The AMS Public Awareness office has a wonderful series of short bursts of mathematics, which are available as posters titled “Mathematical Moments“. Mike Breen comes up with the topics and writes the texts for the posters. He also has a knack for finding clever titles. Many of the posters have to do with some piece of mathematics that has relevance for everyday life. All of them portray interesting, usually deep mathematics in a way that most non-mathematicians can appreciate. They are also very popular in Mathematics Departments†. In what follows, I want to highlight a few of the Mathematical Moments and use MathSciNet to dig more deeply into their subjects. If you would like printed copies of Mathematical Moments, contact the AMS Public Awareness Office at email@example.com.
Mike takes a Moment to talk about the Borromean rings, which then provides a hook into the general topic of knot theory. The Borromean rings are three loops arranged in so that no two of them are linked, but the group of three is linked. If you look closely, you will notice that the logo of the IMU is a clever representation of the Borromean rings. As with quite a few of the Math Moments, there is an accompanying podcast with a mathematician who has worked on the topic. In this case, the mathematician is Colin Adams, author of The Knot Book. Adams is known to many people not just for his mathematics, not just for his writing, but also for his performance skills, which are often on display at the Joint Mathematics Meetings.
And here is a deep result involving the Borromean rings:
In this paper Freedman is moving forward from his major paper on the topology of four-manifolds. An important case for the extension has to do with the Borromean rings!
This is the Moment to address some mathematics related to Facebook. In particular, it is about how large, complex networks often exhibit structure. The notion of “six degrees of separation” is a manifestation of one type of structure. And this structure is what makes the Kevin Bacon game fun. In mathematics, we have Erdős numbers, which you can compute using MathSciNet. There is also a Erdős–Bacon number. Mathematicians have looked into the regularity of large networks. One such mathematician is Jon Kleinberg, who is Mike Breen’s guest in the podcast. Kleinberg has written about one hundred papers, and cowrote a good book on networks:
Easley, David(1-CRNL-EC); Kleinberg, Jon(1-CRNL-C)
Networks, crowds, and markets. Reasoning about a highly connected world.
Cambridge University Press, Cambridge, 2010. xvi+727 pp. ISBN: 978-0-521-19533-1
Another interesting work on the subject is
Collective dynamics of ‘small-world’ networks
Duncan J. Watts and Steven H. Strogatz
Nature 393, 440-442 (4 June 1998)
This is a harder read than the Easley and Kleinberg book, but a creative version of the paper is available on the web. The web version uses some of the methods of graphic novels to express the content of the paper. Give it a try! An easier read by Strogatz is his very successful book, Sync.
Keeping the beat
This Moment is a look at the mathematics involved in cardiac electrophysiology. An early contributor to the subject was George Ralph Mines (1886–1914). Mines is remembered for at least two contributions to electrophysiology, but is also notable for (probably) killing himself through self-experimentation. Despite the seeming intractability of the problem, there are workable mathematical models for the electrical activity in the heart. The starting point of most of them is the work of Alan Hodgkin and Andrew Huxley in the 1950s. The AMS Notices published a good introduction to the subject written by John W. Cain:
Cain is interviewed by Mike Breen for the podcast associated to the Moment.
There are various reasons that modeling electrical activity in the heart should be difficult: you can’t easily do experiments on the heart of a living patient; measurements are often indirect; the topology of the heart is complicated. Even so, it is possible to know a lot. James Keener is a mathematician at the University of Utah who has studied the problem and has coauthored one of the standard books on mathematics and physiology:
There is plenty more that mathematics has to say about physiology. To see what MathSciNet has on the subject, search on the Mathematical Subject Class 92C30 = Physiology (general).
Packing it in
Mike spends a Moment talking about packing problems, specifically bin packing problems. To get a sense of how hard packing problems can be, it took several centuries to prove that the “obvious” optimal packing of spheres in is really the most efficient arrangement, a conjecture due to Kepler in the 17th century and finally resolved by Thomas Hales in 2005. Trying to find optimal solutions to packing more general shapes gets much harder very fast. If you have a bunch of irregular shapes, the simple guess is to start with the big objects, then try to fit in the smaller pieces. This simple approach turns out to get you within 20% of the best solution.
Suppose you want to try packing tetrahedra. (You might want to do this if you are a materials scientist, or if you are a mathematician.) The search for best packing of tetrahedra became something of a race. The tale is compellingly told by Jeff Lagarias and Chuanming Zong in an article in the AMS Notices. As far as I know, the densest packing to date is that discovered by Chen, Engel, and Glotzer, with a density of 0.856347.
The optimal packing of spheres in two dimensions is the hexagonal packing, which was known to be best possible for centuries. Hales solved the sphere-packing problem in dimension three. But what about higher dimensions? Well, not much is known. However, very recently, the problem was solved in dimensions 8 and 24 by Maryna Viazovska. These two dimensions each host a special lattice that provides the best packing. In dimension 8, the lattice is the $E_8$ lattice. In dimension 24, it is the Leech lattice. Packing problems are often solved with the help of number theory. In these two cases, the number theory was quite advanced, involving modular forms. (For more on modular forms, see the paper by Zagier.) There were several write-ups in the media, including this article by Erica Klarreich. The $E_8$ paper is available on the arXiv here. The dimension 24 case, which was done by Viazovska and Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko, is on the arXiv here.
Packing problems are not just limited to physical objects. They come up in materials sciences, in information sciences, in scheduling theory, and in many other areas of optimization.
There are many more Mathematical Moments available here. I invite you to sample a few, then use MathSciNet to dig more deeply into the subject.
I have visited lots of mathematics departments, and many of them have large collections of the Mathematical Moments posters. For instance, for a long time, the department at Pomona College had a complete set on their walls. Unfortunately, Mike created more posters than Pomona could fit. Recently, Pomona College completely rebuilt the building for the department. I haven’t visited yet. Perhaps they have found room for all 122 (and counting) of the Mathematical Moments!