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]
trefethendotnet 
Did you succeed? (I came across your blog when I googled “manhattan tilt east”. I had recalled that it was 29 deg., but your summation was the most poetic: “My feeling is that we get through life with all kinds of useful approximations, and until better data comes along, I’m going to take the view that their alignment with the truth, on average, is around 29 degrees.”) I might steal that!