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All this useless beauty

6 May, 2017

Nonense prevails, modesty fails,
Grace and virtue turn into stupidity,
While the calendar fades almost all barricades
To a pale compromise;
Now our leaders have feasts on the backsides of beasts
– They still think they’re the gods of antiquity –
If something you missed didn’t even exist,
It was just an ideal, is it such a surprise?

Those who find themselves ridiculous

21 February, 2017
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I sing myself to sleep

A song from the darkest hour

Secrets I can’t keep

Inside of the day

My make-up may be flaking

27 December, 2016

From the diary of Josiah Bounderby, aged 72 and 3/4

10 March, 2016
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I note that in Simon Jenkins’s most recent iteration of his contrarian arguments against the teaching of higher maths, he doesn’t include the following in his list of things which students should go into the world with: practice in recognizing repetition, sophistry, and ulterior motives; appreciation of false dichotomies; analysis of sources.

To be honest, I’m more annoyed with the Grauniad for continuing to indulge him, than with his views…

Those were the days, my friend

28 June, 2015
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While looking through various notes and expository articles on Vergne’s website, I came across this reminiscence concerning the first research themes that she published in:

Cependant j’étais fière de ce premier travail. Ces journées passées à écrire pour la première fois un texte personnel et qui fut publié (dont l’inimportance m’était peu apparente) m’avaient procuré un grand plaisir. C’est ce même plaisir de produire quelque chose de personnel que je retrouve identique à travers les années : travailler dans mon bureau lorsque tout est calme, taper sur le clavier de mon ordinateur, agrafer les minces liasses de feuilles couvertes de calculs précieux sortant de l’imprimante, mes documents en ordre autour de moi, la corbeille à papiers vite débordante.

The papers that feel like this rather make up for the ones that feel like chores, or “salvage jobs”…

Us and them: thoughts the day after

9 May, 2015
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If a power was to lift him up,
Make him rich, would he admit it was luck?
Or say he’d earned it, claim a state of grace,
Just like the rich in this hateful place?

I’m willing if not able, I am stretched to my limit,
It won’t take very much more to break my mind;
I watch a stable life drift by, it’s out of my range —
This morning paid me back my fear and let me keep the change

A woman drove her Saturn into the black water
Killed herself and her two kids strapped in the back seat
She’d lost her job, and didn’t want her kids to be poor

How useful is it that the Gelfand representation is a left adjoint?

30 January, 2015

(The question won’t get answered in this post, but it has been bugging me sufficiently that I may as well throw the question online, admit my ignorance, and see if anyone has any suggestions or critiques.)

Recently, on MathOverflow, I offered the following example of an adjunction that comes up in the theory of commutative unital Banach algebras. Let CHff be the category of compact Hausdorff spaces and continuous maps between them; and let unCBA be the category of unital commutative Banach algebras, with the morphisms being the continuous unital algebra homomorphisms between the objects.

There is a functor C from CHffop to unCBA, defined on objects by taking C(X) to be the usual algebra of continuous complex-valued functions on X, and defined on morphisms in the obvious way. Years ago I remembered convincing myself that not only does the functor C have a left adjoint, but one can define/describe the left adjoint as being the functor \Phi: \hbox{unCBA} \to \hbox{CHff}^{\rm op} which assigns to a unital commutative Banach algebra A its character space \Phi_A. Here \Phi_A is defined to be the set of characters (=non-zero multiplicative functionals from A to the ground field \bf C), equipped with the relative weak-star topology that this set inherits from the dual Banach space A^*.

What started to nag at me, after mentioning this example on MathOverflow, is that nothing in this description seems specific to the choice of complex scalars; in other words, it looks like one would obtain the same corresponding adjunction if one worked with unital commutative Banach algebras over \bf R rather than over \bf C. The choice of complex scalars is important because without it one does not get the Gelfand-Mazur theorem, and without that one does not get the fact that all maximal ideals in unital commutative Banach algebras have codimension one, and without that one does not get the following key feature of the Gelfand representation {\cal G}_A: A \to C(\Phi_A):

if a\in A and ${\cal G}_A(a)$ is invertible in C(\Phi_A), then a is invertible in A.

So the question arises: just what does one get from knowing the Gelfand representation arises as a left adjoint? What traction does it give us on the well-known examples and theorems in the theory of commutative Banach algebras? (This has been on my mind on and off for several years, because there are various possible generalizations and extensions of the Gelfand representation, either by passing to the noncommutative world or by looking at more general classes of ideals, not just the maximal ones; and I had hoped that the “left adjoint” perspective could be used as a guide when examining which of these versions is going to lead to a good theory. But if the categorical perspective I’ve outlined above can’t lead us to Gelfand-Mazur, then perhaps a rethink is needed.)