
Now let 
But now we can exploit the fact that 
We have 
Cards on the table, or the man behind the curtain
I have to confess that the phrasing of the argument above wasn’t the first that came to mind when I read GK’s comment. Lurking in the background — above and, I suspect, in his approach also — is the concept of numerical range. The numerical range of an 
and it is clear that 
which is in effect what we proved above.
The reason I should have remembered this is that a short while back I was reading up on some aspects of the numerical range for operators on infinite-dimensional spaces. The definition is the obvious one, and what is interesting is that we still have
-
for every bounded operator
;
-
for every normal operator
Note that in infinite dimensions the spectrum of
Via The Accidental Mathematician, I see that Margaret Wente – who, from what little I understand of the Canadian broadsheet establishment, is employed to be a Courageous Contrarian doughtily sticking up for the big guys against the little ones – is having a pop at the ivory tower. Or, at least, people she thinks are idling away in an ivory tower.
The whole thing needs to be read, if you want to savour le plein gôut of the vinegar and condescension – but inasmuch as it has a thesis, I suppose it’s summed up in this para:
The universities say the problem is money. If only they had more of it, they could do a better job of educating undergraduates. There’s just one catch. Educating undergraduates is just about the last thing most professors want to do.
Good to see that the Globe and Mail employs columnists with such good mind-reading skills, isn’t it?
Since I’m not one of the strawmen professors that Wente is cack-handedly disparaging, and since I don’t have time* to ferret around for counter-arguments and data, I’ll merely direct any readers of this post to the Accidental Mathematician’s own rebuttal, for the moment at least.
I’m mildly interested to see that one of Wente’s quoted sources is a (full) professor at my old stomping grounds, who on a quick look has more than earned his stripes and so is admittedly worth listening to if he has particular criticisms. It’s not clear to me that Clifton’s quotes – which to me say that people would rather do what they enjoy than what they regard as drudgery – actually support the more snide and deliberately vague charges that Wente levels. But then again, it’s not clear that Wente has anything to substantiate her charges, except `anecdata’ and a vague feeling that These Boffins Need To Be Accountable To The Consumer, Dammit. In fact, as I reread the article in search of the evidence for her assertions, it just isn’t there – just a string of assertions and resentment.
(And yes, this post is a transparent attempt to compensate for my failure to keep up with blogging on the Shavgulidze project. Haven’t forgotten about it, though; it’s just starting to sink towards the bottom of the intray…)
[*] lack of time due to effort to juggle a couple of research projects and prepare for my teaching this coming week. Somehow I don’t think Wente’s week is going to be that much more arduous than mine, or that of the full profs in my department, most of whom are teaching heavier loads than I am.
Just a quick note to say that the planned post on function spaces and a few related bits and pieces has been postponed, owing to less interesting but sadly more pressing work-related duties. Probably nothing until Wednesday at the earliest.
By way of sheepish apology, here’s an instance of how to Explain Maths Properly:
Well, the Shavgulidze preprint [henceforth [ETS]] and Brin’s notes [henceforth [MGB]] have been sitting in the intray this week – “real life” intervened in the shape of chores, forms, lost property and other such tedious necessities. So it’s only in the last couple of days that I’ve sat down to read the calculations and definitions…
Having now skimmed over [MGB] and taken a cursory look at [ETS], the current blogging plan starts as follows:
- A few basics on normed spaces.
Although everything can be done on an as-and-when-needed basis, several of the basic preliminaries make more sense to me (given my mathematical background) as special instances of a general outlook. Moreover, this discussion should lead into… - Amenability for right topological groups?*
As remarked in [MGB], the argument in [ETS] is surprising or odd-looking for group theorists, because it works with a definition in terms of a left-invariant mean on a function space, rather than families of almost invariant subsets. Note that the former definition can have technical advantages: sometimes it allows quicker/slicker proofs of certain results (for instance, see Prop. 5 in these expository notes by Terence Tao).
So, I thought it might be worth a post looking at amenability from the function-space perspective, and saying something about how we take into account a `compatible’ topology on the group when one is given to us. - Un soupçon de `F’
Here, since I’m really an outsider and mere spectator when it comes to geometric group theory, I’m not sure I’ll be able to say much about Thompson’s group `F’, except to point to other sources of various degrees of technicality. But I will at least try to give an indication of why the question of its (non-)amenability has occupied people on and off for the last 40 years or so.
After this, there seems to be a choice of emphasis or ordering. I want to spend a post saying something about Wiener measure, although I don’t know if this will just be confined to those properties used in [ETS], or whether I’ll try to throw in some of the more general context.†
If I get this far and still have enough energy/enthusiasm for the project, then I may try to write a post presenting (an approximate version of) my take on the detailed calculations in [ETS]. My current intuition is that the messy estimates on iterated integrals of unpleasant looking formulas must have some probabilistic interpretation, which might not make the proof any shorter but might make it seem less opaque for Bears of Little Brain such as myself. Detailed checking can be found in [MGB], and some of those involved are posting updates here.
[*] This comes with a question mark, because I’m not sure at time of writing if there exists an established definition in the generality I’ve claimed. (Amenability for locally compact groups has many well-known and much studied equivalent definitions; amenability for general topological groups has a less studied but well-established definition. But for right topological groups, I don’t know if the obvious candidate has been borne out by experience.)
[] Unfortunately, the rather solid set of notes I had when I first learned such things is now in someone else’s house, or possibly a nearby ditch, on the other side of the Atlantic, so the post may end up being rather hand-wavy and nebulous, depending on how much time I have to check the details in other sources.
Since I’ll be doing some teaching this coming semester – for the first time in a while – I thought it was long overdue to try and kickstart this weblog back into some pretence at activity. But on a different note, given the increasing visibility/credence given to mathematical blogging in recent months, it seems worth having another go at that, too.
So over the next couple of weeks, the goal is to work through this fairly recent preprint of Shavgulidze and blog my experiences and reactions to it. Actually, I’ll probably follow the notes being written up by Matt Brin et al. most of the way, but given that my background is complementary to that of the geometric group theorists, the emphasis may be different.
At some point, I also want to write up a bit of background and history to the problem that Shavgulidze’s paper solves – even if it ends up being mostly a collection of links – and say a bit about how I first developed a spectator’s interest in’t. For the moment, though, I can’t do much better than this quick account by Danny Calegari.
Lack of energy and lack of ideas for things to say. Sorry.
You can at least see the conference photo from its webpage. It’s been quite fun and amiable – as this series usually is – but, as we approach the end, I’m less and less able to focus on details in the talks.
