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A monthly survey of math news
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This month's topics:

The latest largest prime

A press release from GIMPS (the Great Internet Mersenne Prime Search) was picked up by the BBC on January 20, 2016: "Largest known prime number discovered in Missouri," a title which led Evelyn Lamb to comment on Slate (January 22) that it sounds "a bit like this new prime number ... was found in the middle of some road." The BBC posting shows the beginning of a scroll-through of all of M74207281, as this number ($2^{74207281}-1$) is called, with perky musical accompaniment. There's twenty minutes' worth; the complete file is available from the press release: click on "22,338,618 digits" to savor it at your leisure.

On Slate the coverage is more extensive, and quite enthusiastic. "Here's What's So Exciting About the New Largest Prime Number Mathamaticians Discovered" is the link from their fron page. As Lamb tells us, "GIMPS ... is a large distributed computing project in which volunteers run software to search for prime numbers." She tells us what Mersenne primes are ("The M in GIMPS and in M74207281 stands for Marin Mersenne, a 17th-century French friar [but much more -TP] who studied the numbers that bear his name. Mersenne numbers are 1 less than a power of 2. Mersenne primes, logically enough, are Mersenne numbers that are also prime. The number 3 is a Mersenne prime because it's one less than $2^2$ which is 4") and what we've lost by concentrating on the Mersennes: she uses the prime number theorem to estimate that "There are about $10^{22,338,610}$ primes less than M74207281, and approximately all of them are between it and the next-largest known prime." She also explains the reason for that concentration: there is a special primality test "that can determine whether a number is prime without actually factoring it," that only works for Mersenne numbers. For a lucid description of this test, the Lucas-Lehmer algorithm, listen to Matt Parker on Numberphile.

The story also ran in the New York Times on January 22. Kenneth Chang's "New Largest Prime Number? It's Really, Really Long" starts: "The largest known prime number, newly discovered, is almost five million digits longer than the previous record-holder." Chang quotes George Woltman, who founded the GIMPS project after he retired 20 years ago: "I've always been interested in prime numbers. I had a lot of time on my hands." And he spoke with Curtis Cooper (University of Central Missouri) whose computer found M74207281: "one of the early enthusiasts, joining Gimps in 1997. He has the program currently installed on 800 PCs on the university's two campuses. Dr. Cooper does research in the mathematical realm of number theory and teaches computer science classes. 'This kind of marries the two fields together,' he said."

A nice philosophical point: Chang also tells us that the UCM computer (PC No. 5 in Room 143) "churned for 31 days before completing its calculation that $2^{74207281}-1$ is a prime," and reporting the result, on September 17, 2015. But because of a "glitch on the server" no human being was informed. The official discovery date is January 7, 2016 when the message was picked up during routine maintenance.

Undecidability in physics

"Undecidability of the spectral gap," by Toby S. Cubitt, David Perez-Garcia and Michael M. Wolf, was published in Nature, December 10, 2015. From their abstract: "The spectral gap --the energy difference between the ground state and first excited state of a system-- is central to quantum many-body physics. Many challenging open problems ... are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum many-body system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable."

"The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding 'halting problem'."

physical system

A schematic representation of the physical system built from the Robinson tiling. Image from Nature 528 207-211, used with permission

An interesting mathematical aspect of the construction is authors' use of planar tilings as the link between Turing machines and physics. On the one hand they were able to translate the geometry of a tiling into the configuration of an (ideal) physical system for which a problem, equivalent to the mass-gap problem, was also equivalent to the completion problem for the tiling: can a given partial tiling can be extended to the entire plane? On the other hand they exploited work done by Hao Wang, Robert Berger and Raphael Robinson in the 1960s and 70s on the decidability of of the completion problem. In particular they cite Robinson's 1971 article (Inventiones Math. 12 177-209), which gives a specific way of representing a Turing machine by a collection of tiles with matching rules. Robinson states: "We see that the tiling of the plane can be completed with the tiles mentioned, after the initial tile has been laid down, if and only if the Turing machine never halts." Since Turing had proved that the halting problem is undecidable, the same must hold for the completion problem of the corresponding tiling.

tiles

The six tiles invented by Raphael Robinson, which can tile the plane but only aperiodically. Because these tiles are essentially square-shaped (unlike Penrose tiles), one row can be interpreted as the state and input of a Turing machine; then the next row, determined by the matching rules, givs the next state. Public domain image from Wikipedia.

tiling

The trace of a Robinson tiling. The brown marks on the tiles fit together to give the blue and brown interlocking squares. (More detail in this image by Chaim Goodman-Strauss). Since smaller squares group four by four with larger ones, this pattern cannot be periodic. In the image from Nature, only the blue squares (odd-numbered sizes) are shown.

The article was highlighted in a "News and Views" piece by Davide Castelvecchi in the same issue of Nature: "Paradox at the heart of mathematics makes physics problem unanswerable," showing photographs of Gödel and Turing. "Gödel's incompleteness theorems are connected to unsolvable calculations in quantum physics."

What do marine biologists think of math?

If we really want to know, there's a piece in the Careers section of Nature (December 10, 2015) that spells it out. Chris Woolston, a freelance writer, starts with "These days in science, there's no escape from maths in any scientific discipline, even in one like marine biology, historically lighter on sums than, say, molecular biology or quantitative genetics. But nobody should let maths jitters deter them if their call is to study ocean life." He discusses this point with some active marine biologists. A sample:

  • Milton Love (Marine Science Institute, UCSB). "'I failed eighth-grade algebra' he says. ... He ended up squeaking through a lower-level calculus course, and went on to build a fulfilling career in research without ever feeling comfortable with the numerical side of his work. 'I always managed to finesse the whole thing,' he says." (We see a picture of him "research[ing] fish from a manned submersible") "Love says that many of the required or core maths courses for both undergraduate and graduate students seem designed more to weed out degree candidates or to complete a rite of passage than to prepare students for scientific careers. 'The first couple of years as a biology major has nothing to do with a career in biology,' he says. 'It's not about calculus and physics.'"
  • Tammy Horton (National Oceanography Centre, Southampton, UK) "often shares a not-so-secret confession with her students. 'I'm very honest,' she says. 'I say I'm rubbish at maths. A lot of them breathe a sigh of relief.' As it happens, Horton's speciality, the taxonomy of small deep-sea crustaceans, does not require much quantitative skill. ... 'I'm very lucky that I don't have to use much maths,' she says. 'A lot of marine biologists use a huge amount of maths, and it's getting more mathematical all the time.'"
  • Steve Haddock (Monterey Bay Aquarium Research Institute, Moss Landing, California), co-authored of Practical Computing for Biologists. Haddock says that "Statistics programs such as PERMANOVA, and the increasingly popular R, have levelled the playing field." A warning: "Scientists who do not feel comfortable with numbers need at least to develop an intuitive sense of the problem that they are trying to address, he says, so that they know which part of the program to use. And, he adds, they need to have at least a general feeling for the data so that they can sort out the plausible results from the outlandish. 'If you can't do all of the calculations, you should at least be able to make a ballpark estimate,' he says.
    "From his own experience and conversations with other scientists, Haddock believes that many biologists get counterproductive instruction that erodes their confidence with numbers. 'I would blame math anxiety more on their educational history and less on their innate abilities,' he says. He recalls, for example, a poorly run biology statistics class in his graduate programme [where] the instructor started by mathematically deriving the rationale for the t-test ... which they were unlikely to understand and even less likely to use in the future."
  • Elena Sarropoulou (Hellenic Centre for Marine Research, Crete, Greece). "'I tell all undergrads and grad students to take a statistics class and to learn the programming language Python,' she says. 'Just the basics ...' She maintains that marine biologists ... do need to know enough [mathematics] to be able to design an experiment with the appropriate sample size and other parameters to address the problem that they are trying to solve."

Woolston lists some "Resources for mathophobes" including the website StackExchange, Vi Hart and free online audio-visual tutorials from the Elementary Maths for Biologists course, Cambridge, UK. In case none of those helps, he ends the list with a link to E.O. Wilson's Great Scientist $\neq$ good at math from the Wall Street Journal and the quote: "Many of the most successful scientists in the world today are mathematically no more than semiliterate."

Tony Phillips
Stony Brook University
tony at math.sunysb.edu

Math Digest Math Digest
On Media Coverage of Math

Math Digest includes posts throughout each month, with summaries of math stories and unique insights (and occasionally videos, interviews and podcasts) on math-related topics recently covered by the media. We welcome 2015 AMS-AAAS Media Fellow Rachel Crowell. Here is her first item for Math Digest.

Recently posted:

On slicing pizza, by Rachel Crowell

Imagine serving homemade pizza to a group of children with strong preferences about crust and toppings. All children in the group will be aware of the amount of pizza given to each child. You wonder: Is there a way you can slice the pizza that will result in equal-sized pieces with different properties based on what each child wants?  An article in New Scientist describes how mathematics--specifically monohedral disk tiling--can help you tackle this problem in a few simple steps, as shown in the chart below:

Process for cutting pizza

First, you must decide on an odd number of sides that you want each of your pieces of pizza to have. The curved pieces are described by the number of sides they have. For example, if you want to cut the pizza into five-sided slices, your slices with be 5-gons. This pattern continues. Joel Haddley told New Scientist, "Mathematically there is no limit whatsoever" to the number of sides your slices can have, but the logistics of exceeding 9-gon slices may be challenging.

Once you have decided on the number of sides for your pieces, the next step is to grab your already-cooked pizza and start slicing. Starting at one side of the pizza, cut in a curved pattern to the opposite side of the pizza. Starting from another side of the pizza, make another cut to the side of the pizza opposite the side where you started your second cut. Repeat this step until the number of curved cuts you have made is equal to the number of sides you want your slices to have. For example, if you want 7-gon slices, make seven such cuts. Next, divide each slice in half. The resulting number of identically-shaped slices will be four times the number of sides your slices have. For example, if you cut a pizza into 7-gon slices using this method, you will end up with 28 identically-shaped pieces. If you want to make the pizza-consumption experience even more thrilling for your young guests, you can create eccentrically-shaped pieces by slicing a wedge out of one corner of each shape.

Haddley told New Scientist that while he has used the results of his work with colleague Stephen Worsley to slice real pizzas, he is unsure if there are other applications of their results.  For now, we can all at least rest assured that there are ways to divide a pizza equally among people who want their slices to have different properties.

See "Mathematicians invent new way to slice pizza into exotic shapes," by Jacob Aron. New Scientist, 8 January 2016. Read Haddley and Worsley’s publication "Infinite families of monohedral disk tilings."

--- Rachel Crowell

Also now on Math Digest: Legal battle over Grothendieck's papers, slicing pizza, Picasso and Pollock exhibits, Ada Lovelace and more.

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Citations for reviews of books, plays, movies and television shows that are related to mathematics (but are not aimed solely at the professional mathematician). The alphabetical list includes links to the sources of reviews posted online, and covers reviews published in magazines, science journals and newspapers since 1996.

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