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]

When I’m trying to recall a name, sometimes I take notes of my guesses along the way so I can enjoy following the trail again once I’ve reached the destination. This morning my guesses were Baratchart, D’Alembert, Constable and Dalrymple on the way to Perlmutter. The other day it was Studebaker on the way to Spiegelhalter.

[22 May 2025]

Do butterflies remember when they were caterpillars?

Well, we humans do. As a kid you have no idea of the real meaning of male and female, and you may store up memories that need reanalysis in adulthood. I remember thinking that the words “boy” and “girl” simply marked genders, like a girl cat or a boy dog. I remember asking my parents, who said no, the words aren’t used like that.

A song on the radio yesterday brought back another one of these memories from across the metamorphosis. As a kid I loved Harry Belafonte’s record with the line “I had to leave a little girl in Kingston town.” I remember thinking, how sad, that’s probably his daughter.

[15 April 2025]

Often at a meal among academics I marvel that we are 6, 7, or even 8 people all born in different countries. I don’t think I’ve ever reached 10 or more.

Here’s a variant of the same game, and surely my best score for a lifetime. I’m at a conference this week with 38 participants, and they all have different first names! I feel I’ve won the lottery.

Akil, Anil, Annan, Astrid, Barbara,
Cade, Chris, Daan, Dan, Evan,
Grady, Heather, Heike, Karl, Kaitlynn,
Keaton, Kyle, Levent, Lihong, Lucas,
Mark, Mike, Nick, Petar, Sam,
Sean, Simon, Serkan, Stefano, Steffen,
Tatyana, Tobi, Toby, Tony, Victor,
Vladimir, Yuji, Zoran

[10 April 2025]

I’ve just received an email from the Royal Society with subject “Nominate a brilliant mathematician for the Sylvester Medal”. They could have said “outstanding”, or “top”, or “world-leading”, but they chose “brilliant”. Would that have been the choice for a medal in physics, or in biology?

There’s a footnote in An Applied Mathematician’s Apology on this theme:

I’ve served on committees to award fellowships for pure mathematicians. To explain why a candidate is deserving, a referee will begin by making an attempt to describe the substance of their achievements, but this is hard. Pretty soon the letter moves away from substance and resorts to asserting how brilliant the candidate is. Every discipline judges people in part by their brilliance, but no other takes it as far as mathematics.

[4 February 2025]

As we well know, many things seem continuous though their composition is discrete. I give lectures about this, starting with the bouncing molecules that make up our gases and liquids. The cover image of Trefethen’s Index Cards illustrates how impossible it can be, in borderline cases, to decide which is the right truth.

Atop your head is another good example. In English and Spanish your hair is singular, in French and German it’s plural.

[25 November 2024]

Global warming is always in the news, and we are told that average temperature has gone up 1.3 C or 1.4 C since preindustrial times. We hear constantly about these numbers.

I’ve never heard it mentioned that such small figures don’t seem to fit our own experience. When I was a kid we skated on the pond all winter; now it usually doesn’t freeze. The Dutch no longer skate on their rivers the way they used to. The ski industry is in trouble worldwide because of lack of snow. Europe, Asia, and the US have deadly summer heat waves lasting weeks, far above the temperatures we grew up with. Novembers and Marches are warm where they used to be cold. How can all this fit with an average increase as small as 1.4 C?

I’ve been asking around to try to find the answer. A common expert’s reaction is: young man, you don’t understand statistics. I push past that, and provisionally, I have learned two surprising things especially from https://data.giss.nasa.gov/gistemp/maps/:

• There has been more warming on land than over ocean;
• There has been more warming in the north than in the tropics and the south.

On the maps, these two effects are quite striking. Put them together, and it seems the warming experienced by many of us has in fact been closer to 3-4 C.

[22 November 2024]

I find that my apprehensions about the years ahead fall in four categories.

Substantitve local: What damage will be done to the USA?

Substantive global: What damage will be done to the world?

Symbolic local: How can we now deny that we the American people have chosen this monster?

Symbolic global: What does this say about democracy?

Events have a way of surprising us. Who knows, maybe not all my fears will come to pass. It will be interesting to revisit this note in 2028 and beyond.

[10 November 2024]

“Two roads diverged,” wrote Frost, and the traveller more or less arbitrarily has to choose one. The mathematical line comes later. “Knowing how way leads on to way, I doubted if I should ever come back.”

Frost is describing here the essence of chaos theory, specifically, the exponential divergence of trajectories. “Way leads on to way” reminds us that the bifurcations will keep coming. The possibilities branch out to almost infinite complexity, doubling at each junction in this rural New England image.

[6 October 2024]