Eponyms in mathematics

Let S be the standard Smith class of normalized univalent Matcuzinski functions on the unit disc, and let B be the subclass of normalized Walquist functions. We establish a simple criterion for the non-Walquistness of a Matcuzinski function. With this technique it is easy to exhibit, using standard Hughes-Williams methods, a class of non-Walquist polynomials. This answers the Kopfschmerzhaus-type problem, posed by R. J. W. (“Wally”) Jones, concerning the smallest degree of a non-Walquist polynomial.

This fake abstract of a paper is from Merv Henwood and Ivan Rival, Eponymy in Mathematical Nomenclature: What's in a Name, and What Should Be? (PDF), from the Mathematical Intelligencer in 1980. It sounds to me like slight caricature -- but only slight. Henwood and Rival point out that such names are lazy. Names have at least two important functions -- to describe and to label -- and eponyms only label.

Perhaps such abstracts would be more common in areas which are small enough that all the major players talk to each other. I imagine that Smith, Matcuzinski, Walquist, etc. know each other.

Also of interest is David Rusin's list of eponyms occurring in the MSC classification. These names in general seem a bit less obscure than the names one would find in the abstract of a random paper, which isn't surprising as they're names of concepts big enough to get areas named after them.

(And can someone confirm or refute the story that Banach, in the paper in which he introduced Banach spaces, called them "spaces of type B" in an effort to get them named after himself? I've heard this one a few times but always unsourced.)

Pizza seminars

In Penn's math department, there is a "Pizza Seminar". It is on Fridays at noon, only graduate students in the math department are allowed to come, and there is free pizza. Each week a graduate student gives a talk that is intended to be accessible to most graduate students; sometimes they focus on some accessible piece of their research, but more often these talks are expository. (Occasionally, maybe three times a term, a professor speaks -- but still only graduate students are allowed to attend the talk.)

If you Google "pizza seminar", most of the results are similar series in math or closely allied fields (physics, computer science). Is there some reason that such a format wouldn't work well in more distant fields, or is this just historical accident?

On foods of genus one

It seems that some people describe the torus as the shape of a bagel, and others as the shape of a doughnut.

I wonder if this is somehow correlated with geography; bagels are more common in some places, doughnuts in others.

Kiyoshi Ito dead

Kiyoshi Ito (of Ito calculus fame) is dead.

(When? I'm not sure. The New York Times says Monday, the 17th, but the Japan Times published an obituary on Saturday the 15th which said he died "Monday" -- so I'm guessing the 10th.)

See the obituaries, the MacTutor biography and Wikipedia article, or this Notices article upon his receipt of the Gauss Prize for an idea of his contributions.

Folklore for divergent series

From Emanuel Kowalski, quoting what is apparently folklore: "The only 'truly' divergent series is the harmonic series."

The idea is that one can assign a value somehow to basically any other divergent series; see the link for a more thorough explanation.

(But don't tell the calculus students! They already can't remember the harmonic series is divergent.)

And I usually think of the harmonic series as being "equal" to log n, although of course log n isn't a number. So I amend the folklore, somewhat facetiously: "there are no divergent series."

Freakonomics compiles math quotes

An interesting list of quotes about mathematics, as compiled by readers of the Freakonomics blog.

Here's one that's new to me and that resonates with me at the moment, having spent the afternoon creating a document which is essentially a list of facts, to which I'll add the proofs later. It's attributed to Poincare, although a bit of googling seems to indicate he said it about science, not mathematics.
"Math is built with facts as a house is built with bricks, but a collection of facts cannot be called mathematics anymore than a pile of bricks can be called a house."
What I have now is a pile of bricks. I have the mortar that holds them together, but it's not in the same place.

You'll also find the text of the poem which begins "I’m sure that I will always be a lonely number like root three", featured in Harold and Kumar Escape From Guantanamo Bay.

Erdos and grapefruits?

From Paul Graham, How to Start a Startup (and no, I'm not looking to):

People who don't want to get dragged into some kind of work often develop a protective incompetence at it. Paul Erdos was particularly good at this. By seeming unable even to cut a grapefruit in half (let alone go to the store and buy one), he forced other people to do such things for him, leaving all his time free for math.

I don't use this strategy. But I do use the strategy of cooking things in ridiculously large batches, which isn't much more work than small batches; then when I want something to eat I just fire up the microwave.

(I'm not sure if I can cut a grapefruit in half; to do that without struggling might require some special knife I don't have. I haven't tried, because I don't like grapefruit.)

Ranking graduate departments

On Some Random Webpage Full Of Spam, I came across what purports to be US News' 2009 ranking of graduate programs in mathematics. (I feel bad about linking to this, because it just helps the spammers, but I'm doing it anyway.)

If I remember correctly, this ranking is produced by surveying people in mathematics departments at various schools and asking them to rank other institutions. That's it.

It seems to me that a more sensible way of ranking mathematics departments would be to start with the assumption that a better department is, by definition, one which has its students get hired by better departments. This could work, at least to do the high end of the ranking, because the departments doing the hiring are often the same departments that have graduate students; eventually I expect the process I'm alluding to would converge on a ranking. I'm not sure what you'd do to deal with ranking "lesser" programs where many students are not hired by departments which themselves have doctoral programs. This just aggregates what people think about reputation, but in a way that's more principled than just asking some shadowy panel thinks.

Of course, in the end these sorts of rankings are not particularly valuable, so I don't want to pursue this any further.

(Disclaimer: my memory of how these rankings work comes from standing in a bookstore four years ago and copying down that year's rankings on a scrap of paper. I think the statue of limitations has passed on that, so I'll admit to it.)

The Journal Of Stuff I Like

Chad Orzel writes about The Journal of Stuff I Like. Basically, the idea is that you're reading papers anyway; just make a list of the ones you found worthwhile and put it where people can read.

This certainly seems like it could be a useful thing. If you liked one paper I liked, you might like other papers I like.

The dual, of course, would be to compile a list of "people who liked this paper". And from there it's a short step to "people who liked this paper also liked...", and if there's some centralized system keeping track of things, amazon.com-style recommendations, which have on occasion found me books that I liked but wouldn't have known about. (It's a short step conceptually; I'll admit that I don't know too much about how big of a step it is computationally.) But even without these extra steps in the previous paragraph, it could still be worthwhile.

I might start a list. Like I said, it wouldn't be hard. I already have a list of papers I've read, which is reasonably complete over the time period I've been keeping it.

The blackboard of anonymous questions

There are blackboards in the common room of my department. They don't see much use.

We were joking at tea today that people ought to write problems they want solved or questions they have on the blackboard, and other people could write the answers. Surely this has been tried somewhere. How did it work?

You mean there are people who *don't* write everywhere?

Harnessing Biology, and Avoiding Oil, for Chemical Goods, today's New York Times. I studied a fair bit of chemistry as an undergrad, so this is of interest to me academically. Basically, a lot of synthetic goods are made out of compounds with lots of carbon, which can eventually be traced back to petroleum; as you may have noticed, petroleum and its derivatives have gotten more expensive recently. So even if you were never a chemist you should still care.

The photo at the top of the article, though, is what got my attention. It's captioned "The scientists use the glass shield as a board on which to write chemical formulas", and I feel like there's the implication that they're doing this to conserve scarce resources (coming from the captions on the other photos). No! It's just that scientists of any sort write things everywhere -- every chemistry lab I was ever in had this property. I wonder what they'd think of mathematics departments. (One professor that I know often has about four different calculations going on simultaneously on the whiteboard of his office; they overlap each other, but they're in different colors, so he can tell them apart. I can't do that.)

In the hall of the dormitory floor I lived on as an undergrad we had several blackboards. They were often filled with mathematics of one sort of another. Of course, they were also often filled with transcriptions of the silly or obscene things some of us had said. I kind of wish I'd written them down... but let's face it, they were probably pretty embarrassing and are best left where posterity can't see them.

It might be interesting to see pictures of well-known mathematicians' blackboards...

Smale's problems

A lot of people refer to the Clay Mathematics Institute's seven "Millennium Prize Problems" as an analogue of Hilbert's problems for the 21st-century.

In 1998, Stephen Smale produced a list (of 18 problems) as well. (There is significant overlap with the Clay problems: both lists contain Riemann, P =? NP, Poincaré, and Navier-Stokes.) Two of them are Hilbert problems (the Riemann hypothesis and Hilbert's 16th problem). The list seems a bit biased, though, in that Smale made contributions to many of the problems mentioned; Smale acknowledges this as one of the criteria for forming his list, and the document isn't meant to stand alone; the essay was written in response to a query of Vladimir Arnold, and Smale was not the only person Arnold asked. One has to wonder if it would be possible for anyone to really be able to survey all of mathematics intelligently in the way I'm told Hilbert did. (I've read Hilbert's address but I don't know enough of the history to be able to assess whether it really covers all of mathematics at the time.)

In Smale's discussion of the Poincaré conjecture, after pointing out that a big part of the importance of the Poincaré conjecture is that it helped make manifolds respectable objects to study in their own right,he states:

I hold the conviction that there is a comparable phenomenon today in the notion of a "polynomial time algorithm". Algorithms are becoming worthy of analysis in their own right, not merely as a means to solve other problems. Thus I am suggesting that as the study of the set of solutions of an equation (e.g. a manifold) played such an important role in 20th century mathematics, the study of finding the solutions (e.g. an algorithm) may play an equally important role in the next century.

This introduces the discussion of P =? NP, although the reason people study algorithms is not to answer that question; but one often hears statements like Smale's statement on Poincaré's conjecture, or statements that Fermat's Last Theorem is more important for the development in number theory that it spurred than for the result itself.

The unexamined life?

From Bill James answers all your baseball questions, a long interview posted at the Freakonomics blog:

Q: Has looking at the numbers prevented you from actually just enjoying a summer day at the ballpark? Have we all forgotten the randomness of human ballplayers? By reducing players to just their numbers can we lose sight of the intangibles such as teamwork, friendships, and desire.

A: Does looking at pretty women prevent one from experiencing love? Life is complicated. Your efforts to compartmentalize it are lame and useless.

This is yet another example of the "people who think about things are strictly better off than people who don't" meme -- roughly speaking, the usual justification for this is that we can turn off the thinking when we want to. But can we? I know I can't just turn off the part of my brain that is constantly counting things or figuring odds of things, and there are moments when that does hurt my quality of life. I think in the end I come out ahead -- and most mathematicians probably would agree with me, otherwise they wouldn't be mathematicians -- but it is not so simple.

Stuff mathematicians like?

You probably by now know about the blog Stuff White People Like, which talks about stuff white people like.

By "white people", the blog doesn't mean all white people, but rather white urban twenty- and thirty-somethings with money to burn.

Stuff white people like (according to the blog) that I'd guess a lot of mathematicians also like include coffee (if you don't know why, you're probably not a mathematician), "Gifted" children (lots of mathematicians probably were), Apple products (mostly judging from what sort of computer colloquium and seminar speakers are running, although this could be skewed by the fact that Apple users, being in the minority, are probably more likely to bring their own machine), not arts degrees (arts degrees apparently make people more interesting to talk to at parties, a concept entirely foreign to us), graduate school (although it seems to be humanities PhD programs that they're referring to), and bad memories of high school.

There are also a lot of less-good imitations: Stuff Educated Black People Like, Stuff Lesbians Like, Stuff Gay Guys Like, Stuff College People Like, Stuff Asian People Like, Stuff White Trash People Like, Stuff Unimaginative Bloggers Like, and perhaps others.

But what about "Stuff Mathematicians Like"? I know that I wouldn't do a good job with it -- but someone should. I'll get you started -- the set of stuff mathematicians like includes coffee, Rubik's Cubes, saying things are trivial, being proud of the uselessness of one's work, and the prefix "co-". (So what is "ffee"? Okay, there's another thing mathematicians like -- bad jokes.) Feel free to name other elements of this set.

Top 5 Reasons It Sucks to Be an Engineering Student

Top 5 Reasons It Sucks to Be an Engineering Student, from Wired, via Slashdot.

I think a lot of the same ideas -- poorly written textbooks, relatively non-inflated grades, etc. -- apply to mathematics education (at least in some places) as well. By the way, I'd argue not for grade inflation in math and science courses, but for grade deflation in the rest of the curriculum, because when it gets to the point where there are only, say, three grades that one can reasonably give (A, A-minus, or B-plus) that's just silly. (I even think it's silly for graduate classes; in some courses in my program everyone gets A's. Why even bother making the professor fill out the form with the grades on it? I'd support graduate classes not having grades, which from what I understand is the norm at some places.)

Thankfully, in five weeks I will be done taking classes (for credit) forever.

Names of mathematicians in the MSC

Dave Rusin has a list of names of mathematicians which appear in the MSC scheme. (That's the "Mathematics Subject Classification." I almost called it the "MSC classification" but managed to stop myself.) For reference, here's the 2000 MSC. (Rusin's file treats the 1991 version.) The idea is that these must be Important Mathematicians because some subfield of mathematics has been named after them, for the most part.

Rusin claims that Abel is the only person usually given the honor of having his name lowercased ("abelian", not "Abelian"). I have a sense that "euclidean" also occurs, though I may be thinking of the French usage.

The most amusing one, to me, is that "Monte Carlo" is named after a person -- Charles III of Monaco -- who was as far as I know not a mathematician at all. Amerigo Vespucci is also on the list; 01A12 is the history of mathematics and mathematicians of indigenous cultures of the Americas. If I had been compiling the list I'd probably have missed that.

Rusin also calls Fibonacci a "rabbit-farmer". I'd laugh, except I'm pretty sure at least one of my students also thinks this.

The Unreasonable Effectiveness of Mathematics in the Natural Sciences

Eugene Wigner's famous essay The Unreasonable Effectiveness of Mathematics in the Natural Sciences is available online.

Is Liouville's number interesting

Mark Dominus writes about uninteresting numbers. In particular he claims that Liouville's number,

is uninteresting; the only thing that's interesting about it is its transcendentality, and that's not a big deal, because almost all real numbers are transcendental. (In fact, "almost all" seems too weak here, to me, although it is technically correct in the measure-theoretic sense.)

But I think this number is interesting. Why? Because an expression that gives it can be written with a very small number of characters (bytes of TeX code, strokes of a pen, etc.) and most numbers can't be.

Of course, this means every number anyone's ever written down is interesting, by this definition -- even if it took them, say, ten thousand characters to define it. Say we work over a 100-symbol alphabet; then there are at most 100¹⁰⁰⁰⁰ or so numbers which can be defined in less than that many characters! (There are multiple definitions for the same numbers; most things one could write are just monkeys at a typewriter, and so on -- but this is an upper bound.) But this number is finite! The complement of a finite set is still "almost all" of the real numbers.

That last paragraph, I don't actually believe. But any number that can quickly be written down is "interesting" in some (weak) sense.

(By the way, "Liouville" is not pronounced "Loo-ee-vill". And I'm told that the city of Louisville in Kentucky is not pronounced this way either.)

edit, 10:22 pm: Take a look at Cam McLeman's The Ten Coolest Numbers. (The list, in reverse order, is: the golden ratio φ = 1.618..., 691, 78557, π²/6, Feigenbaum's constant δ = 4.669201..., 2, 808017424794512875886459904961710757005754368000000000, the Euler-Mascheroni constant γ = 0.577215..., the Khinchin constant K = 2.685252..., and 163.)

Journal of the Empty Set, or how to fail your defense

Mark Dominus writes about a notional "Journal of the Empty Set", which would publish papers on results about mathematical objects that don't actually exist. He asks: "But on the other hand, suppose you had been granted a doctorate on the strength of your thesis on the properties of objects from some class which was subsequently shown to be empty. Wouldn't you feel at least a bit like a fraud?"

There's a (perhaps apocryphal) story that's made the rounds in my department about one possible way to fail your thesis defense. Namely, it's said that a student went to their defense and began by defining the sort of group they were studying. There was a rather long list of conditions, and one of the examiners quickly proved there was no such group. The student therefore failed.

The good thing about combinatorics? That sort of thing doesn't happen. Usually we can actually construct explicit examples of the things we're talking about.

Rejecta Mathematica

Rejecta Mathematica "is a new, open access, online journal that publishes only papers that have been rejected from peer-reviewed journals (or conferences with comparable review standards) in the mathematical sciences. We are currently seeking submissions for our inaugural issue."

Not surprisingly, the second FAQ is "Is this some kind of joke?" -- the answer is in the negative.

Basically, the point is that papers in which people reprove known results, try seemingly promising techniques and show they don't work, and so on are of value. Papers will be accompanied by a letter from the author saying why the journal to which the paper was originally submitted rejected it.

It's an open-access journal, and the articles will be available online. I can't tell if there's a print version; the site seems to neither confirm nor deny this. It'll be interesting to see what comes out of this.

(Also, the journal's URL is math.rejecta.org; this seems to imply a possible branching out into other fields if this experiment goes well.)