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]