Crouzeix’s conjecture

I am on BA 285, one of thirty mathematicians flying to San Jose to spend a week together at the American Institute of Mathematics trying to prove Crouzeix’s conjecture. The conjecture asserts that a certain quantity arising in linear algebra is ≤2. So far, it’s known to be ≤1+√2 (about 2.41). 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]