{"id":1012,"date":"2016-05-09T09:10:58","date_gmt":"2016-05-09T14:10:58","guid":{"rendered":"http:\/\/blogs.ams.org\/beyondreviews\/?p=1012"},"modified":"2018-03-14T08:34:29","modified_gmt":"2018-03-14T13:34:29","slug":"mathematical-moments","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/beyondreviews\/2016\/05\/09\/mathematical-moments\/","title":{"rendered":"Mathematical Moments"},"content":{"rendered":"

\"Mathematical<\/a>The AMS Public Awareness office has a wonderful series of short bursts of mathematics, which are available as posters titled “Mathematical Moments<\/a>“. \u00a0Mike Breen<\/a>\u00a0comes up with the topics and writes the texts for the posters. \u00a0He also has a knack<\/a> for\u00a0finding clever titles. \u00a0Many of the posters have to do with some piece\u00a0of mathematics that has relevance for everyday life. \u00a0All of them portray interesting, usually deep mathematics in a way that most non-mathematicians can appreciate. \u00a0They are also very popular in Mathematics Departments\u2020<\/a>. \u00a0 In what follows, I want to highlight a few of\u00a0the Mathematical Moments and use\u00a0MathSciNet to dig more deeply into their\u00a0subjects.\u00a0If you would like printed copies of Mathematical Moments, contact the AMS Public Awareness Office at paoffice@ams.org<\/a>.<\/p>\n

Being Knotty<\/h2>\n

\"Borromean-3\"<\/a><\/p>\n

Mike takes a Moment<\/a>\u00a0to talk about the Borromean rings<\/a>, which then provides\u00a0a hook into the general topic of knot theory. \u00a0The Borromean rings are three loops arranged in \"mathbb{R}^3\" so that no two of them are linked, but the group of three is linked. \u00a0If you look closely, you will notice that the logo of the IMU<\/a> is a clever representation of the Borromean rings. \u00a0As with quite a few of the Math Moments, there is an accompanying\u00a0podcast<\/a> with a mathematician who has worked on the topic. \u00a0In this case, the mathematician is Colin Adams<\/a>, author of\u00a0The Knot Book<\/a>. \u00a0Adams is known to many people not just for his mathematics, not just for his writing, but also for his performance skills<\/a>, which are often on display at the Joint Mathematics Meetings<\/a>.<\/p>\n

MathSciNet has thousands of entries<\/a> on knot theory, including\u00a0some\u00a0directly on the topic of\u00a0the Borromean rings<\/a>. \u00a0Here are a couple that are pitched at a general mathematical audience<\/p>\n

MR1601839<\/a>
\nCromwell, Peter<\/a>(4-LVRP-PM); Beltrami, Elisabetta<\/a>(I-PISA); Rampichini, Marta<\/a>(I-MILAN)
\nThe Borromean rings.
\nMath. Intelligencer 20 (1998), no. 1, 53\u201362.<\/p>\n

MR3356112<\/a>
\nChamberland, Marc<\/a>(1-GRIN-MS); Herman, Eugene A.<\/a>(1-GRIN-MS)
\nRock-paper-scissors meets Borromean rings.
\nMath. Intelligencer 37 (2015), no. 2, 20\u201325.<\/p>\n

And here is a deep result involving\u00a0the Borromean rings:<\/p>\n

MR0872484<\/a>
\nFreedman, Michael H.<\/a>(1-UCSD)
\nAre the Borromean rings A-B-slice?
\nSpecial volume in honor of R. H. Bing (1914\u20131986).
\nTopology Appl. 24 (1986), no. 1-3, 143\u2013145.<\/p>\n

In this paper Freedman is moving forward from his major paper on the topology of four-manifolds<\/a>. \u00a0An important case for the extension has to do with the Borromean rings!<\/p>\n

Finding Friends<\/h2>\n

\"Facebook<\/a>This is the\u00a0Moment<\/a>\u00a0to address some mathematics related to\u00a0Facebook<\/a>. \u00a0In particular, it is about how large, complex networks often\u00a0exhibit structure. \u00a0The notion of “six degrees of separation<\/a>” is a manifestation of one type of\u00a0structure. \u00a0And this structure is what makes the Kevin Bacon game<\/a> fun. \u00a0In mathematics, we have Erd\u0151s numbers<\/a>, which you can compute using MathSciNet<\/a>. There is also\u00a0a Erd\u0151s\u2013Bacon number<\/a>. Mathematicians have looked into the regularity of large networks. \u00a0One such mathematician is Jon Kleinberg<\/a>, who is Mike Breen’s guest in the podcast<\/a>. \u00a0Kleinberg has written about one hundred papers, and cowrote a good book on networks:<\/p>\n

MR2677125
\n<\/a>Easley, David<\/a>(1-CRNL-EC<\/a>); Kleinberg, Jon<\/a>(1-CRNL-C<\/a>)
\nNetworks, crowds, and markets.\u00a0Reasoning about a highly connected world.
\nCambridge University Press, Cambridge, 2010. xvi+727 pp. ISBN: 978-0-521-19533-1<\/p>\n

Another interesting work on the subject is<\/p>\n

Collective dynamics of ‘small-world’ networks<\/a>
\nDuncan J. Watts and Steven H. Strogatz
\nNature<\/em>\u00a0393<\/strong>,\u00a0440-442 (4 June 1998)<\/p>\n

This is a harder read than the Easley and Kleinberg book, but a\u00a0creative version of the paper<\/a> is available on the web. \u00a0The web version uses some of the methods\u00a0of graphic novels to express the content of the paper. \u00a0Give it a try! \u00a0An easier read by\u00a0Strogatz is his very successful book, Sync<\/a><\/em>.<\/p>\n

Keeping the beat<\/h2>\n

\"Photo<\/p>\n

This\u00a0Moment<\/a>\u00a0is a look at the mathematics involved in cardiac electrophysiology. \u00a0An early contributor to the subject was George Ralph Mines<\/a> (1886\u20131914). \u00a0Mines is remembered\u00a0for at least two contributions to electrophysiology, but is also notable\u00a0for (probably) killing himself through self-experimentation. \u00a0Despite the seeming intractability of the problem, there are workable mathematical models for the electrical activity in the heart. \u00a0The starting point of most of them is the work of\u00a0Alan Hodgkin and Andrew Huxley in the 1950s. \u00a0 The AMS Notices<\/em> published\u00a0a good introduction to the subject written by John W. Cain:<\/p>\n

MR2807520<\/a>
\nCain, John W.<\/a>(1-VACW)<\/a>
\nTaking math to heart: mathematical challenges in cardiac electrophysiology.<\/span>
\nNotices Amer. Math. Soc.<\/em><\/a> 58 <\/a>(2011), <\/a>no. 4,<\/a> 542\u2013549.
\n[Full-text<\/a>]<\/p>\n

Cain is interviewed by Mike Breen for the\u00a0podcast<\/a>\u00a0associated to the Moment.<\/p>\n

There are various reasons that modeling electrical activity in the heart should be difficult: \u00a0you can’t easily do experiments on the heart of a living patient; measurements are often indirect; the topology of the heart is complicated. \u00a0Even so, it is possible to know a lot. \u00a0James Keener<\/a> is a mathematician at the University of Utah who has studied the problem and has \u00a0coauthored one of the standard books on mathematics and physiology:<\/p>\n

MR1673204<\/a>
\nKeener, James<\/a>(1-UT)<\/a>; Sneyd, James<\/a>(1-MI)<\/a>
\nMathematical physiology.<\/span>
\nInterdisciplinary Applied Mathematics, 8.<\/a> Springer-Verlag, New York,<\/em> 1998. xx+766 pp. ISBN: 0-387-98381-3
\n92C30 (92-01)<\/a><\/p>\n

\u00a0There is plenty more that mathematics has to say about physiology. \u00a0To see what MathSciNet has on the subject, search on the Mathematical Subject Class\u00a092C30<\/a> =<\/strong>\u00a0Physiology (general).<\/p>\n

Packing it in<\/h2>\n

\"Cube<\/a><\/p>\n

Mike spends a Moment<\/a> talking about packing problems, specifically bin packing problems. \u00a0To get a sense of how hard packing problems can be, it took several centuries to prove\u00a0that the “obvious” optimal packing of spheres in\u00a0\"mathbb{R}^3\" is really the most efficient arrangement, a conjecture due to Kepler<\/a> in the 17th century and finally resolved by Thomas Hales<\/a> in 2005<\/a>. \u00a0Trying to find optimal solutions to packing more general shapes gets much harder very fast. \u00a0If 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. \u00a0This simple approach turns out to get you within 20% of the best solution.<\/p>\n

Suppose you want to try packing tetrahedra. \u00a0(You might want to do this if you are a materials scientist, or if you are a mathematician.) \u00a0The search for best packing of tetrahedra became something of a race. \u00a0The tale is compellingly told by\u00a0Jeff Lagarias<\/a> and Chuanming Zong<\/a> in an article in the AMS\u00a0Notices<\/em><\/a>. \u00a0As far as I know, the densest packing<\/a>\u00a0to date is that discovered by Chen<\/a>, Engel<\/a>,\u00a0and Glotzer<\/a>, with a density of\u00a00.856347.<\/p>\n

The optimal packing of spheres in two dimensions is the hexagonal packing, which was known to be best possible for centuries. \u00a0Hales solved the sphere-packing problem in dimension three. \u00a0But what about higher dimensions? \u00a0Well, not much is known. \u00a0However, very recently, the problem was solved in dimensions 8 and 24 by\u00a0Maryna Viazovska<\/a>. \u00a0These two dimensions each host a special lattice that provides the best packing. \u00a0In dimension 8, the lattice is the $E_8$ lattice<\/a>. \u00a0In dimension 24, it is the Leech lattice<\/a>. \u00a0Packing problems are often solved with the help of number theory. \u00a0In these two cases, the number theory was quite advanced, involving modular forms<\/a>.\u00a0(For more on modular forms, see the paper<\/a> by Zagier<\/a>.)\u00a0 There were several write-ups in the media, including this article<\/a> by\u00a0Erica Klarreich. \u00a0The $E_8$ paper is available on the arXiv here<\/a>. \u00a0The dimension 24\u00a0case, which was done by Viazovska and Henry Cohn<\/a>, Abhinav Kumar<\/a>, Stephen D. Miller<\/a>, and Danylo Radchenko<\/a>, is on the arXiv here<\/a>.<\/p>\n

Packing problems are not just limited to physical objects. \u00a0They come up in materials sciences, in information sciences, in scheduling theory, and in many other areas of optimization.<\/p>\n

 <\/p>\n

\n

There are many\u00a0more Mathematical Moments available here<\/a>. \u00a0I invite you to sample a few, then use MathSciNet to dig more deeply into the subject.<\/p>\n

\n
\u2020<\/h5>\n

I have visited lots of mathematics departments, and many of them have large collections of the Mathematical Moments posters. \u00a0For instance, for a long time, the department at Pomona College<\/a> had a complete set on their walls. \u00a0Unfortunately, Mike created more posters than Pomona could fit. \u00a0Recently, Pomona College\u00a0completely rebuilt the building<\/a> for the department. \u00a0I haven’t visited yet. \u00a0Perhaps they have found room for all 122 (and counting) of the Mathematical Moments!<\/p>\n

<\/h4>\n

\u00a0<\/span><\/p>\n

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The AMS Public Awareness office has a wonderful series of short bursts of mathematics, which are available as posters titled “Mathematical Moments“. \u00a0Mike Breen\u00a0comes up with the topics and writes the texts for the posters. \u00a0He also has a knack … Continue reading →<\/span><\/a><\/p>\n

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