updates on lack of updates

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.

R.I.P. Israel Moiseevich Gelfand, 1913-2009

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.)

A simple remark on the spectrum of certain matrix sums

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 and be normal matrices, with spectra and respectively. Then the spectrum of is contained in .

Proof. Since is normal, there exists an orthonormal basis of , which consists of eigenvectors for . Let’s denote this basis by and let the corresponding eigenvalues be .

Similarly, there is an orthonormal basis and scalars such that for all .

Now let be an eigenvalue of , and let be a corresponding eigenvector of unit length. We have

But now we can exploit the fact that and each have a complete set of orthonormal eigenvectors. In particular, writing we have

We have (again, using the orthonormality of the ) and so . An exactly similar argument, this time using the , tells us that . Hence lies in the sum of these two convex hulls, as claimed.

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 complex matrix is the set

and it is clear that is contained in for every pair of matrices. Now, by considering appropriate eigenvectors, one sees that every eigenvalue of is contained in . Also, if is a diagonal matrix, then the same calculation that was made above shows that is contained in the convex hull of , and since the numerical range is unchanged if we conjugate by a unitary matrix, it follows that for every normal matrix . In particular, if and are normal matrices then

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

1. for every bounded operator ;
2. for every normal operator .

Note that in infinite dimensions the spectrum of might contain points which are not eigenvalues, and so the argument above with eigenvectors doesn’t work anymore.

Want to know why (some) op-ed columnists write bilge?

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.

delays expected

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:

excuses, excuses

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:

1. 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…

2. 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.
3. 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.

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[*] 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.

Semi-expository project: amenability of Thompson’s group F

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.

Absence of conference blogging

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.

An allergy? You get an allergy, you’re a scientist

Apologies for the post title, which is probably only comprehensible if (a) you remember a certain advert for British Telecom, and (b) you tolerate really, really lame puns…

So, in response to a polite challenge/request towards the tail of this comment thread, here is my attempt at describing something I’ve been working on in my research, in at most 500 words, using as little technical jargon as possible. It was written in an hour or so while procrastinating over the actual work itself, and hasn’t been subbed in any way, so is no doubt far from best possible in style or in selection.

As I will point out in a subsequent post, when I give the “version for mathematicians”, most of what follows is of course a lie in the strictest sense; however, they are untruths not intended to deceive, but to serve as analogies. Which I hope you get, even if you’re not a scientist.

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Mathematics often deals with symmetries of an object or construct: that is, the reversible mappings from that object to itself which preserve the given structure of interest. Thus, at least in the context of Euclidean geometry, the symmetries of a circle are the rotations.

If we drop the word “reversible”, and consider *partial* symmetries, then matters become more complicated, and potentially more interesting. In special infinite-dimensional settings, certain partial symmetries, though not themselves symmetries, may be regarded as shadows of symmetries in yet higher dimensions. More precisely, given our object X and a partial symmetry S, it may be possible to find a larger object Y in which X embeds, and a “genuine” symmetry T of Y, such that T takes X to itself and coincides with S on X. One then hopes to understand S by examining T.

A partial symmetry S of this kind is in some sense “residually reversible” (non-standard terminology); it may not be reversible, but we can make it so by passing to a larger ambient object. Such a partial symmetry cannot “destroy information” or “collapse structure” (because if it did, this property would pass to the putative extension T, and this is forbidden since T is reversible). The prototypical example arises if we take X to be a collection of strings, which may have infinite length, but which have a fixed starting point (one may think of the decimal expansions of numbers between 0 and 1, as a very loose model). Then our “residually reversible” mapping S is simply the operation “shift everything one place to the right” (which in our decimal-expansion model corresponds to “divide by 10″). The required “dilation” of S and X to T and Y is obtained by “enlarging X to the left” (which in our decimal-expansion model means “consider all positive numbers, not just those between 0 and 1″).

One of my current research projects examines certain restricted settings where one can state and prove results of the following form: any partial symmetry which is residually reversible is in fact a full, “genuine” symmetry. Crudely speaking, there are no “one-sided shift operators” of the sort described in the previous paragraph: or, put another way, any partial symmetry which is not a full symmetry must either “destroy structure”, or at least “crumple it” in some way that loses information. The interest in doing this is because, in some sense, residually reversible partial symmetries are usually easy to find in non-commutative, infinite dimensional settings (that is: we have infinite degrees of freedom; and the order in which we do things matters). The examples I am looking at are highly non-commutative, yet do not admit these kinds of partial symmetry; and this is evidence for a certain rigidity in the objects being considered, which is in my view still not fully understood, and which merits further attention.

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That was apparently just over 480 words, according to the text editor. Bonus points for anyone who can infer from this what the actual piece of research is on, even vaguely. I don’t think there’s enough information above, but you never know.

(I should point out that while I find 500 words or fewer (ahem) an interesting challenge, I’d be leery of depending on it as a criterion for judging the worth of a project, topic or discipline, largely for reasons similar to those laid out in these comments. My own speculative guess is that while bad examples may betray flaws in the proposal or project, “good” ones are largely just an advertising sell. But then, as someone who is personally more interested when the current Charles Simonyi chair for PUoS talks about zeta functions and algebraic curves, than when he talks about symmetries in hyperspace, I’m obviously^[1] not the target audience.)

[1] In case my intended frivolity above is taken too seriously: I am immensely cheered that du Sautoy has taken up the Simonyi chair, and find his efforts to engage lay readers of various ages a Very Good Thing. Even if he is a Gooner.

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update 31-03-09: small typo fixed (there was an `S’ that should have been a `T’)

something stirs and something tries, and starts to climb towards the light

Have been getting sufficiently vexed with people saying daft things online, or not bothering to do their homework before mouthing off, that blogging may resume sooner rather than later. However, actual mathematical posts (like a long-planned but infrequently revisted post on the general theme of parity arguments, signature and other invariants) will have to wait.

Next up, if I can get organized enough to write it, will be something on the off-topic theme of checking the secondary sources. Probable mentions of Bill Hartston and Susan Greenfield — there is a connection, I promise, though it’s tenuous.