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The traditional celebration is Chinese food and a movie, so in that spirit: AB and I tried a new kind of dumpling on our recent trip to New York, sheng jian bao. These are soup dumplings, but with a thicker, crispier wrapper than the xiao long bao I’m more familiar with. Excellent!
As for a movie, we all re-watched Trading Places last night. Except it turns out Dr. Mrs. Q has never actually seen Trading Places! Arguably the very greatest Christmas movie.
A $5 million grant for my old friend Richard Rusczyk and the MATHCOUNTS foundation. Great news! Richard founded Art of Problem Solving, which is the only online math course platform I’ve ever found to be worth a damn. I love the way they emphasize depth over acceleration. When you do algebra with them, you don’t do the high school algebra curriculum in six weeks, you spend the whole year but you do a hell of a lot more algebra. That’s the way it should be! The point of learning math, if you love math, isn’t to get to the highest course number at the earliest age possible; it’s to learn deeply, richly, broadly, and well. I have taught too many grad students who can say “a representation is a functor from a groupoid with one object to the category k-Vect” but who have never sat down and worked out the character tables of a dozen different groups.
I also like the way AOPS is resolutely monomedia. No voice, no video. Just text! That makes sense. Math is made of words.
I met Richard when we were in high school, both in training for the Math Olympiad. He was from Alabama and he taught me how to say “my bad” for “sorry about that.” I’d never heard it! In 1988 nobody outside the South said “my bad.” I was impressed. I started saying “my bad” when I got home, just one of my many 1988 affectations.
“The American people have just one fixed idea and, since that dominant thought is money, it follows that their spirit has a certain uniform and unpleasant dryness… prodigality is an unforgivable crime in the eyes of these austere money-makers who worship the Almighty Dollar. Hard-working, sweating for gain, they have little pity for idle spenders; but they are indulgent to the clever bankrupt who enriches himself by his ruin and who knows how to keep public confidence even after his shipwreck. Instead of saying with scorn, “He is a crook,” they say admiringly, “He is quite a fellow! He is smart!”
(Ernest Duvergier de Hauranne, from Huit Mois en Amérique, 1864 — translation by Ralph H. Bowen.)
Maybe this is just a blog where I post unimpeachably classic songs about begging to be saved.
Has it ever been done better than this?
I do believe that we know right from wrong, and not because somebody instructed us. We do not always act on this knowledge, we’re not always even inclined to try. But we know!
This piece of marketing doggerel from McDonald’s, concerning their Xmas-special Grinch meal, is the most foully defiling attempt to imitate Dr. Seuss’s verse I’ve ever encountered.
How I know zombies aren’t real is that, if zombies were real, we would already have read about the McDonald’s PR employee who wrote this abomination being brutally eviscerated by a vengeful Dead Seuss.
The Baltimore Orioles have signed Madison Mallards legend Pete Alonso to a 5-year contract. $155 million. The biggest deal since (ulp) Chris Davis. He’ll wear Rafael Palmeiro and Anthony Santander’s #25. And Jay Gibbons’.
The fanbase is over the moon that we finally signed somebody. I get it, after so many years with no big contracts. But I also don’t get it. One of the Orioles podcasters I listen to said “This guy gets you 40, maybe 50 home runs a year.” Does he? Last year he hit 38, the year before that 34. He’s hit 50 once and only once, his rookie year in 2019. He does hit home runs! More than anybody else on the Orioles. But beyond that? He sorta kinda gets on base and he’s a terrible defensive first baseman. He’s not likely to be the Orioles’ best player. He would have been third best last year, after Gunnar Henderson and Trevor Rogers. A team whose best player is Pete Alonso is not a contending team.
And yet, for all this, I’m cheered by this deal. There is no Mookie Betts or Juan Soto or Shohei Ohtani to sign right now. And the Orioles with Alonso are incrementally better than the Orioles without Alonso. What you have to do to be good, in 2025, is be willing to spend a lot of money to incrementally improve your team, with guys, who, if not superstars, are at least plausible regular stars. The Orioles have not been willing to do that for a long time, and now, maybe, they are.
Also, of course, the team now has five guys, Rutschmann, Basallo, Mountcastle, Mayo, and Alonso who are limited to three positions, C,1B,DH. Alonso being here means one or more of Mountcastle and Mayo probably get traded for pitching. So if you spend the money to get Alonso playing every day at 1B (and one thing I’ll say for the guy, he does play every day) the improvement over Mayo playing there every day maybe isn’t that great; but if instead of Mayo sitting on the bench you turn him into a #2 or #3 starter, now you’re starting to get some real value for your $31m a year. (You know, now that we have more righthanded punch in the lineup, does Taylor Ward become a trade chip for pitching? Maybe we could talk to the Angels, they have a young guy Grayson Rodriguez with a spotty injury history but a lot of upside…)
But look, even if we do add pitching: you take the players on the 2025 Orioles, you add a poor defender who wears #25 and hits 40 HR and an ace starting pitcher, and you know what you have? The 2024 Orioles. Who were not that great! We did the tank, we did the rebuild, we have most of an entire lineup of players who were high draft picks and blue-chip prospects just a few years ago. If those guys play like they were supposed to play, this is a really good team. If they play like they played in 2025, $155m is not gonna help.
Just a couple more notes on this setup so I don’t forget.
First of all, I set this up as a problem about G acting on G/H, i.e. G acting transitively on a set. But the transitivity is actually not critical. The group could be trivial! If we call the set Σ, then G could be the trivial group, and we’d just be asking for large subsets S of Σ such that S^m was contained in some specified subset R of Σ^m. There are perfectly questions like that! I’ve seen this come up for instance when Σ is k and R is an algebraic subvariety of k^m. So maybe you could think of the setup of the blogpost as “questions like that where the problem admits a really big automorphism group, maybe so big that the elements of Σ are homogeneous for it.” I do think the transitivity cuts you down to a particularly congenial and natural class of such problems; “problems with no moduli” if you like.
Yet one more note. I think one could work with more general tensor categories than the category of sets. For instance, one could take V to be a representation of G, and let R be a subrepresentation of V^{⊗m}, and ask: how large can a subspace (not a subrepresentation) W of V be with the property that W^{⊗m} lies in R? And tensor powers are actually not necessarily the most natural thing to do to V; probably rather than choosing m and tensoring m times, we should choose a partition λ and apply the λ Schur functor. I assumed that once again lots of standard problems would fall out of this setup, but to be honest, I couldn’t think of any. Can you?
I guess here’s one baby case. Let V be the standard permutation representation of S_n. Then Sym^d V is homogeneous forms in n variables with S_n permuting coordinates. Let R be the subrepresentation of forms F such that the coefficients of X_i^d sum to 0. Then a subspace W of linear forms such that Sym^d W lies in R is just (I think) a linear subspace of the diagonal hypersurface sum x_i^d = 0 in P^{n-1}, and such a subspace has dimension at most n/2 or so I think (pair up coordinates to make x_i^d = – x_{i+1}^d; maybe you have to be a little more clever if d is even and -1 doesn’t have a square root?)
I remember Larry Agran because he was the dad of a college classmate of mine and he ran an outsider campaign for the Democratic nomination for President in 1992. I happened to look him up and learned that he’s still in politics, 30 years later, and is the current mayor of Irvine, CA. Better yet — he is serving his sixth term as mayor, no two of which have been consecutive. That has to be some kind of record. Paul Soglin was born about three months after Agran. He has served eight terms as Mayor of Madison, in three contiguous blocks, and the part I like is — no two of those blocks were in consecutive decades! He served as a new UW-Madison grad student radical in the 1970s, a Clintonian New Democrat in the 1990s, and as a grumpy old cuss in the 2010s.
Quomodocumque