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This month's topics:
Emmy Noether in The New York TimesBack on March 26, 2012, the New York Times ran an article about Emmy Noether on the front page of their Science section. "The Mighty Mathematician You've Never Heard Of", by Natalie Angier, focuses on the "20thcentury mathematical genius Amalie Noether" and on "the depths of her perverse and unmerited obscurity." Angier concentrates on Noether's Theorem, which "united with magisterial concision two conceptual pillars of physics: symmetry in nature and the universal laws of conservation," and which "undergirds much of today's vanguard research in physics, including the hunt for the almighty Higgs boson." After recounting Noether's background and early career, Angier returns to Noether's Theorem, with the nice example of a bicycle wheel: "If a bicycle wheel is radially symmetric, if you can spin it on its axis and it still looks the same in all directions, well, then, that symmetric translation must yield a corresponding conservation. By applying the principles and calculations embodied in Noether's theorem, you'll see that it is angular momentum, the Newtonian impulse that keeps bicyclists upright and on the move." Angier also records the "most profound" consequence of the theorem: "a symmetry of timelike the fact that whether you throw a ball in the air tomorrow or make the same toss next week will have no effect on the ball's trajectoryis directly related to the conservation of energy." Angier is very brief on Noether's mathematics: she published "groundbreaking papers, sometimes under a man's name, in rarefied fields of abstract algebra and ring theory." [ From AlKhwarizmi to Emmy Noether is the subtitle of B. L. Van der Waerden's A History of Algebra. And Noether was in at the birth of modern, functorial algebraic topology: she recognized that the objects to be studied were not just Betti numbers and torsion coefficients but homology groups. TP] Grisha Perelman in PlayboyBrett Forrest's story of how he staked out Gregory Perelman's Leningrad appartment, accosted the mathematician on the street and managed to exchange a few words with him appeared in the JulyAugust 2012 issue of Playboy. The title: "Shattered Genius." Forrest went to Leningrad on assignment for the magazine. As he explains it, his preliminary study had led him to some measure of admiration for Perelman, who gained fame and then notoriety by proving the Poincaré conjecture and then turning down first the Fields medal and next the $1 million prize offered for the solution by the Clay Mathematics Institute. "I liked his style. The more he did, the more I liked. I admired him for his renunciation of the modern world's expectations, his devotion to labor, his results. He had not solicited fame or reward in proving the Poincaré, so why should he be required to react to public notice? His will was free, his results pure, and therein lay his glory." Nevertheless, Forrest booked passage to Leningrad, found out where Perelman lived, rented an apartment nearby, and a car, and spent days parked across from Perelman's front door, waiting for his chance. The meeting, when it finally happens, is almost friendly (Forrest works hard with what sounds like considerable charm) but ultimately Perelman retreats into his apartment. The story ends on a philosophical note: "We don't have to figure out everything. The unknown has its own value." The "abc conjecture" in Nature, Science, and the New York Times"Proof claimed for deep connection between primes" is Philip Ball's piece in Nature News, September 10, 2012. "The usually quiet world of mathematics is abuzz with a claim that one of the most important problems in number theory has been solved." The problem is the socalled $abc$ conjecture, and the putative solution is contained in four papers posted by Shinichi Mochizuki on his Kyoto University website. As Ball explains it, the $abc$ conjecture relates to the prime factorizations of three integers satisfying $a+b=c$. It is stated in terms of the squarefree part $\mbox{sqp}(n)$ of an integer $n$: this the product of the distinct prime factors of $n$. Ball gives the example $\mbox{sqp}(18) =2\times 3 =6$. "The conjecture states that for integers $a+b=c$, the ratio $\mbox{sqp}(abc)^r/c$ always has some minimum value greater than zero for any value of $r$ greater than $1$." Ball consulted Brian Conrad, a number theorist at Stanford, who told him that "the conjecture encodes a deep connection between the prime factors of $a, b$ and $a+b$." (Fermat's Last Theorem is only one of its implications). Conrad mentions the "huge investment in time" it will take to check such a long and intricate proof, but that Mochizuki's reputation for depth and thoroughness may encourage mathematicians to try. "The exciting aspect is not just that the conjecture may have now been solved, but that the techniques and insights he must have had to introduce should be very powerful tools for solving future problems in number theory." Science published a short item on their "News of the week" page, September 14, 2012. It refers to the potential lure of Mochizuki's Interuniversal Teichmüller theory (his name for the complex of ideas). "If initial assessments of Mochizuki's claim hold up, there'll be an influx of mathematical prospectors primed [sic] to use his proof to mine their own intellectual gold." The New York Times joined in with Kenneth Chang's piece, "A Possible Breakthrough in Explaining a Mathematical Riddle," in the Science Times for September 17. "Numbers, addition, multiplication the basic stuff of gradeschool arithmeticare suddenly the excited talk of cuttingedge mathematicians." Chang emphasizes the difficulties ahead. "Dr. Mochizuki's new mathematical languageon his web page, Mochizuki describes himself as an "interuniversal geometer"is at present incomprehensible even to other top mathematicians." Chang talked to Minhyong Kim (Oxford and Pohang UST in South Korea) who contributed some additional explanation of the conjecture. Consider $81 + 64 = 145$. "The conjecture says roughly that if there are prime numbers that divide either $a$ or $b$ too many times, then their presence has to be 'balanced out' by largish primes that divide $c$ only a few times. We see 3 divides 81 four times, and 2 divides 64 six times. But then, 145 equals 5 times 29, so you get the larger primes 5 and 29 dividing 145 just once." The Museum of Mathematics in NatureA rendering of the upper level of the Museum of Mathematics, due to open in New York City on Saturday, December 15, 2012. Nature's Jascha Hoffman interviewed Glen Whitney for a "Q&A" piece in the September 6, 2012 issue. "Mathematician Glen Whitney left a job in finance to set up the Museum of Mathematics (MoMath), which is due to open in Manhattan, New York, on 15 December. He wants to spread the word that mathematics is a beautiful discipline and all around us, from the geometry of soap bubbles to the algorithms that control traffic lights." We learn that Glen went to Harvard, taught at Michigan and worked with Jim Simons at Renaissance Technologies. "It was exciting and intellectually demanding, but I wanted to do something beneficial to society at large." Hoffman asks why Glen is focussed on the public image of mathematics. "I believe this attitude ['I was always terrible at maths'] stems primarily from the emphasis on rote procedures and people paying too little attention to making connections with everyday life and the world around them. We need a cultural institution to combat this prejudice." What will a visitor find at your museum? "Handson exhibits showing how mathematics can be tangible, openended and fun. In the new museum, we will have exhibits on everything from the beautiful patterns created by video feedback to the probabilities of making a free throw in basketball." "There is maths everywhere."
Tony Phillips 
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