My work deals with approximating complicated functions f(x) by simpler ones. You might think that the goal was just to match the data accurately. But no, the deeper aim is to obtain a model that can predict how f will behave at other values of x, not yet seen. For this purpose, simple approximations do better than complicated ones. And since optimality means maximal simplicity by some measure, it follows that optimal approximants are better than suboptimal ones not just because they are more efficient at capturing the data.

The classical example goes by the name of the Runge phenomenon. A polynomial that interpolates the data exactly may be useless between the sample points. It has merely fit the data, not understood it. In the current era, we know that machine learning algorithms must avoid “overfitting”.

Henceforth this will be my take on Occam’s razor. Complexity does not just make a model less elegant. It lessens its power.

[22 November 2025]

In 1991 I suggested in one of these notes that if there are a lot of habitable planets out there, yet still we see no evidence of extraterrestrial life, this may indicate that the emergence of life on a habitable planet is more exceptional than we imagined.

It occurs to me that the same argument applies locally. Why has life emerged on Earth just once in billions of years? For I think that is the consensus of biologists: all life on this planet shares the same chemistry and stems from one original event. Experts may investigate details, arguing perhaps that the early Earth had conditions more favorable to appearance of life than the later one, but these are details. Earth is our archetype of a planet favorable for life. If life has appeared here just once, that’s a weighty data point.

[8 November 2025]

Biking is not allowed in Harvard Yard (“Please walk your wheels”), but many students do it anyway. If I walk through the Yard on a typical day, about four cycles or scooters whiz by. I don’t like this, and when I can, I raise my arm and say “Walk your wheels!” Some riders swear at me, most stay silent, almost all keep riding. I’ve never seen anyone else but me ask a rider to stop.

This upsets me more than you can imagine, and more than I can understand — to the point where I may start going around the Yard to avoid these encounters. I am trying to figure out why I am so upset. Partly it’s a feeling of political hopelessness. If people don’t care enough about social harmony to follow even this easy rule, what hope is there of collaboration to counter the bigger things going wrong in our country? So biking in the Yard becomes a symbol for me of the self-centeredness of all humanity, and of the careerism of Harvard undergraduates, and of paralysis in the Trump era.

Yet I think a bigger part of it is a sense of incomprehension and alienation. I simply cannot get my head inside the pattern of thinking that leads undergraduates to decide to break the rule. These are not troublemakers living on the edge, they are Harvard students, the very definition of society’s winners! What story are they telling themselves? My incomprehension makes me feel aged and remote from them. As another bike speeds by, I feel I am an old man on the way out, being replaced by an alien generation.

[8 November 2025]

It’s funny how, in the year you turn 70, so do a lot of your friends. I have a strange sense of the befores and the afters. Greg and Chris have made the transition, and so have I, but Nat is still just 69.

As many have noted, 1955 was also a big year for baby boys on the larger stage. John Roberts (Jan 27), Steve Jobs (Feb 14), Alan Garber (May 7) and the Pope (Sept 14) are behind us. Bill Gates (Oct 28) is up next.

In my own world of scientific computing, we have Volker Mehrmann (Apr 24) and Bill Gropp (Sept 23) as well as youngsters Randy LeVeque (Sept 30) and Jim Demmel (Oct 19).

[28 September 2025]

With the next academic year about to begin, I have filed away my email correspondence folders for 2024-25. It turns out that during the year, I kept records of interchanges with 629 different people.

[25 August 2025]

Here in St. Vith they speak German and French. English is fine too. You don’t have to choose.

The WC in the Bäckerei Fonk shows

with some words below, in English.

Whatever.
Just wash your hands.

[14 August 2025]

I have a PhD in Computer Science from Stanford, and it doesn’t get much better than that.

The other evening Kate was out of town and I decided to watch the 10 o’clock news. I discovered, I didn’t know how to turn on the television.

[7 July 2025]

A gentleman takes off his hat when he step indoors. My mother taught me this, and it is deeply ingrained. It’s been my lifelong habit.

Times have changed and guys wear caps indoors all the time. This made me think, my mother’s rule is obsolete. I should stop being weak and change my habit and leave the cap on.

But I couldn’t do it. So I thought further. It’s no big deal, why not follow her rule anyway, though society doesn’t care? Once again I have reverted to taking off my cap as I step indoors, and I feel better.

And then I realize, this is a model of belief in God. People’s mothers bring them to church as kids, and as adults they have their doubts. At some level they know the belief probably doesn’t make sense, but what’s the big deal? It’s more comfortable to keep thinking as mother taught us.

[25 July 2023]

The theory of Sobolev spaces has grown into a central part of the study of partial differential equations, to the point where many authors are unable to discuss a PDE except in this language. Sobolev spaces quantify the smoothness of functions — their “regularity”. Thus the discussion of a PDE in these terms requires one to attend to the smoothness of its solutions all the time, even when that isn’t the point of the investigation.

I see an analogy. The nouns in languages like French and Italian have genders, so it is impossible to mention a fork or a knife without specifying if it is feminine or masculine. Of course, the French know that the femininity of a fork and the masculinity of a knife are artifacts, but they have to get the artifacts right if they want to speak correctly. And sometimes grammatical gender can be helpful, when one is discussing people or animals that have a biological gender.

Is there a cost? Well, you decide. Memorizing la fourchette and le couteau feels like extra work to outsiders learning a language, introducing distinctions that may seem distracting or comical; but children growing up with the language barely notice. Many mathematicians are like that these days, native speakers of Sobolev spaces.

[25 May 2025]

The AAA algorithm computes rational approximations with amazing speed and reliability. Things are now easy that would have required major effort before, and in practice were not done at all.

We have no proof that the method works. So it is perhaps not surprising that rather often, mathematicians attending my talks express incredulity that I spend my time working on a method that has no theorem to justify it.

The incredulity sounds reasonable enough until you dig deeper. Suppose those people had the theorem they say is so important. Reassured, would they then use AAA to compute rational approximations? I think not. Their talks (which are awfully unlike mine) reveal an implicit view that the purpose of mathematics is not (A) to do things, but (B) to prove one could do things in principle. (A) may be important, but it is not mathematics. More like engineering.

I yearn for a world where mathematicians recognize both (A) and (B) as their business.

[21 May 2025]