Crouzeix’s conjecture

I am on BA 285, one of thirty mathematicians flying to San Jose, California, to spend a week together at the American Institute of Mathematics trying to prove Crouzeix’s conjecture. The conjecture asserts that a certain quantity that arises in linear algebra is ≤2. So far, it’s known to be ≤2.41 (1+√2). This week, a man-year of time and a man-year of money will be spent trying to prune away that last 20%.

Direct consequences if we pull it off? Next to none. Nothing really depends on 2.41 being improved to 2.

The point is the indirect, the intellectual consequences. With luck, a week from now the theorem will be proved and the proof will have made use of a new idea or two that may lead on to further advances is the future. This is how mathematics — science — grows. If we succeed this week, the time and money will have been well spent.

[30 July 2017]


Being left-handed, being gay

The Wimbledon match between Federer and Zverev yesterday got me thinking. Federer is right-handed and Zverev is left-handed, and that’s all there is to it. Nobody wonders what made Zverev left-handed, or if he could be talked out of it. The tabloids do not print rumors that Federer has left-handed tendencies he keeps darkly to himself.

The analogy of left-handedness with gayness goes pretty deep. So far as I can tell, both are understood to have complex causes mixing genes, development in utero, and other factors hard to disentangle. Like so many analogies, this one draws you in with a complex skein of similarities and differences. A similarity is that not everybody is perfectly left-handed or right-handed, just as not everybody is perfectly gay or straight. A difference is that gayness comes with an obvious evolutionary cost, whereas left-handedness seems on the face of it to be evolutionarily neutral. And what about our cultural responses to these syndromes? My opening paragraph may suggest that left-handedness is accepted without a ripple, but it is not so simple in Saudi Arabia, India, or China.

[9 July 2017]

Two white noise paradoxes and Einstein 1905

There’s a paradox of white noise: in its standard idealization it has equal energy at all frequencies, hence infinite energy. Curiously this paradox is linked to two of Einstein’s great papers from his annus mirabilis.

The ultraviolet catastrophe. The first paradox was a big problem implicit in 19th century physics. Statistical mechanics predicted that a cavity should support electromagnetic waves of all frequencies, each with an equal amount of energy. So at each instant a cavity should radiate infinite energy! Planck’s quantization, it was later realized, resolved this difficulty by positing that in fact, higher frequencies have less energy in them. Einstein investigated more deeply the implications of quantization for the behavior of light photons, for which he won the Nobel Prize.

Brownian motion.  Meanwhile another work of Einstein’s that year eventually led to today’s standard idealization of Brownian motion, which has become a central subject in mathematics. The idealization is this: Brownian motion is the indefinite integral of white noise. But white noise has infinite variance! This time the physicists are not concerned, since bouncing molecules don’t go down to the infinitesimal scale, but the paradox still needs a mathematical resolution if we are to work with these ideas rigorously. In effect mathematicians take the view that white noise itself is never measured, only integrals of it, and these are finite because of sign cancellations. The higher the frequency, the greater the cancellation. But the details make stochastic analysis technical and forbidding.

Did Einstein notice that blackbody radiation and Brownian motion are linked in this way?

[6 May 2017]

Familiar music is not as loud

I’m sitting in a coffee shop in Boise, Idaho. I spend a lot of time working in coffee shops. But here the music is too loud, and it has been hard to work.

Yet just now they’ve switched from current songs I don’t know to an oldie that I know very well, “All along the watchtower” by Jimi Hendrix. How amazing — it’s still loud, but somehow less intrusive. I can work! This is an effect I’ve noticed before. My theory is that my brain is familiar with this track and knows there is no new information to be processed, so it tunes it out.

[14 March 2017]

What they teach us in school

The other night at a dinner for graduate students at Balliol, I had a German on my left and a Frenchman on my right. The question came up, what happened around 1870 involving Germany and France?

The German student was unaware of any events from this period.

The French student was aware of one event from 1870: the founding of the French Third Republic. He did not know of any connection of this with Germany.

(In fact, in 1870-71 Germany conquered France.  Napoleon III surrendered in September, and Paris fell in January.)

[1 February 2017]

The purest love of all

The love of parents for their children, it is well known, is the purest love of all. There is nothing we would not do for them, and we keep giving to the end of our lives without thought of what we get in return.

The irony is that from an evolutionary point of view, love of one’s children is obviously selfish, since they carry forward one’s genes. It is so perfectly in our self-interest to take care of our children that natural selection would not dream of letting us deliberate about the matter! No, our love is hardwired. We deeply want to give to them, and who cares what it costs?

And because it’s hardwired, that’s why it feels selfless. This is the love that bypasses our brains and inhabits our bones.

[19 January 2017]