What's Happening in the Mathematical Sciences, Volume 10

This year’s collection of articles on recent developments in mathematics is among the best of the series. The range and variation of topics is unusually large. There are nine articles, six by Mackenzie and three by Cipra. These two are mathematical freelance writers, both skillful in conveying the essence of mathematical ideas and why they might be of interest.

The articles are generally accessible at the undergraduate level, at least in the early going. (They follow the model of a good colloquium talk: the beginning is accessible to everyone in the audience.) Some of the articles get into more advanced material, but they do so gracefully and give less experienced readers a sense of how the work continues to develop.

The topics range from number theory (gaps in the primes) to climate modeling, a new tiling of the plane by convex pentagons, sports analytics, origami and a good deal more. Not everything would be equally interesting to everybody. The following are a few things that caught my interest.

The article on climate effectively describes issues in understanding the past and present climate, how mathematicians can contribute to the effort, and the value of focused smaller scale climate modeling. It also discusses tipping points — gradual changes in system parameters that cause an abrupt shift from one stable equilibrium to another.

A partitioning or fair-sharing challenge, technically known as the Kadison-Singer problem, led its investigators on a merry chase into the theory of partitions, frame theory, and a brand new theory of interlacing polynomials. What might eventually result is a much more efficient algorithm for what some signal processing experts call the de- interleaving problem, the task of efficiently disentangling a gaggle of separate signals.

Although it solves no major world problems, the story of the discovery of a new pentagonal tiling is fun to read. Regular pentagons cannot tile the plane, but fourteen families of convex pentagons that can tile the plane were known. Two mathematicians at the University of Washington at Bothell devised an algorithm to search for new pentagonal tilings and ran it on a high performance computing cluster. The computer program produced a list of candidates that needed to be hand-checked. Last July a fifteenth family of convex pentagonal tilings was discovered. It was pleasingly simple with familiar angles and a nice decomposition into right trangles.

This is a great collection, well worth investigating.

One suggestion for future collections in this series: include at least a short list of references for each article. Some of the articles in the current collection include a reference or two in the text, but a bit more would really help those who would want to explore further.

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Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.