Mathematics reaps the benefits of physical insight

A few days ago, slashdot linked to this essay by M. du Sautoy in Seed on the relationship between the Riemann Hypothesis and quantum physics. I've been consistently impressed by Seed's provision of scientific reporting that neither glosses over significant concepts nor drowns the non-specialist in technical details. I think it answers the scientist's perpetual complaint against journalists' relationship with science, as summarized by this recent opinion piece in the Notices of the AMS. Now, if only more media outlets would follow Seed's excellent example.

On a tenuously related note; the author's website is by far the most elaborate I've ever seen among academics. It's refreshing to see such an open rejection of the reverse snobbery so common among our ilk.

A formula for misery

R. Stevens does it again; this time, he illustrates why Math is Life.

The other side of the arXiv

It turns out that some decent papers are posted to arXiv:cs, where authors are not listed alphabetically.

Convex Separation from Optimization via Heuristics
L. M. Ioannou, B. C. Travaglione, D. Cheung

It's good to see convex geometry presented in a decidedly algorithmic setting, since it really thrives there.

No opportunity for subtle wordplay

The fourth Abel prize has been awarded to L. Carleson of the Royal Institute of Technology, Sweden "for his profound and seminal contributions to harmonic analysis and the theory of smooth dynamical systems." The Slashdot story mentions that "[h]is theorems have been helpful in creating [the] iPod," and includes a decent layman's explanation of Prof. Carleson's contributions to analysis. Essentially, he proved that using Fourier series to encode functions is theoretically sound, and not just "effective in practice."

I name my Apple products after mathematicians; my iPod shuffle is Diaconis, my Airport Express is Heddy (after H. Lamar), and my iPod (with video), after careful consideration, was christened Fourier, owing to its algorithmic dependence on Fourier series. Perhaps Carleson would have been a better choice. Well, there will probably be an iPod in need of a name in the future.

Watershed

The project I mentioned in some previous posts is now on the arXiv.

On the local structure of doubly laced crystals
P. Sternberg

It's good to have publicly visible work.

Non, c'est pas une Weezer logo

As part of a run of strips on superheroes, R. Stevens produced a comic in which a character points out that "... Justice League is not a reputable science journal." Buzzwords from mathematical physics puncuate the retort that follows.

I know what the golden "W" in the comic's header stands for, but I can't help but wish it was meant to recognize E. Witten as the strongest driving force in string theory.

Many scientifically concerned individuals bemoan the lack of public interest in science and mathematics. Perhaps what we need is better personal brand management, beginning with a broad campaign of brand identification.

Here, I'll start.

What's a five-letter word for "Surfaces of genus zero with three boundary cycles"?

Any situation can be made humorous (or more so, if it's already funny) by the addition or subtraction of pants or monkeys. At least, so said a dear friend of mine several years ago. I'm inclined to believe him.

As far as I know, there is no mathematical object known as a monkey; until last night, I thought the same was true of pants. As this nomenclature is due to W. Thurston, it has been suggested that they should be called Thurstonian pant-pairs.

A colleague of mine has written a paper on the subject.

Heegaard Splittings and the Pants Complex
J. Johnson

A nice round number

If you write dates with the month before the day, today is 3/14, or more affectionately, π day. Those who are especially meticulous celebrate at a moment about half a minute past 1:59. Yesterday, the Independent ran an article celebrating the wonderful properties and history of this number. Unfortunately, the following sentence appeared near the end of the story:

Pi, you see, is always going to be represented by an approximation because, like all irrational numbers, its digits never really end.

This is perfectly acceptable for a pre-calculus audience, but anyone who has been exposed to infinite sums deserves to see some of the miraculous closed-form expressions for π, such as the following, famously due to Ramanujan.

Happy π day!

Physics and Sociology, helping each other

The history of Mathematical Physics includes many examples of both of its constituent disciplines providing insight to the other. The mathematical tools of differential equations, linear algebra, and manifolds have allowed for precise descriptions of physical models; at the same time, formulas such as the Riemann-Roch Theorem and the ongoing X=M Conjecture could never have been discovered without deep physical insight.

In this article (which I found by way of Slashdot) we see an example of Physics informing the study of social networks. The authors point out that Physics is, in some sense, just returning the favor Sociology lent it about 100 years ago by inspiring the use of probabilistic models for gases, rather than relying on determinism.

Personally, I want to know if a mechanistic relationship exists between statistical particle models and large groups of people. Sadly, such a connection is missing from the current stage of this research; not surprising, since it is still so new. Once that bridge is built, however, a new paradigm for complex systems will likely emerge. There's no telling what we might be able to do then.

Look, biological imaging is actually useful

A group led by David Shapiro of SUNY Stony Brook has developed a new algorithm for image reconstruction from X-ray diffraction microscopy. And as has been reported on Cornell's news site, it is easily adapted to quickly solve sudoku puzzles.

So, will a sizable number of young Americans realize that combinatorialists solve puzzles like sudoku, and decide that they want to pursue discrete math as a major, or even a career?

For future reference

It's always a joy (if a slightly vain one) to see one's work cited, as in this paper which makes an important step in the program to understand the representation theory of quantum affine algebras:

Paths and tableaux descriptions of Jacobi-Trudi determinant associated with quantum affine algebra of type D_n
W. Nakai, T. Nakanishi

I've received a few citations before, but the sense of recognition that comes from seeing my name on the page is still strong. According to researchers who are considerably my senior, this feeling is very slow to fade.

It reminds me of a point of advice made by Gian-Carlo Rota at a 1996 conference in his honor:

8 Give lavish acknowledgments

I have always felt miffed after reading a paper in which I felt I was not being given proper credit, and it is safe to conjecture that the same happens to everyone else. One day, I tried an experiment. After writing a rather long paper, I began to draft a thorough bibliography. On the spur of the moment, I decided to cite a few papers which had nothing whatsoever to do with the content of my paper, to see what might happen.

Somewhat to my surprise, I received letters from two of the authors whose papers I believed were irrelevant to my article. Both letters were written in an emotionally charged tone. Each of the authors warmly congratulated me for being the first to acknowledge their contribution to the field.

One may conclude that the marginal cost of liberal citation is overwhelmed by the potential for improved collegiality.

We don't need no (constructivist) education!

In the world of mathematical research, the word "constructivism" refers to a debate addressing the existence of objects that cannot be explicitly constructed. If such assumptions are disallowed, some aspects of the infinite become very difficult to deal with. However, in the field of mathematics education (which is simultaneously close to and distant from mathematical research), constructivism instead applies to an approach to learning based on individual experience.

For a little more than a decade, so-called Math Wars have raged across the country. Textbooks and curricula advocating an exclusively constructivist approach to mathematics have been adopted by many boards of education both on the state and local levels. A very insightful comparison between a constructivist and a deductionist approach to the Pythagorean Theorem was written by G. D. Chakerian and K. Kreith of the UC Davis Department of Mathematics when the Math Wars reached the secondary schools of Davis.

What I find saddest about this situation is that Mathematics education does benefit from taking constructivism into account. The best teachers from whom I've ever had the pleasure to learn made excellent use of examples to be worked out by the student, often when only a partial understanding of the mathematical theory had been presented. However, these instructors made absolutely certain to declare clearly to the students the general principles governing the behavior of these examples. Examples present students with data; theorems provide a framework for making sense of these data. Without teaching students the theorems that control the behavior of the numbers that surround us, we tell them either that a few examples suffice to determine general behavior, or that every situation is a special case and general principles cannot inform our understanding. I'm not sure which is a more dangerous lesson to learn.