Invited Addresses, Sessions, and Other Activities

Math Mistakes that Make the News, Heather A. Lewis (Nazareth College)
In her presentation, Heather Lewis presented three examples of mistakes (each resulting from simple math errors) that were reported in the media. Perhaps the most dramatic was the story of the “Gimli Glider,” referring to an Air Canada Boeing 767 that was to fly from Montreal to Edmonton on July 23, 1984. Because the plane’s fuel gauge was not working—a factor that did not prevent the plane from being flown across Canada according to regulations at the time—the pilots had to calculate how much additional fuel they would need. Specifically, they knew they needed 22,300 kilograms of fuel for the trip; they knew they already had 7682 liters of fuel; and they knew the conversion factor (which was also printed in the airplane’s documents) for converting kilograms to liters. They performed the conversion: 22,300 kilograms divided by 1.77 kilograms per liter equals 12,598 liters. Then they subtracted the amount of fuel already in the plane to determine how many liters to add: 12,598 – 7682 = 4916. The problem? The conversion factor of 1.77 actually converts pounds to liters: the pilots were used to using the imperial system but the Boeing 767 was Air Canada’s first all-metric airplane. That meant that the plane actually had 22,300 pounds of fuel, which is just over 10,000 kilograms of fuel. You can see where this is going: The pilots were forced to land their plane on the runway of an air base near a town by the name of Gimli. Unfortunately, they did not know that the base had been decommissioned—and did not see the people attending the Winnipeg Sports Car Club “Family Day” on the runway—until it was too late: in the end, they literally glided in. (There were a few injuries to people getting off the plane, but no one was killed.)

---Claudia Clark

Boole and Hamilton: An Unanswered Question, Charlotte K. Simmons (University of Central Oklahoma)

Simmons began her presentation with a brief biography of mathematicians William Hamilton (above, right) and George Boole (above, left). On the surface, the two men were very different. Hamilton began preparing for a career at Trinity College at the age of three, and had completed his first paper while still an undergraduate. Boole, on the other hand, was almost entirely self-taught. The first paper he submitted to the Royal Society was awarded the first gold medal in mathematics presented by that society, after nearly being rejected because the author was poor and unknown. However, Simmons notes, the two had much in common: They were both highly original mathematicians working in related fields; they lived within 200 miles of each other for several years; and they shared many views. In spite of this, the two had little to do with each other. Simmons describes an attempt Boole made to get Hamilton’s opinion on his work Laws of Thought: After receiving no reply for a year, Boole’s second letter received a terse acknowledgement, a response that was highly unusual for the normally obliging Hamilton. The lack of a relationship between the two men remains a mystery.

---Claudia Clark

Geometry from Data Points
Homological Methods for the Study of Data Sets, Shmuel Weinberger (University of Chicago)
Is there geometry in a set of data points? If so, how can it be detected and put to use? Shmuel Weinberger explored these questions in his talk, presented in a special session on "Mathematics of Information and Knowledge", organized by AMS President James Glimm. Weinberger pointed out that topological methods for understanding data have an advantage over statistical methods, in that topology does not change with small perturbations in the data. He gave a series of examples to show the potential of topological methods in data analysis. One example is the work of Gunnar Carlsson (Stanford University) and his collaborators, which has found that Klein bottles arise often in data from images of natural scenery. Another example is from a problem explored by Robert Ghrist (University of Pennsylvania) and others, in which sensors are dropped randomly onto a planet of unknown shape. The question here is, How can one gauge whether the sensors cover the planet well enough? The common feature of the examples Weinberger presented is that they use homological techniques to extract information about the geometry of data sets. Weinberger posed the question, Can one recover the homology of a manifold from a finite sampling of it? In principle, the answer is no, but in recent work, Weinberger---together with P. Niyogi, and S. Smale, and also with Y. Barishnikov---has shown that it is possible to recover some information about the homology of a well-conditioned manifold in certain cases when the data is not too noisy and when one samples a sufficiently large number of points. Their work gives a lower bound on the probability that the homology calculated in this way is actually the correct homology.

---Allyn Jackson

MAA Invited Address: Stacking Bricks and Stoning Crows, Peter Winkler (Dartmouth College)
Peter Winkler began this entertaining lecture with the following problem: A crow lands in the middle of a brick wall. The only way to get the crow to fly away is by throwing stones at it until it reaches the end of the wall. Assuming that the crow will jump randomly one step left or right when a stone is thrown at it, how many stones must be thrown before the crow flies away? Because this problem can be described as a simple random walk, it will take about n² stones, where 2n is the length of the wall. Now suppose the crow returns to the center of the wall at night, making it impossible for us to know where the bird is after our first throw. How many stones will it take to make the crow fly away? Given that you need at least n² stones, and that n³ will certainly be enough, how many stones do you need? By defining two new potential functions, and applying the Schwartz inequality, Winkler showed that the number of stones must be greater than or equal to n³/4.

Winkler then turned to a related problem: what is the optimal overhang of a stack of n bricks of length 1, assuming no friction and only vertical forces? Starting with the classical solution— harmonic stacks, which can achieve a maximum overhang of 1/2 ln n—he displayed various balanced stacks that show increasingly greater overhang. These include the “3-diamond” and “5-diamond” structures as well as “spinal” stacks, which have an overhang of order ln n, and are the optimal structure for stacks containing 3 to 19 bricks. But the best overhang is achieved by the brick-wall “parabolic,” “vase,” and “oil lamp” constructions: each has an overhang of order n^1/3, an exponential improvement over the spinal stack overhang! Using an argument that is based on the idea that stacking bricks is like stoning crows, Winkler and his colleagues were able to prove that a stack of n bricks cannot overhang by more than an order of n^1/3: “it takes order n³ bricks to make the stack lean out to n.” To read more about the ideas presented in this lecture, see the papers Overhang by Mike Paterson and Uri Zwick and Maximum Overhang by Mike Paterson, Yuval Peres, Mikkel Thorup, Peter Winkler, and Uri Zwick. You may also look for them in a future edition of American Mathematical Monthly.

---Claudia Clark

MAA Special Film Presentation: The Story of Maths, Part I: The Language of the Universe, Robin Wilson (The Open University)
The Story of Maths is a series of four engaging and accessible one-hour programs that explore the development of mathematics through the ages and around the world. These programs were written by Oxford University professor Marcus du Sautoy and co-produced by the BBC and the UK’s Open University. The Language of the Universe, the first program in the series, explores the mathematics of ancient Egypt, Mesopotamia, and Greece. The story begins at the Nile River, where the need to administer agricultural settlements in the ancient world led to a need to count, hence a need for numbers. Du Sautoy describes the Egyptians as brilliant problem solvers. Evidence for this can be found in the Rhine Papyrus, which demonstrates the ancient Greek’s unusual methods for solving practical problems involving multiplication, division (how to divide 9 loaves equally among 10 people), and determining the area of a circle with a diameter of 9 (it is approximated to be close to the area of a square with sides of length 8). In another document—the Moscow Papyrus—a formula for calculating the volume of a pyramid can be found.

Moving on to Damascus, we learn that the ancient Babylonians, like the Egyptians, focused on solving practical problems (such as calculating the area of a piece of land), evidenced by the clay tablet records that have been found. Unlike the Egyptians (whose number system was based on the decimal system), they used a base-60 number system: to count they used 12 knuckles on one hand and 5 fingers on the other. Also unlike the Egyptians, their number system recognized place value. This led them to invent a sign that indicated an empty place in the middle of a number, although, du Sautoy notes, “it would be over 1000 years before this little placeholder would become a number in its own right.” There is also evidence that they had some understanding of the Pythagorean relationship, 1000 years before Pythagoras. In Greece, du Sautoy offers some of the Greeks contributions to mathematics, the greatest of which, he says, is the “power of proof.” He discusses the legend of Pythagoras, who is credited, “rightly or wrongly, with beginning the transformation from mathematics as a tool for accounting to the analytic subject we recognize today.” He also presents a geometric proof of the Pythagorean Theorem, and describes a legend surrounding the discovery (and tragic consequences to one eager discoverer) of irrational numbers. He finishes this program speaking about Plato and the five solids, Euclid and the contents of The Elements, and introduces some of Archimedes' work, including his method for calculating the volume of solid objects. The Open University has posted more information about the series, and the 10-week course designed to accompany it.

---Claudia Clark

Summaries of three talks in the AMS-MAA Special Session on History of Mathematics:

Mrs. Bean’s Young Ladies: Mathematics Education in Early Modern England, Kathryn James (Beinecke Library, Yale University)
Kathryn James, a curator of an early modern books and manuscript collection at Yale University, discussed images from five students’ notebooks from late 17th century/early 18th century England. These manuscripts provide some insight into the relationship of numeracy to the popular culture at the time. What is immediately striking about these notebooks—particularly the first two, owned by students of one Mrs. Elizaeth Bean—is the beauty of the writing and the ornamental motifs that appear on the pages. Indeed, these were not working notebooks, but decorative folios that the owner could keep and display. Another feature of these notebooks is the application of basic mathematical skills to “important and complicated facets of daily life,” such as calculating weights, volumes, and currencies. Yet another interesting feature that James pointed out in the Elizabeth Bean notebooks: They are filled with aphorisms.

Toward the end of her presentation, James listed five conclusions that she has drawn about mathematics education in early modern England from these notebooks:

• arithmetic and writing were “categorized together as related arts whose boundaries overlapped”;
• the mathematical curriculum (including mathematical definitions, basic skills, and their application to everyday problems) was “quite consistent”;
• mathematics was taught using word problems related to business;
• mathematics was taught “as an art for both young men and young women and taught in much the same manner;” and
• mathematics was “part of a broader culture of edification, of moral improvement, in which the arts of arithmetic lead to social virtue and well brought up youth.”

Several pages from the first notebook, entitled “Mary Sarjent her book scholler to Eliz Bean Mrs. in the art of writing and arithmetick,” are available at the Beinecke Library website.

---Claudia Clark

Computing Devices, Mathematics Education and Mathematics – Sexton’s Omnimetre in Its Time, Peggy Aldrich Kidwell (National Museum of American History, Smithsonian Institution)

As part of her presentation on mathematical aids and their impact on mathematics, engineering, and education, Peggy Kidwell discussed a type of circular slide rule known as Sexton’s Omnimetre. This instrument was invented by Albert Sexton of Philadelphia and copyrighted in 1898. It could be used to perform a host of arithmetic and trigonometric operations, including multiplication, division, logarithms, squares and cubes of numbers, sines, tangents, secants, and versed sines. At the time of its invention, Kidwell noted, “the principle of the slide rule had been known for some 250 years.” Indeed, Kidwell presented images of several rules created for specialized tasks, such as calculating the volume of barrels, and therefore, excise taxes. Sexton’s decision to invent this slide rule was instigated by his reading of an article in the Journal of the Franklin Institute, in which the author referred to circular slide rules and encouraged the use of slide rules by young engineers. Sexton decided to invent a circular slide rule that would be less expensive than existing models and be able to perform more calculations. Six models of the omnimetre were offered, differing in size, material, and types of calculations performed. The Smithsonian’s omnimetre is one of the “improved third edition,” seemingly the “only edition to sell in later years.” It was owned by George Dankers, “a naval architect, [who] acquired his omnimetre when he joined the U.S. Navy’s Preliminary Ship Design Branch in 1938. He used it for thirty years, in the design of ships from PT boats to aircraft carriers.” The last advertisement for an omnimetre that Kidwell has found is from 1958, and slide rules stopped being manufactured in the mid-1970s with the introduction of the handheld electronic calculator. To learn more about the Mathematical Collection at the Smithsonian Institution, read “Mathematical Treasures of the Smithsonian Institution” by Allyn Jackson, Notices (May 1999).

---Claudia Clark

Theology, or Mathematics?
Hilbert and the origin myth of modern mathematics, Colin McLarty (Case Western University)
"This is not Mathematics, this is Theology!" This assessment of David Hilbert's 1888 proof of the finite basis theorem was made by Paul Gordan, a mathematician, a contemporary of Hilbert's, and the doctoral advisor of Emmy Noether. Colin McLarty of Case Western Reserve University traced the origin of Gordan's statement and attempted to understand what Gordan meant by it. The proof was contained in a paper that Hilbert submitted to Mathematische Annalen and that Gordan refereed at the request of editor Felix Klein. In his referee report, Gordan expressed disappointment in Hilbert's proof and recommended against publication (Klein published the paper anyway, with no changes). Gordan objected that Hilbert's proof was not at all clear. But Hilbert had found the proof in discussions with Gordan, and Gordan was convinced it was correct. McLarty's research shows that Gordan never complained about the lack of a constructive way to solve the problem. To the contrary, he was sure that further work on Hilbert's ideas would lead to constructive solutions. Both Hilbert and Gordan continued to work on the problem, and eventually an explicit algorithm for the solution was found. Gordan and Hilbert were both satisfied with the outcome. Nevertheless, Gordan's "theology" remark was so striking that many people took it as a sign of a major rift between the two mathematicians. What is more, the remark took on the aura of a myth in which the mathematical god Hilbert defeats his lesser doubting colleague Gordan. The main propagator of this misinterpretation was probably Eric Temple Bell. McLarty points out that Bell's book Development of Mathematics (1940) mentions the Gordan-Hilbert episode three times---and this in a book that calls itself a brief summary of 6000 years of mathematical development.

---Allyn Jackson

MAA Invited Address: Geometreks, Ivars Peterson (MAA)

Promising that “this presentation is about seeing—looking at things in a variety of different ways,” Ivars Peterson spent the next 45 minutes identifying the mathematics in well over 100 images, many from art and architecture, and many from around Washington D.C. He began with tilings, including a convex pentagonal tiling that appears on the floor of the front hallway of the Mathematical Association of America headquarters in Washington. He showed several examples of symmetry, including a skylight with 18-fold symmetry in the conference hotel. Examples of curves followed, including the St. Louis arch (an inverted catenary) and the Eiffel Tower, which, according to one source, has the shape of “one exponential function for the lower part and another exponential function for the upper part.”

Both images courtesy of Ivars Peterson.

Peterson also presented images that illustrated statistical concepts, such as the distribution of hand marks along a 19-degree vertical corner of the National Gallery of Art’s East Building. He showed works of art that were inspired by different numbers—pi and binary numbers—as well as sequences of numbers, including the Fibonacci numbers. The audience saw the influence of combinatorics in artist Sol LeWitt’s etching “Straight Lines in Four Directions All Their Possible Combinations” and his incomplete open cube sculptures. These were followed by more sculptures, built from shapes that included tetrahedrons and möbius strips. The presentation ended at the MAA headquarters with a discussion of Helaman Ferguson’s beautiful Umbilic Torus sculpture. Peterson has posted images that appeared in the presentation.

---Claudia Clark

Why Politics Matters

A highlight of the end of the Joint Meetings was a panel called "Mathematics and Public Policy", organized by the MAA. Although some participants had left the meetings and those who remained were tired out, the panel drew a sizable audience. It was organized by Philippe Tondeur, professor emeritus at the University of Illinois at Urbana-Champaign and former director of the Division of Mathematical Sciences (DMS) at the National Science Foundation. Tondeur took advantage of the location of the meeting in Washington, DC, to invite two members of Congress, Vernon Ehlers of Michigan and Jerry McNerney of California, as well was Daniel Ullman of George Washington University, who recently served as an AMS Congressional Fellow for one year (as often happens with busy congressmen, Ehlers was unable to attend). McNerney, the only Ph.D. mathematician in Congress, led off the panel by describing his own background and what a big change it was for him to enter politics. Rather than rational argument, what matters most in politics, he said, is who can intimidate others, who is the most aggressive, who has the most seniority, etc. And yet, political decisions are too important to ignore, and the disinterested viewpoint of mathematicians is very much needed. "We want people who can look at a problem without the trappings of partisanship," he said. In discussing education, McNerney noted that although young people recognize the importance of science and are concerned about issues like global warming, many college students major in areas like political science or communications. A big challenge, he said, is to encourage young people to look at mathematics and science as inspiring and as offering excellent, interesting careers.

The second speaker was Douglas Arnold of the University of Minnesota, who recently finished a term as director of the Institute for Mathematics and its Applications, and is beginning a term as SIAM president. Arnold looked back on the DMS "Odom Report", which appeared ten years ago. This hard-hitting report argued for increased federal support for mathematics and encouraged mathematicians to reach out to other disciplines. Arnold concluded that, in the ten years since the appearance of the Odom Report, a lot of progress has been made on both fronts. In particular, he noted how different the Joint Mathematics Meetings are than they were ten years ago---there are many more sessions on topics at the interface of mathematics and other sciences, including sessions on major challenges like climate change, computation, and energy. "Our community has been very responsive to moving in interdisciplinary directions," he concluded. He also found that core funding at the DMS had risen substantially in the past ten years. As the final speaker, Ullman noted that when policy issues arise, mathematicians typically focus on one question: How much money are we getting from the government? In fact, he argued, mathematicians need to take a different approach and put the emphasis on showing policymakers that mathematics is important in addressing national needs. "People who believe their work has no connection to the human condition should not expect funding," Ullman said. But he also noted that even very esoteric areas of mathematics are connected to problems people care about, and mathematicians need to illuminate these connections.

---Allyn Jackson

Gregory Margulis, AMS Colloquium Lectures: Homogeneous dynamics and number theory

Maria Chudnovsky, MAA Invited Address: Perfect graphs--Structure and recognition

Doug Arnold, Joint AMS-MAA Invited Address: Stability, consistency, and convergence: Modern variations on a classical theme

Ken Ono, AMS Invited Address: Unearthing the visions of a master: The web of Ramanujan's mock theta functions in number theory

More highlights of the 2009 Joint Mathematics Meetings.