Showing papers in "Notices of the American Mathematical Society in 2008"

Cross-cultural analysis of students with exceptional talent in mathematical problem solving

Abstract: A t a conference held in January 2005, Lawrence Summers, then president of Harvard University, hypothesized that a major reason for the paucity of women mathematicians among the tenured faculty of elite research universities in the USA might be sex-based differences in “intrinsic aptitude” for mathematics, especially at the very high end of the distribution [36]. This commonly held belief is largely based upon data from standardized tests such as the quantitative section of the Scholastic Aptitude Test (SAT) I. This test, designed to determine mathematical proficiency of USA eleventh and twelfth graders, identifies students who have mastered grade-level material, but does not distinguish the profoundly gifted, that is, those who are four or more standard deviations above the mean, from the merely gifted who also score in the ninety-ninth percentile on this exam. To identify students who perform above grade level, the Study of Mathematically Precocious Youth (SMPY) administered the SAT I to children younger than thirteen years of age. The SMPY defined children as highly gifted in mathematics if they scored at least 700 (on a 200 to 800 scale) on the quantitative section of this test. Using this criterion, Benbow and Stanley reported in 1980 large gender differences in “mathematical reasoning ability” [4]. They concluded that “sex differences in achievement in and attitude towards mathematics result from superior male mathematical ability . . . [it] is probably an expression of a combination of both endogenous and exogenous variables.” Since these tests lack questions that require creative thinking and insight into higher-level mathematical concepts, they do not identify children with extremely high innate ability in mathematics, that is, ones who may go on to become top research mathematicians. They cannot differentiate between profoundly and moderately gifted children, regardless of age at which the examinations are administered. Thus, the SMPY identified thousands of children who, while quite bright and ambitious, were not necessarily profoundly gifted in mathematics. The SMPY also failed to identify many children with extreme ability in mathematics who lacked one or more of the socio-economically privileged environmental factors necessary to be recognized by this mechanism. Coincidentally, the ratio of boys to girls identified in the SMPY has dramatically declined during the past quarter century from the high of 13:1 originally reported in 1983 [5] to 2.8:1 in a 2005 report [6]. The fact that 29% of Ph.D.’s awarded to USA citizens in the mathematical sciences went to women in the 2006–2007 academic year [30] supports the idea that this latter ratio is a more accurate reflection of current Titu Andreescu is professor of science/mathematics education at the University of Texas, Dallas. His email address is txa051000@utdallas.edu.

Formal proof -- getting started

Abstract: A List of 100 Theorems Today highly nontrivial mathematics is routinely being encoded in the computer, ensuring a reliability that is orders of a magnitude larger than if one had just used human minds. Such an encoding is called a formalization, and a program that checks such a formalization for correctness is called a proof assistant. Suppose you have proved a theorem and you want to make certain that there are no mistakes in the proof. Maybe already a couple of times a mistake has been found and you want to make sure that that will not happen again. Maybe you fear that your intuition is misleading you and want to make sure that this is not the case. Or maybe you just want to bring your proof into the most pure and complete form possible. We will explain in this article how to go about this. Although formalization has become a routine activity, it still is labor intensive. Using current technology, a formalization will be roughly four times the size of a corresponding informal LTEX proof (this ratio is called the de Bruijn factor ), and it will take almost a full week to formalize a single page from an undergraduate mathematics textbook. The first step towards a formalization of a proof consists of deciding which proof assistant to use. For this it is useful to know which proof assistants have been shown to be practical for formalization. On the webpage [1] there is a list that keeps track of the formalization status of a hundred well-known theorems. The first few entries on that list appear in Table 1.

Möbius transformations revealed

Abstract: M öbius Transformations Revealed is a short film that illustrates a beautiful correspondence between Möbius transformations and motions of the sphere. The video received an Honorable Mention in the 2007 Science and Engineering Visualization Challenge, cosponsored by the National Science Foundation and Science magazine. It subsequently received extensive coverage from both traditional media outlets and online blogs. Edward Tufte, the world’s leading expert on the visual display of information, came across the video and reported on his blog “Möbius Transformations Revealed is a wonderful video clarifying a deep topic... amazing work...” But the film has also attracted a far less expert audience. As of this writing, it has been viewed nearly 1.5 million times on the video-sharing website YouTube and is rated as the number three top favorite video of all time in YouTube’s educational category. Over 11,000 viewers have declared it among their favorites, which makes it one of the YouTube top favorites of all time. From the more than 4,000 written comments left by YouTube viewers it is clear that many of them had little mathematical background, and some were quite young. To view Möbius Transformations Revealed, visit the website http://umn.edu/ ̃arnold/moebius/. In this article we discuss some of the technical details behind the video and offer a “behind the

Your hit parade: The top ten most fascinating formulas in Ramanujan's lost notebook

Abstract: A t 7:30 on a Saturday evening in March 1956, the first author sat down in an easy chair in the living room of his parents’ farm home ten miles east of Salem, Oregon, and turned the TV channel knob to NBC’s Your Hit Parade to find out the Top Seven Songs of the week, as determined by a national “survey” and sheet music sales. Little did this teenager know that almost exactly twenty years later, he would be at Trinity College, Cambridge, to discover one of the biggest “hits” in mathematical history, Ramanujan’s Lost Notebook. Meanwhile, at that same hour on that same Saturday night in Stevensville, Michigan, but at 9:30, the second author sat down in an overstuffed chair in front of the TV in his parents’ farm home anxiously waiting to learn the identities of the Top Seven Songs, sung by Your Hit Parade singers, Russell Arms, Dorothy Collins (his favorite singer), Snooky Lanson, and Gisele MacKenzie. About twenty years later, that author’s life would begin to be consumed by Ramanujan’s mathematics, but more important than Ramanujan to him this evening was how long his parents would allow him to stay up to watch Saturday night wrestling after Your Hit Parade ended.

Hardy's "Small" Discovery Remembered.

Abstract: The most beautiful mathematics to Godfrey Harold Hardy was that which had no application. For Hardy, mathematics was purely for intellectual challenge. He justified the pursuit of pure mathematics with the argument that its very “uselessness” meant that it could not be used to cause harm. Hardy went so far as to describe applied mathematics as “ugly”, “trivial”, and “dull” [1]. Despite Hardy's aversion to applied mathematics, he had a profound impact on biology. Mathematicians tend not to realize his contribution, which was downplayed by Hardy himself. At the same time, biologists tend not to appreciate his mathematical brilliance. This note is intended to recognize G. H. Hardy's “little” discovery as a contribution to genetics and to revisit a classic paper [2] that has shaped the field for the past century. Godfrey Harold Hardy, a graduate of Trinity College, Cambridge, began his mathematics career in the early 1900s as a fellow at Trinity [3]. He lectured in mathematics for a number of years and published many papers of such significance that he was considered Britain's leading pure mathematician. Hardy was also responsible for bringing the Indian mathematical genius Srinivasa Ramanujan to England, where they published many papers together and developed the field of number theory. While visiting an ill Ramanujan on one occasion, Hardy mentioned that he had traveled in cab number 1729 and “hoped it was not an unfavorable omen” to which Ramanujan replied “...it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways” (it is expressible as 1729 = 13 + 123 or 93 + 103, now known as the Hardy-Ramanujan number). In 1908, Hardy published a paper in Science that changed the field of population genetics, entitled “Mendelian Proportions in a Mixed Population”. The findings were later known as the Hardy-Weinberg Law (Equilibrium) because the same principle was published by Wilhelm Weinberg in the same year [4]. This principle offered a simple solution for the question of how genetic diversity is maintained in a population. For Hardy, the law was trivial and obvious, and he was reluctant to acknowledge its applications. But one hundred years later, the Hardy-Weinberg Law remains a cornerstone of modern computational genetics. Preservation of genetic diversity in a population requires stability, or equilibrium, of the genotype distribution from one generation to the next. The following is an outline of Hardy's stability condition for Mendelian proportions [2]. This condition holds for a closed system, where the population mates randomly, or for purposes of simplicity, where every individual mates with every other individual once, and where each mating yields a single offspring, with no selection, mutation, migration, or death. Genotypes A1A1 A1A2 A2A2 2nd generation frequency A1A1 P11a×P11a P11a×2P12a P11a×P22a (P11+P12)2a2 A1A2 2P12a×P11a 2P12a×2P12a 2P12a×P22a 2(P11+P12)(P12+P22)a2 A2A2 P22a×P11a P22a×2P12a P22a×P22a (P12+P22)2a2 View it in a separate window Recall that in the mammalian genome, each gene is represented by two of many possible variants, called alleles. For example, the gene, or locus, for eye color can exist in many forms, including a “blue” allele coding for blue eyes and a “brown” allele coding for brown eyes. If we consider only the blue and brown alleles, an individual may have two copies of the blue allele (homozygous for the blue allele), one copy of each allele (heterozygous), or two copies of the brown allele (homozygous for the brown allele). An individual who is homozygous for the blue allele can be designated as having an “A1A1” genotype, while an individual who is homozygous for the brown allele is designated as having an “A2A2” genotype. Consider the following case for alleles A1 and A2, P11 = the number of individuals with genotype A1A1 (the homozygous A1 case) 2P12 = the number of individuals with genotype A1A2 or A2A1 (the heterozygous case) P22 = the number of individuals with genotype A2A2 (the homozygous A2 case) Therefore, we can write the first generation proportions of individuals as P11 : 2P12 : P22. Let the total number of individuals in the first generation be represented as a, where a = P11 + 2P12 + P22. The second generation proportions of individuals can be derived from the table seen above. Therefore, we can write the second generation proportions of individuals as (P11+P12)2:2(P11+P12)(P12+P22):(P12+P22)2 Let the total number of individuals in the second generation be represented as b, where b = (P11 + P12)2 + 2(P11 + P12)(P12 + P22) + (P12 + P22)2. Hardy's stability condition requires that the proportion of individuals with any given geno-type remains constant across generations and is therefore established as follows P11a=(P11+P12)2b (1) 2P12a=2(P11+P12(P12+P22)b (2) P22a=(P12+P22)2b (3) Solving the above system simultaneously yields P122=P11×P22, the stability condition for the two-allele case. This provides a null hypothesis for biologists investigating the distribution of genetic characteristics in a population. Since a2=(P11+2P12+P22)2=P112+4P12P11+4P122+2P11P22+4P12P22+P222=(P11+P12)2+2(P11+P12)(P12+P22)+(P12+P22)2=b, this demonstrates that a2 = b, as required by the assumption of a closed system. Let the haplotype frequency of allele A1 = p, and the haplotype frequency of allele A2 = q, recalling that a haplotype is a combination of alleles at multiple genetic loci that are transmitted together from one generation to the next. Accordingly, and p=2P11+2P122a, and q=2P22+2P122a, where homozygous individuals are counted twice, heterozygous individuals counted once, and the number of haplotypes is twice the population size. Hence, p=P11+P12a, and q=P22+P12a. Since p + q = 1, the modern population genetics interpretation, p2 + 2pq + q2 = 1 must hold, which states that Pr(A1A1) = p2, Pr(A1A2) = 2pq, and Pr(A2A2) = q2. We can extend the two allele case to a general case as follows. Under the same assumptions as above and additionally that a given gene includes k alleles, say A1, A2, · · · , Ak and Pij is the number of people with the genotype AiAj, where i, j can be any real number. The following generations will approach values in proportion to those suggested by counting all possibilities of mating (the way Hardy did). Thus, if in addition the values satisfy (or nearly satisfy) the stability condition Pij2=Pii×Pij, then we can assume that the probabilities for each genotype in each generation will be the same as the probabilities for each genotype in the preceding generation. That is, the proportion of individuals with each genotype will stay the same, while genetic diversity will be maintained in a predictable way. This formulation is the k allele analog of the Hardy-Weinberg Law. Certain violations of the closed system assumption lead to departure from Hardy-Weinberg Equilibrium. One such violation is non-random mating, where preferential mating according to genotype may occur in the population. We will not explore the mathematical details, but we can represent this case as follows: Pr(A1A1) = p2 + pqF,Pr(A1A2)2) = 2pq(1 − F), and Pr(A2A2) = p2 + pqF, where F is defined as the inbreeding=coe cient [5]. F can be thought of as the probability that two alleles are identical due to parents passing on the same allele to their progeny. It is therefore also a measure of the degree of parental relatedness. When 0 < F < 1, homozygosity in the population increases, which may reduce health and reproductive fitness. The modern interdisciplinary approach has gathered great minds to work on some of the most challenging problems in biology. Hardy's contribution has greatly influenced the growing field of computational genetics. Today, applied mathematics is a critical component of genetics and has the potential to revolutionize the field and profoundly impact modern medicine. If Hardy were alive today, it would be interesting to know whether he would join these minds or remain steadfast in his pursuit of pure mathematics. Despite his disdain for applied mathematics, Hardy was one of the greatest contributors to contemporary mathematical biology, and at this one hundred year anniversary, his “small” discovery will be remembered as such a great contribution.

Is the public hungry for math

Abstract: At the end of 2007 CNN compiled a list of the top news stories of the year. A third of the stories were about celebrities behaving badly (Anna Nicole Smith's death from a drug overdose, Barry Bond's steroid use, Paris Hilton's drunk driving conviction , Britney Spears's child custody battle, Michael Vick's arrest for dog-fighting, etc.). None of the stories concerned mathematics, of course, or even science or technology. Sadly, the list accurately reflects the choices the news media makes throughout the year in deciding what stories are of interest to the American public. Call me a starry-eyed optimist, but I beg to differ. While the director's office at the Institute for Mathematics and its Applications—which has been my headquarters for the last seven years—is admittedly not the best place to gauge the interests of John Q. Public, some of the IMA's activities and some of my own involve public outreach, and in a broad segment of the public I regularly observe curiosity about contemporary math research and how it impacts our world, and a hunger for the sort of intellectual stimulation that comes from exploring mathematics. The IMA hosts a public lecture series called Math Matters. Four times a year I recruit a distinguished mathematician who is also a superb expositor and lay before him or her a daunting challenge: to get across a message about math—how exciting it is and how central a role it plays in understanding our world and shaping our lives—to a very diverse and mostly nonexpert audience ranging from high schoolers to retirees, and to do so in fifty entertaining minutes to boot. The speakers have consistently risen to the challenge and the audience for the series has steadily grown, so that it now numbers several hundred. You can view most of the lectures in the IMA's video library available from our website. Recently my optimistic thesis received remarkable support from an unlikely source: YouTube. YouTube—for any readers of this column who are completely removed from popular culture—is a Google-owned website at which about a hundred million user-contributed videos can be viewed. In June 2007, a colleague, Jonathan Rogness, and I completed a short mathematical video entitled Möbius Transformations Revealed and posted it to YouTube. (You can read more about our video in an article to appear in the November 2008 issue of the Notices , including the little-known theorem it demonstrates.) As I …