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!
I was wondering what you think about a claim that I sometimes hear in this context – that one of the problems is that universities train too many Ph.D. students. That with a smaller number of math Ph.D. students the above will be less of a problem, and also that this way there will be a smaller number of people dealing with less “serious/important” topics (whatever this means exactly).
This question is certainly relevant to the “adjunct issue”. I heard it before, but always found it somewhat confusing. Specifically to the US, with its market based system, who exactly is supposed to decrease the number of Ph.D.’s? The student themselves should realize how useless in the doctoral degree and stop applying? The individual professors should refuse to accept graduate students? The universities should do this together, in some kind of union? The government? All these questions are a bit different and need untangling.
I was going to write a brief reply, but after Adam asked this question I found a yet another example of lazy journalism by Slate’s “education columnist” Rebecca Schuman who argues:
It is, simply put, irresponsible to accept so many Ph.D. students when you know graduate teaching may well be the only college teaching they ever do.
Of course, Dr. Schuman already has a Ph.D. (from our neighbor UC Irvine) — she just wants others not get one, perhaps to avoid her own fate of an adjunct at University of Missouri. Needless to say, I cannot disagree more. Let me explain.
Universities are not allowed to form a cartel
Let’s deal with the easy part. If the American universities somehow conspired to limit or decrease the number of graduate students they accepts, this would be a classical example of anti-competitive behavior. Simply put, the academia would form a cartel. A textbook example of a cartel is OPEC which openly conspires to increase or decrease oil production in order to control world energy prices. In the US, such activity is against the law due to to the Sherman Act of 1890, and the government/courts have been ruthless in its application (cf. European law to that effect).
One can argue that universities are non-profit institutions and by definition would not derive profit should they conspire, but the law makes no distinction on this, and this paper (co-authored by the celebrity jurist and economist Richard Posner) supports this approach. And to those who think that only giants such as Standard Oil, AT&T or Microsoft have to worry about anti-trust, the government offers plenty of example of going after small time cartels. A notable recent case is Obama’s FTC going after Music Teachers National Association, who have a non-poaching of music students recommendation in their “code of ethics”. Regardless what you think of that case, it is clear that the universities would never try to limit the number of graduate students in a similar manner.
Labor suppy and demand
As legions before her, Schuman laments that pospective grad students do not listen to “reason”:
Expecting wide-eyed, mind-loving intellectuals to embrace the eventual realities of their situations has not worked—yes, they should know better, but if they listened to reason, they wouldn’t be graduate students in the first place. Institutions do know better, so current Ph.D. recruitment is dripping with disingenuousness.
But can you really be “wide-eyed” in the internet era? There is certainly no shortage of articles by both journalists and academics on the “plight” of academic life – she herself links to sites which seem pretty helpful informing prospective graduate students (yes, even the link to Simpsons is helpful). I have my own favorites: this, that, that and even that. But all of these are misleading at best and ridiculous at worst. When I mentioned them on MO, José Figueroa-O’Farrill called them a “parallel universe”, for a good reason.
You see, in this universe people make (mostly) rational decisions, wide-eyed or not. The internet simply destroyed the information gap. Faced with poor future income prospects, graduate students either choose to go elsewhere or demand better conditions at the universities. Faced with a decreasing pool of candidates the universities make an effort to make their programs more attractive, and strive to expand the applicant pool by reaching out to underrepresented groups, foreign students, etc. Eventually the equilibrium is reached and labor supply meets demand, as it always has. Asking the universities (who “do know better”) to have the equilibrium be reached at a lower point is equivalent to asking that Ph.D. programs become less attractive. And I thought Schuman cares…
Impact of government actions
Now, when it comes to distorting of the labor market, the government is omnipotent and with a single bill can decrease the number of graduate students. Let’s say, the Congress tomorrow enacts a law mandating a minimum wage of $60,000 a year for all graduate students. Of course, large universities have small armies of lawyers and accountants who would probably figure out how to artificially hike up the tuition for graduate students and include it in their income, but let’s assume that the law is written to prevent any loopholes. What would happen next?
Obviously, the universities wouldn’t be able to afford that many graduate graduate students. The number of them will plunge. The universities would have to cut back on the TA/recitation/discussion sessions and probably hire more adjuncts to compensate for the loss. In time, this would lower the quality of education or lead to huge tuition increases, or mostly likely a little bit of both. The top private universities who would want to maintain small classes will become completely unaffordable for the middle class. Meanwhile the poorer state universities will commodify their education by creating huge classes with multiple choice machine testing, SAT-style, and further diminishing student-faculty interaction. In fact, to compensate for their increasing cost to universities, graduate students will be asked to do more teaching, thus extending their time-to-degree and decreasing the graduation rates.
Most importantly, this would probably have no positive effect on decreasing competition for tenure track jobs, since the academic market is international. In other words, a decreasing american supply will be immediately compensated with an increasing european supply aided with inflow from emerging markets (ever increasing in quantity and quality production of Ph.D.’s in Asia). In fact, there is plenty of evidence that this would have sharply negative effect on prospects of American students, as decreased competition would result in weaker research work (see below).
In summary, who exactly would be the winners of this government action? I can think of only one group: lazy journalists who would have many new reasons to write columns complaining about the new status quo.
The out of control academics
Let’s go back to Schuman’s “it is [..] irresponsible to accept so many Ph.D. students” quote I mentioned above, and judge in on moral merits. Irresponsible? Really? You are serious? Is it also irresponsible to give so many football scholarships to college students if only a few of them can make it to the NFL? Is it also irresponsible to have so many acting schools given that so few of the students become movie stars? (see this list in my own little town). In the previous post I already explain how graduate schools are apprenticeship programs. Graduate schools give students a chance and an opportunity to succeed. Some students do indeed, while others move to do something else, sometimes succeeding beyond expectations (see e.g. this humorous list).
What’s worse, Schuman implicitly assumes that the Ph.D. study can only be useful if directly applicable to obtain a professorship. This is plainly false. I notice from her CV that she teaches “The World of Kafka” and “Introduction to German Prose”. Excellent classes I am sure, but how exactly the students are supposed to use this knowledge in real life? Start writing in German or become a literary agent? Please excuse me for being facetious – I hope my point is clear.
Does fewer students means better math? (on average)
In effect, this is Adam’s speculation at the end of his question, as he suggested that perhaps fewer mathematics graduate students would decrease the number of “less ‘serious/important’ topics”. Unfortunately, the evidence suggests the opposite. When there is less competition, this is a result of fewer rewards and consequently requires less effort to succeed. As a result, the decrease in the number of math graduate students will lead to less research progress and increase in “less important” work, to use the above language.
To see this clearly, think of sports. Compare this list of Russian Major League baseball players with this list by that of Japanese. What explains the disparity? Are more Japanese men born with a gift to play baseball? The answer is obvious. Baseball is not very popular in Russia. Even the best Russian players cannot compete in the american minor leagues. Things are very different in Japan, where baseball is widely popular, so the talented players make every effort to succeed rather than opt for possibly more popular sport (soccer and hockey in Russian case).
So, what can be done, if anything?
To help graduate students, that is. I feel there is a clear need to have more resources on non-academic options available for graduate student (like this talk or this article). Institutionally, we should make it easier to cross register to other schools within the university and the nearby universities. For example, USC graduate students can take UCLA classes, but I have never seen anyone actually doing that. While at Harvard, I took half a dozen classes at MIT – it was easy to cross register and I got full credit.
I can’t think of anything major the universities can do. Government can do miracles, of course…
P.S. I realize that the wage increase argument has a “fighting straw men” feel. However, other government actions interfering with the market are likely to bring similarly large economic distortions of the academic market, with easily predictable negative consequences. I can think of a few more such unfortunate efforts, but the burden is not on me but on “reformers” to propose what exactly do they want the government to do.
P.P.S. We sincerely wish Rebecca Schuman every success in her search for a tenure track appointment. Perhaps, when she gets such a position, she can write another article with a slightly sunnier outlook.