### Tony Phillips' Take Math Digest Reviews Blog on Math Blogs

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# This month's topics:

## Szemerédi's Theorem in the Guardian

Endre Szemerédi's 2012 Abel Prize was reported in Nature ("Mathematician's 'irregular mind' scoops Abel award," by Philip Ball, March 21, 2012) and was picked up April 30 on the GRRLSCIENTIST blog hosted by the Guardian. One of Szemerédi's best-known achievements is his proof of a 1936 conjecture of Erdös and Turán (now known as Szemerédi's Theorem): for every density $0 < d < 1$ and every integer $k$ there exists a number $N(d, k)$ such that every subset of $\{1, ..., N\}$ of cardinality $dN$ contains an arithmetic progression of length $k$, provided $N > N(d, k)$. The blog briefly refers to this theorem (and gives an incorrect account of RSA encryption, not really related to the matter at hand) but links to a nice Numberphile video where James Grime (Cambridge) and David Hodge (Nottingham) walk us through it. The Numberphile site, "videos about numbers & stuff" made by Brady Haran, currently offers some 40 short, informative and entertaining videos.

## Topological analysis of basketball

The 452 2010-2011 NBA players appear as nodes in this network. Edges join players who are similar "with regard to points, rebounds, assists, steals, blocks, turnovers and fouls, all normalized to per-minute values in the 2010-2011 season. Further, the network is colored by a player's points-per-minute average, with blue being low and red being high." Images courtesy of Muthu Alagappan and Ayasdi. Higher resolution image.

This image was posted on the Wired website on April 30, 2012 (the quote is from their caption). The context was "Analytics Reveal 13 New Basketball Positions" by Wired's regular contributor Jeff Beckham; he was reporting on work by Muthu Alagappan, a senior at Stanford. Alagappan ("a basketball fan") has been working with Stanford's Applied and Computational and Algebraic Topology research group and its commercial sibling Ayasdi. He applied Ayasdi's topological data analysis algorithms to a year's worth of NBA statistics; as Beckham puts it, "For as long as basketball has been played, it's been played with five positions. ... . A California data geek sees 13 more hidden among them ... ." In fact Alagappan interprets the network as defining functional positions, to which he assigns the names on the display. This allows him to analyze the composition of individual teams from a new point of view.

The same network, with vertices from Mavericks players (left) and Timberwolves players (right) colored red. Higher resolution images: Mavericks, Timberwolves.

The exercise shows significant differences between teams. Beckham: "Alagappan proved the title-winning Mavs had a solid diversity of 'ball handlers' and 'paint protectors,' giving them the ability to put a balanced lineup on the floor with few weak spots. The ... Timberwolves, on the other hand, had too many players with similar styles and a dearth of 'scoring rebounders' and 'paint protectors,' leaving them vulnerable along the front line."

The Mavericks were also subjected to mathematical analysis by Jennifer Fewell and Dieter Armbruster, as reported in this column last January.

## Functional MRIs of Math Anxiety

"The Neurodevelopmental Basis of Math Anxiety," by Christina Young, Sarah Wu and Vinod Menon (all at Stanford School of Medicine), appears in the May 2012 issue of Psychological Science. It was picked up in the "Editor's Choice" section of Science for May 11, 2012. From the paper's abstract: "In a functional MRI study on 7- to 9-year-old children, we showed that math anxiety was associated with hyperactivity in right amygdala regions that are important for processing negative emotions. In addition, we found that math anxiety was associated with reduced activity in posterior parietal and dorsolateral prefrontal cortex regions involved in mathematical reasoning. ... Furthermore, effective connectivity between the amygdala and ventromedial prefrontal cortex regions that regulate negative emotions was elevated in children with math anxiety. These effects were specific to math anxiety and unrelated to general anxiety, intelligence, working memory, or reading ability. Our study identified the neural correlates of math anxiety for the first time, and our findings have significant implications for its early identification and treatment."

## Goldbach Conjecture progress in the Scientific American and Le Monde

The May 2012 Scientific American ran an item by Davide Castelvecchi: "Goldbach's Prime Numbers" (posted online with a different title). Castelvecchi reports on Terence Tao's announcement, described on his blog, that every odd integer larger than 1 is the sum of at most five primes. Castelvecchi: "The weak Goldbach conjecture says that you can break up any odd number into the sum of, at most, three prime numbers (numbers that cannot be evenly divided by any other number except themselves or 1). For example: $35 = 19 + 13 + 3$ or $77=53+13+11."$ This is distinguished from the strong conjecture (which, Castelvecchi tells us, is actually due to Euler): every even number larger than 2 is the sum of two primes. Castelvecchi summarizes recent work on the conjectures: "Mathematicians have checked the validity of both statements by computer for all numbers up to 19 digits, and they have never found an exception." In the weak case, I. M. Vinogradov proved in the 1930s that the conjecture is true for sufficiently large numbers. "Tao combined the computer-based results valid for small-enough numbers with the result that applies to large-enough numbers. By improving earlier calculations with 'lots of little tweaks,' he says, he showed that he could bring the two ranges of validity to overlap -- as long as he could use five primes." Castelvecchi reports that Tao hopes to go on to prove the complete weak conjecture, but that the strong conjecture seems much farther out of reach. He concludes: "Thus, a quarter of a millennium after Goldbach's death, no one even has a strategy for how to solve his big challenge."

The item was also picked up by David Larousserie in the online Le Monde on May 22, 2012. With the strong conjecture in mind, Larousserie remarks: "For a mathematician, taking small steps forward does not necessarily mean getting closer to the goal." He interviews Olivier Ramaré, the previous record holder ($\leq 6$ primes, 1995), who tells him "It may be a thousand years before we see a proof." But, "You never know." [My translation. TP]

## New Math Institute in Senegal

Martin Enserink reports in the May 4, 2012 Science on "the newest branch of the African Institute for Mathematical Sciences (AIMS)," which opened recently in M'bour, Senegal. (The original center is in South Africa). Funding higher education in Africa has been a hard sell partly because, as Enserink tells us, such projects "aren't among the UN's Millenium Development Goals." Nevertheless, the Institute at M'bour has been supported by Canadian funds and by the Senegalese government. "Students come for months of day-and-night immersion in high-level mathematics." The institute was strategically placed in M'bour, far enough (2.5 hours) from the "famed music and night-life scene" of Dakar. "Perhaps the main attraction here is the lecturers: internationally renowned mathematicians, who each come to spend 3 full weeks. They're extremely accessible; discussions often continue during the meals and into the night." Current plans are to open institutes in Ghana and in Ethiopia. "With more money, there could be a fifth AIMS, and a sixth, and so on, leading to a pan-African network of thousands of well-trained alumni." Enserink quotes Neil Turok, head of the Perimeter Institute in Waterloo, and the motivating spirit behind the Institutes: "It would cost \$100 million over the next 10 years -- that's about 0.003% of Africa's total aid budget."

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