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This month's topics:Ramanujan, Ono in Scientific American"The Oracle" (subhead: "Mathematician Ken Ono has solved longstanding puzzles using insights hidden in the unpublished papers of Indian prodigy Srinivasa Ramanujan"), by Ariel Bleicher, appears in the Scientific American for May, 2014. Bleicher tells us how six equations buried in a manuscript of Ramanujan's came to Ono's attention in 1989, and changed the direction of his mathematical career. The equations "drew a parallel between modular forms [Ono's specialty] and ... partition numbers." (The $n$th partition number $p(n)$ "counts the combinations of positive integers that sum to a given integer $n$. For example, $~p(4)$ is 5: $ 1 + 1 + 1 + 1,~ 1 + 1 + 2,~ 2 + 2,~ 1 + 3$ and $~4$.") Ramanujan had published in 1919 "one of the first real breakthroughs" in the study of the numbertheoretical properties of $p(n)$. He and G. H. Hardy had commisioned the computation of the first $200$ values by a "calculation wizard named Percy Alexander MacMahon (aka Major MacMahon);" the results came arranged in a 5column table, starting with $p(0)=1$. Ramanujan noticed that all the entries in the fifth column were divisible by $5$, and that starting with $p(5)$ every seventh entry was divisible by $7$; he proved that these patterns persist forever: $p(4+5k)\equiv 0 ~\mbox{mod}~5$ and $p(5+7k)\equiv 0 ~\mbox{mod}~7$; he also observed that $p(6+11k)\equiv 0 ~\mbox{mod}~11$, but remarked: "It appears that there are no equally simple properties for any moduli involving primes other than these."
The first few rows of MacMahon's table of the partition numbers $p(n)$ for $n=0, 1, 2, \dots$, showing Ramanujan's congruences: ${\bf p(4+5k)\equiv 0 ~\mbox{mod}~5}$, $~\color{red}{ p(5+7k)\equiv 0 ~\mbox{mod}~7}$, $~\color{blue}{p(6+11k)\equiv 0 ~\mbox{mod}~11}$. Adapted, with two additional rows, from Scientific American 310 No. 5 (May 2014). The six equations that were brought to Ono's attention in 1989 led him to understand that there was a very deep connection between the partition function and modular forms: "Ramanujan's six statements linked the two fields in a profound way that no one had anticipated." and in fact, "if he thought of the partition function as a modular form in disguise, he could show that they were true. Another thought immediately followed: ... with a few adjustments, the theories he had developed about modular forms could be powerful tools not just for verifying Ramanujan's genius but also for unearthing deeper secrets about the partition function. ... In this way, Ono was able to prove that partition congruences are not rare at all. Mathematicians had assumed there were few beyond 5, 7 and 11. But in fact, as Ono discovered, there are infinitely many." After Ono's first round of discoveries, Bleicher tells us, "The problem of how to predict partition congruences lay dormant for five years, until postdoctoral fellow Zachary A. Kent arrived at Emory in the spring of 2010." Together, Ono and Kent, "little by little, ... built in their minds a labyrinthine superstructure into which the partition numbers could be neatly arranged." The congruences "obey a fractal structure they repeat in nearidentical patterns at different scales, like the branches of a snowflake." Working together with Amanda Folsom (Yale) the collaborators "were able to prove that partition congruences appear in a calculable manner. They exist for every prime and every prime power. Beyond 11, though, the patterns get much more complex, which is probably why Ramanujan never worked them out." Bleicher quotes George E. Andrews (Penn State): "It's a dramatic and surprising discovery. I don't think even Ramanujan could have dreamt it." [This article exemplifies how hard it is to present mathematical research to a large audience. Scientific American avoids equations and when introducing an abstract concept like function feels obliged to explain: "describes a relation between two things: it takes a given input $x$ and spits out the corresponding output $f(x)$." Modular forms, which are mentioned 12 times, are only described as "abstract twodimensional objects revered for their remarkable symmetry." Their deep connection to the partition function is stated, but cannot begin to be explained. This part of the story is vivid and compelling on the human side, but one could substitute "potato pancakes" for each occurrence of "modular forms" without diminishing the amount of mathematics that has been communicated. The article has links to the research papers; curious readers might also appreciate an expository work like Amanda Folsom's What is a Mock Modular Form? in the December 2010 Notices.TP] Sperner's Lemma helps split the rent"To Divide the Rent, Start With a Triangle," by Albert Sun, appeared in the April 28 2014 New York Times. (The online version has excellent interactive graphics). "Last year, two friends and I moved into a small threebedroom apartment in Manhattan. ... The bedrooms were different sizes, ranging from small to very small. Two faced north toward the street and had light; the third and smallest faced an alley. The largest had two windows; the midsize room opened onto the fire escape." How to split the rent? "The problem is that individuals evaluate a room differently. I care a lot about natural light, but not everyone does. Is it worth not having a closet? Or one might care more about the shape of the room, or its proximity to the bathroom." Sun discovered a mathematical solution to the problem in a 1999 Monthly article by Francis Edward Su of Harvey Mudd College. Su's algorithm is presented in the following three images and their captions, adapted from the Times article and Su's paper. a. Each point in the triangle represents three percentages adding up to 100%, telling how the occupants of rooms 1, 2, and 3 will contribute to the total rent. Along the bottom edge, for example, room 1 costs 0%, and the cost of room 2 varies from 0% at the far left to 100% at the right. At the midpoint of the triangle, each room costs $\frac{1}{3}$ of the total. b. The triangle is partitioned into a network of subtriangles fine enough so that the price for each room varies, say, by less than $1 between adjacent triangles, and in such a way that the vertices in the network can all be labeled with the initials A, B, C of the renters with every subtriangle getting one of each. c. At each of A's vertices, A chooses a preferred room at that particular pricepoint; similarly for B and C. Renters are supposed to be rational, so that for example at vertices where room 1 costs nothing, that room will be preferred. A "trapdoor" argument given by Su, guarantees that, no matter how the additional choices are made, somewhere in the network will be a triangle labeled 1, 2, 3. (Su explains how the diagram can be dualized so that the proof also follows from Sperner's Lemma). The combination of roomchoices and prices encoded in that triangle should divide the total rent (plus or minus a couple of dollars) in a way agreeable to all three renters. The online New York Times links to an updated version of a rent division calculator, due to Su and Elisha Peterson, which gives a dynamic implementation of the process. Spontaneous hemihelices"Things That Happen When Harvard Researchers Play With Rubber Bands" is the title for a video posted on the Slate website, May 1, 2014. The research in question is presented in a PLOSone article, "Structural Transition from Helices to Hemihelices," by Jia Liu, Jiangshui Huang, Tianxiang Su, Katia Bertoldi, and David R. Clarke, all of Harvard. The article describes the fabrication of prestressed elastomer strips (two lengths are glued together, one relaxed and one stretched) and the behavior of the bistrips when they are gradually allowed to take their "natural" shape. This image, downloaded from the PLOSone article by Liu et al., shows how the prestressed strips are fabricated. The red strip, initially shorter and thicker, is stretched to the length of the blue one; then the two are clued together. When the bistrip is allowed to contract, it curls. doi:10.1371/journal.pone.0093183.g002 The phenomenon the Harvard team discovered is what the title of their paper suggests. "[W]hen the bistrips have a large aspect ratio, they spontaneously twist along their length to form a regular helix. ... [W]hen the aspect ratio is small, we observe the formation of periodic perversions, separating helical segments of alternating chiralities." "Illustration of a helix (top), a hemihelix with one perversion marked by an arrow (middle) and a hemihelix with multiple perversions (bottom). The scale bar is 5 cm, and is the same for each image. These different shapes were all produced in the same way as shown [above] with the same value of prestrain $\chi=1.5$ but with decreasing values of the heighttowidth ratio of the bistrip's crosssection. $L=50cm, w=3mm, h=12,8,2.5mm$)." doi:10.1371/journal.pone.0093183.g001 The authors present a numerical model which replicates their experimental findings; they also attack the problem analytically, and derive equations with solutions which "clearly resemble the hemihelices observed in the experiments." The authors remark that "it is highly probable that the reason hemihelices with multiple perversions have escaped notice in the past has been that most manmade materials, unlike elastomers, would fracture well before these strains could be achieved." This research was widely picked up in the press: for example, the second image above was reproduced in the New York Times ("A Rubber Band With a Surprising Twist," April 29, 2014), in La Repubblica ("C'è una nuova forma geometrica: scoperta l'emielica," April 24) and in the Los Angeles Times ("With a little rubber, scientists make weird, twisted hemihelix," April 24).
Tony Phillips 
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