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What's Happening in the Mathematical Sciences, Volume 10

Dana Mackenzie and Barry Cipra
American Mathematical Society
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

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 ( 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.

Origami: Unfolding the Future by Dana Mackenzie

The ancient Japanese art of paper-folding is going high-tech, as engineers invent new devices that deploy or undeply folding. These inventions lead in turn to challenging mathematical problems about assembly pathways, defects, and curved folds in flat materials.

Prime Clusters and Gaps: Out-Experting the Experts by Dana Mackenzie

Mathematics got its real-life Walter Mitty story in 2013, when Yitang Zhang shocked number theorists with the first finite upper bound on the minimum size of prime gaps. One of the oldest problems in number theory, the Twin Prime Conjecture, may now be within reach.

The Truth Shall Set Your Fee by Barry Cipra

When you pay a stranger, especially online, for help, how can you be sure you're getting honest answers? A new theory in computer science shows how rational self-interest dovetails with the pressing need for trustworthy computation.

Climate Past, Present, and Future by Dana Mackenzie

Change is everywhere you look in Earth’s climate, and always has been. Throughout climate science, mathematical models help sort out what did happen (mass extinctions), what is happening (melting ice sheets), and what might happen (tipping points).

Following in Sherlock Holmes' Bike Tracks by Dana Mackenzie

In a story published in 1905, Sherlock Holmes incorrectly deduced which way a bicycle went, on the basis of its tracks. The subtle relationship between a bike’s front and rear tracks recently helped mathematicians solve another Victorian-era problem on the operation of planimeters.

Quod Erat Demonstrandum by Barry Cipra

A proof is a kind of mathematical poem—and sometimes an epic one, at that. Two recent proofs, each years in the making, show the lengths to which mathematicians will go in the dogged pursuit of truth, including, these days, enlisting computers to double check their logic.

The Kadison-Singer Problem: A Fine Balance by Dana Mackenzie

Great problems come in many disguises. The Kadison-Singer problem, first posed as a problem in theoretical physics, popped up in many other mathematical contexts over more than half a century until it was finally solved in 2013 graph theorists.

A Pentagonal Search Pays Off by Barry Cipra

Finding shapes that tile the plane isn’t hard to do. Finding all of them is trickier. Mathematicians still don't know how many different convex pentagons are capable of tiling the plane. But the list, long stalled at 14, just inched up, thanks to a new algorithm and a computer search.

The Brave New World of Sports Analytics by Dana Mackenzie

In the last few years, professional sports have been swept a new wave of statistical methods, or “analytics.” These methods, coupled with new data sources like video capture, quantify elusive skills and challenge cherished assumptions about in-game strategy.