Holiday season offers endless opportunities to celebrate, relax, rest, reflect and meditate. Whether you are enjoying a white Christmas or a palm tree Chanukkah, a mathematician in you might wonder if there is more to the story, a rigorous food for thought, if you will. So here is a brief guide to the holidays for the mathematically inclined.
1) Christmas tree lectures
I have my own Christmas tree tradition. Instead of getting one, I watch new Don Knuth‘s “Christmas tree lecture“. Here is the most recent one. But if you have time and enjoy binge-watching here is the archive of past lectures (click on “Computer musings” and select December dates). If you are one of my Math 206 students, compare how Knuth computed the number of spanning trees in a hypercube (in a 2009 lecture) with the way Bernardi did in his elegant paper.
2) Algorithmic version of Fermat’s Christmas theorem
Apparently, Fermat’s theorem on sums of two squares first appeared in Fermat’s long letter to Mersenne, written on Christmas Day (December 25, 1640). For background, see Catalan and French language Wikipedia articles. Zagier’s “one-sentence proof” is well known and available here. Long assumed to be mysterious, it was nicely explained by Elsholtz. Still mysteriously, a related proof also appears in a much earlier paper (in French), by a Russian-American mathematician J. Uspensky (ht. Ustinov). Can somebody explain to me what’s in that paper?
Interestingly, there is a nice polynomial time algorithm to write a prime p=1 mod 4 as a sum of two squares, but I could not find a clean version on the web. If you are curious, start with Cornacchia’s algorithm for more general quadratic Diophantine equations, and read its various proofs (advanced, elementary, short, textbook, in French). Then figure out why Fermat’s special case can be done in (probabilistic) polynomial time.
3) Dreidel game analysis
The dreidel is a well known Chanukkah game with simple rules. Less known is the mathematics behind it. Start with this paper explaining that it’s unfair, and continue to this paper explaining how to fix it (on average). Then proceed to this “squared nuts” conjecture by Zeilberger on the expected length of the game (I have a really good joke here which I will suppress). This conjecture was eventually resolved in this interesting paper, definitely worth $25 promised by Zeilberger.
4) Santa Claus vs beautiful mathematics
Most readers of this blog are aware of existence of beautiful mathematics. I can only speculate that a clear majority of them would probably deny the existence of Santa Claus. However, there are millions of (mostly, very young) people who believe the exact opposite on both counts. Having grown up in the land of Ded Moroz, we have little to say on the great Santa debate, but we believe it’s worth carefully examining Santa proponent’s views. Could it be that their arguments can be helpful in our constant struggle to spread the gospel of beautiful mathematics?
We recommend reading “Yes, Virginia, there is Santa Claus“ column (fully available here), which was originally published by the New York Sun in 1897. In fact, read it twice, three times, even four times. I am reluctant to quote from it because it’s short and deserves to be read in full. But note this passage: “The most real things in the world are those that neither children nor men can see.” The new Jewish editor of the Sun reports that the rabbis he consulted think this is “a joyous articulation of faith”. Maybe. But to me this evokes some beautiful advanced mathematics.
You see, when mathematicians try to explain that mathematics is beautiful, they tend to give simple visually appealing examples (like here). But I suggest closing your eyes and imagining beautiful mathematical objects, such as the 600-cell, Poincaré homolgy sphere, Lie group E8, Monster group, or many other less known higher dimensional constructions such as the associahedron, the Birkhoff polytope, Walz’s flexible cross-polyhedron, etc. Certainly all of these can be seen by “neither children nor men”. Yet we can prove that they “are real”. We can then spend years studying and generalizing them. This knowledge alone can bring joy to every holiday season…
HAPPY HOLIDAYS EVERYONE! С НОВЫМ ГОДОМ!
As some of you figured out from the previous post, my recent paper (joint with Martin Kassabov) was accepted to the Annals of Mathematics. This being one of my childhood dreams (well, a version of it), I was elated for a few days. Then I thought – normal children don’t dream about this kind of stuff. In fact, we as a mathematical community have only community awards (as in prizes, medals, etc.) and have very few “personal achievement” benchmarks. But, of course, they are crucial for the “follow your dreams” approach to life (popularized famously in the Last Lecture). How can we make it work in mathematics?
I propose we invent some new “badges/statistics” which can be “awarded” by AMS automatically, based on the list of publications, and noted in the MathSciNet Author’s Profile. The awardees can then proudly mention them on the department websites, they can be included in Wikipedia entries of these mathematicians, etc. Such statistics are crucial everywhere in sports, and most are individual achievements. Some were even invented to showcase a particular athlete. So I thought – we can also do this. Here is my list of proposed awards. Ok, it’s not very serious… Enjoy!
Triple Crown in Mathematics
A paper in each of Annals of Mathematics, Inventiones, and Journal of AMS. What, you are saying that “triple crown” is about horse racing? Not true. There are triple crowns in everything, from bridge to golf, from hiking to motor racing. Let’s add this one to the list.
Other Journal awards
Some (hopefully) amusing variations on the Tripe Crown. They are all meant to be great achievements, something to brag about.
Marathon – 300 papers
Ultramarathon – 900 papers
Iron Man – 5 triple crown awards
Big Ten – 10 papers in journals where “University” is part of the title
Americana – 5 papers in journals whose title may only include US cities (e.g. Houston), states (e.g. Illinois, Michigan, New York), or other parts of American geography (such as Rocky Mountains, Pacific Ocean)
Around the world – 5 papers in journals whose titles have different continents (Antarctica Journal of Mathematics does not count, but Australasian Journal of Combinatorics can count for either continent).
Decathlon – papers in 10 different journals beginning with “Journal of”.
Annals track – papers in 5 different journals beginning with “Annals of”.
Now, imagine AMS awarded badges the same way MathOverflow does, i.e. in bulk and for both minor and major contributions. People would just collect them in large numbers, and perhaps spark controversies. But what would they look like? Here is my take:
enthusiast (bronze) – published at least 1 paper a year, for 10 years (can be awarded every year when applicable)
fanatic (silver) – published at least 10 papers a year, for 20 years
obsessed (gold) – published at least 20 papers a year, for 30 years
nice paper (bronze) – paper has at least 2 citations
good paper (silver) – paper has at least 20 citations
great paper (gold) – paper has at least 200 citations
famous paper (platinum) – paper has at least 2000 citations
necromancer (silver) – cited a paper which has not been cited for 25 years
asleep at the wheel (silver) – published an erratum to own paper 10 years later
destroyer (silver) – disproved somebody’s published result by an explicit counterexample
peer pressure (silver) – retracted own paper, purchased and burned all copies, sent cease and desist letters to all websites which illegally host it
scholar (bronze) – at least one citation
supporter (bronze) – cited at least one paper
writer (bronze) – first paper
reviewer (bronze) – first MathSciNet review
self-learner (bronze) – solved own open problem in a later paper
self-citer (bronze) – first citation of own paper
self-fan (silver) – cited 5 own papers at least 5 times each
narcissist (gold) – cited 15 own papers at least 15 times each
enlightened rookie (silver) – first paper was cited at least 20 times
dry spell (bronze) – no papers for the past 3 years, but over 100 citations to older papers over the same period
remission (silver) – first published paper after a dry spell
soliloquy (bronze) – no citation other than self-citations for the past 5 years
drum shape whisperer (silver) – published two new objects with exactly same eigenvalues
neo-copernicus (silver) – found a coordinate system to die for
gaussian ingenuity (gold) – found eight proofs of the same law or theorem
fermatist (silver) – published paper has a proof sketched on the margins
pythagorist (gold) – penned an unpublished and publicly unavailable preprint with over 1000 citations
homologist (platinum) – has a (co)homology named after
dualist (platinum) – has a reciprocity or duality named after
ghost-writer (silver) – published with a person who has been dead for 10 years
prince of nerdom (silver) – wrote a paper joint with a computer
king of nerdom (gold) – had a computer write a joint paper
sequentialist (gold) – authored a sequel of five papers with the same title
prepositionist (gold) – ten papers which begin with a preposition “on”, “about”, “toward”, or “regarding” (prepositions at the end of the title are not counted, but sneered at).
luddite (bronze) – paper originally written NOT in TeX or LaTeX.
theorist (silver) – the implied constant in O(.) notation in the main result in greater than 1080.
conditionalist (silver) – main result is a conditional some known conjecture (not awarded in Crypto and Theory CS until the hierarchy of complexity classes is established)
ackermannist (gold) – main result used a function which grows greater than any finite tower of 2’s.