Polynomials pure and applied

Algebraic geometry is the purest of pure branches of mathematics, concerned with the intrinsic structure of functions. Its central tool is polynomials, which constitute the very special case of functions you can “write down”.

Chebfun is the most practical of practical branches of mathematics, concerned with machine computing with functions. And its central tool is polynomials too! Mine is the kind of mind that needs to know, are algebraic geometers and Chebfunners interested in polynomials ultimately for the same reason, or different?

I think they are different.  The starting point of algebraic geometry is that, given a function f and a point a, there is a polynomial that exactly matches f and its derivatives at a. The starting point of Chebfun is that, given a function f and an interval [a,b], there is a polynomial that approximately matches f on [a,b] to any prescribed accuracy.

[9 October 2014]

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Safety tips

Some maxims seem so natural one assumes they must have been expressed many times before. Here are two that I’ve shared with Emma and Jacob at various ages.

For cyclists (or drivers, or skiers):
Never do anything surprising.

In the restaurant (or the bar, or at home):
Never drink when you’re thirsty.

[6 January 2018]

Yuji Nakatsukasa meets Russell’s paradox

Over coffee we were discussing how to respond to a referee report and I commented to Yuji, “You’re much nicer than I am.” Yuji being so nice, I pretty much expected him to respond, “Oh, no, that’s not true!” He didn’t say that, however, because one must not contradict the professor.

[2 June 2017]

New Zealand English is missing two words

The Kiwis (as they happily call themselves) are great at slang. A sauvignon blanc is a sav, a pavlova is a pav, and a holiday house a bach. What a friendly place to spend two weeks.

But do you know what’s missing? They have no nicknames for the two great halves of their country: just stodgy “North Island” and “South Island”.

[5 December 2017]

Iceland and USA, Lyon and New York

During a holiday in Iceland some years ago it amused me to learn how the USA manages to have 1000 times the population of Iceland. Approximately speaking it’s 10 times as long, 10 times as wide, and 10 times as densely populated.

I’ve just arrived to live in Lyon after a stay in New York and noticed that these cities have something in common: their heart is an angled peninsula running north-south between two rivers.  And how does Manhattan manage to have 27 times the population of Lyon’s Presqu’île? You guessed it. Approximately speaking it’s 3 times as long, 3 times as wide, and — this time we can say it less abstractly — 3 times as tall!

[8 November 2017]

American guns, French cigarettes

In the USA, roughly speaking, everybody has a gun.  This big awful fact stares you in the face. Europeans find it inexplicable. It’s just so obvious, why don’t the Americans prohibit these killing machines?

In France, roughly speaking, everybody smokes. I’ve arrived for a year, and it’s strange how this fact feels similar.  Americans find it incomprehensible. It’s just so obvious, why don’t the French just quit?

I feel oddly optimistic about these pathologies. Eventually, sloppily, rationality more or less prevails. The Americans will lose their guns one day, and the French will lose their cigarettes. But it will take generations, for habits steep into us, and we come to feel they are part of our identity. I’m sitting outside a bar right now in Vieux Lyon, smoke all around, and the very smokiness adds to that agreeable feeling of Frenchness.

[24 Oct 2017]

New York pedestrians, discrete vs. continuous

Classically, discrete molecular systems like the air in a bicycle tire have been modeled as continua. This would seem like the right thing to do for almost any application. We may know that the pressure in the tire ultimately results from 10^25  molecules bouncing around at random, but for most purposes it would seem crazy to try to follow those motions individually.

Lately, however, more and more mathematical scientists are trying to track statistics of particles. It seems everything nowadays is getting reformulated stochastically. I have been skeptical of this trend. Is it partly just a fad, a way of generating impressive new challenges for our latest computers?

But walking around New York has given me an illustration that sometimes, discreteness in our models really is indispensable. On the sidewalks there are hundreds of pedestrians flowing north, south, east, and west, and it is tempting to think of these flows as continua. You certainly feel like part of a wave when you are flowing along with the crowd! But at each crosswalk, something discrete happens. If the walk light is illuminated, a car wanting to turn right has to wait until the number of pedestrians crossing falls to exactly zero.

If pedestrians were a continuum rather than discrete, their density would always be positive, and no car in New York would ever be able to turn right.

[8 October 2017]