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Beginning this month Brie Finegold will summarize two mathematics blogs. She's covered some in the past, and will now do so regularly. For this issue Brie describes entries from Brian Hayes's blog, Bit-Player, and from Vi Hart's blog. Also this month, Adriana Salerno begins her blog, PhD + epsilon about the fun and challenges of being an early-career mathematician which, unfortunately, means she won't be contributing to Math Digest. Good luck, Adriana!
Bit-Player: An amateur's outlook on computing and mathematics, by Brian Hayes. Brian Hayes, who also writes for the magazine American Scientist, provides frequent and well-written posts on matters accessible to the laymen and enjoyable to the scientist. His most recent post, "The Prime Twins Conjecture" is about his search for the origin of the following statistic he heard of in a TV news report; the odds of a pair of twin sisters celebrating their 100th birthday is 1 in 700 million. First off, Hayes discusses the multiple possibilities for the meaning of this statistic. Not finding a direct source, he reverse-engineered the assumptions one would might make for the odds quoted to be correct, and calculated the odds himself using data he researched and basic probability and computer simulation.
Hayes's sense of curiosity shines through as he conjectures mathematically on events in the everyday world, explores the use of data in research and media, and attempts to understand aspects of new mathematics research. For example, he photographs the shape of the snowball resting on his fence and speculates on reasons for its form. Days later, he writes about a mathematical talk on the modeling of snowflakes. In his post "Googling the Lexicon," he plays with and conjectures uses for a new tool developed by Google and Harvard that compares the written usage of words or phrases over time. In his "When life gives you lemmas, make lemma-ade" he ponders the fate of research mathematics as it becomes specialized-- so specialized that in a talk for mathematicians at the Joint Mathematics Meetings about the Fundamental Lemma, the speaker explained at the very beginning that he would never state the celebrated lemma. Let's hope the bits keep coming.
Vi Hart's Blog You may never think of a snow angel or the Möbius Strip in quite the same way if you visit Vi Hart's Blog. Her most recent video entry, which has garnered over 100,000 hits on Youtube since its posting less than two weeks ago, features the adventures of a triangle living on a Möbius strip. With some very clever usage of upside down/backwards handwriting, and a cliff-hanger ending, she'll have those viewers unfamiliar with the Möbius strip grabbing tape, paper and scissors in a heartbeat.
Mainly composed of video entries, original music clips, and colorful photographs, the blog exposes the beauty of mathematics in a very tangible way that would entertain most mathematicians and teenagers. Hart's mathematical doodling videos (which can be found on the blog) went somewhat viral on Facebook this winter. But her creativity doesn't stop there. The blog documents her leading workshops to build mathematical objects such as a model of hyperbolic space made of balloons. And she recently gave a talk at the Joint Meetings entitled "Hyperbolic planes take off." She may be well on her way to earning the title "the ambassador of mathematics" as she reportedly hopes to according to a recent article in The New York Times (summarized below).
--- Brie Finegold
"Deep Meaning in Ramanujan's 'simple' pattern," by Jacob Aron. New Scientist, 27 January 2011.
"Hidden Fractals Suggest Answer to Ancient Math Problem," by Dave Moser. Wired.com, 28 January 2011.
"Mathematics' Nearly Century-Old Partitions Enigma Spawns Fractals Solutions," by Davide Castelvecchi. Scientific American, 8 February 2011.
There is a good deal of coverage of two results about the partition function. A team of researchers has done work concerning congruences, as well as finding a finite formula for partition numbers. The New Scientist article describes the partition function and some of its history, beginning with Ramanujan in 1918, and quotes the team leader, Ken Ono (Emory University and the University of Wisconsin-Madison), "It's a privilege to explain some of Ramanujan's work." Ono adds, in the Scientific American article, "All of this stuff that we're studying right now for some crazy reason was anticipated by Ramanujan." In the New Scientist article, Trevor Wooley (University of Bristol, UK) says that he is interested in applying the team's methods to other problems: "There are lots of tools involved in studying the theory of partition functions which have connections in other parts of mathematics." Castelvecchi also quotes AMS Past-President George Andrews on Ramanujan: "He was a magical genius, and the rest of us wish we knew how he was able to see so deeply." The two papers, "l-adic Properties of the Partition Function," by Amanda Folsom, Zachary A. Kent, and Ono; and "An Algebraic Formula for the Partition Function," by Jan Hendrik Bruinier and Ono are linked to from the website of the American Institute of Mathematics, which supported the research along with the National Science Foundation.
--- Mike Breen
Articles on a resolved conjecture in enumerative combinatorics:
* "Ein Beweis, für den der Computer ein paar Monate braucht (A proof that took the computer a couple of months)," Die Presse, 25 January 2011;
* "Mathe-Vermutung bewiesen---Hilfsformel eine Million A4-Seiten lang (Math conjecture proved---auxiliary formula runs one million A4 pages)," by Austrian Press Agency. Der Standard, 25 January 2011;
* "Linzer Forschern gelang mit Computer Beweis (Linz researchers reach with computer proof)," Neues Volksblatt, 25 January 2011;
* "295 kilometer lange formel liefert Beweis (295 kilometer formula delivers proof)," OÖNachrichten, 29 January 2011;
* "Spitzenforschung in IT (Top research in IT)," Chefinfo, February 2011;
* "Un théorème pour des empilements de cubes (A theorem for stacking of cubes)," by Maurice Mashaal. Pour la Science, 2 February 2011;
* "Eine solide Grundlage für eine kombinatorische Vermutung (A solid basis for a combinatorial theorem)," by George Szpiro. Neue Zürcher Zeitung, 2 March 2011.
These articles report on a recent result of three mathematicians, who have resolved a conjecture in enumerative combinatorics that remained open for nearly 30 years. The conjecture was proposed independently by George Andrews (who served as AMS President 2009-2010) and David Robbins (after whom the AMS Robbins Prize is named). A team of three mathematicians---Christoph Koutschan, Manuel Kauers, and Doron Zeilberger---solved the problem and have described their proof in an announcement in the Proceedings of the National Academy of Sciences. The conjecture concerns plane partitions, which are arrays of numbers with weakly decreasing entries. Plane partitions can be represented by cubical diagrams like the one shown in the accompanying picture. Kauers et al considered a special type of plane partition, called a "totally symmetric plane partition" (TSPP). A plane partition is a TSPP if its cubical diagram looks the same when it is rotated so as to exchange the coordinate axes. An "orbit" in a TSPP is a set of cubes that remains unchanged under such a rotation. The accompanying picture is an example of a TSPP, and the red cubes form an example of an orbit. What Koutschan, Kauers, and Zeilberger proved is that it is possible to write down an explicit formula that counts the orbits for totally symmetric plane partitions. Although related results had been obtained by others, no one had fully solved the problem until now. Another reason this result is especially noteworthy is that it used computers in a crucial way. As the PNAS article states, "the computations we performed went far beyond what has been thought to be possible with currently known algebraic algorithms, software packages, and computer hardware." [Image: A totally symmetric plane partition. Picture courtesy of Christoph Koutschan.]
--- Allyn Jackson
"Ancient puzzle gets new lease of 'geomagical' life," by Jacob Aron. New Scientist, 24 January 2011.
A magic square is a square array of integers in which each row, column, and diagonal add up to the same number. This article discusses the work of recreational mathematician Lee Sallows, who has invented a new kind of magic square called a "geomagic square". In a geomagic square, the entries in the square array are not numbers but geometrical shapes. The shapes in each row, column, and diagonal can be combined to form a "target" shape. For example, in the accompanying picture, the target shape is a square: The shapes in any row, column, or diagonal of the geomagic square (the inner 3 by 3 grid of shapes) can be assembled to make a square. Note that, in assembling the shapes to reach the target shape, rotations and reflections are allowed. The picture (click for a larger version) shows an example using polyominoes, but any kind of shape is allowed---convex shapes, broken crockery pieces, puzzle pieces, etc. Many other pictorial examples---in addition to a good and clear description of the concept of geomagic squares---are available on Sallows' web site. (Image: A 3×3 geomagic square using polyominoes. Piece sizes were selected with an eye to achieving an unblemished square target of size 6×6. Note how any two pieces opposite about the center form a complementary pair that will fit together to form a 4×6 rectangle. Copyright 2011 Lee Sallows.)
---Allyn Jackson
"ESP Paper Rekindles Discussion About Statistics," by Greg Miller. Science, 21 January 2011, page 272.
In this article, Greg Miller describes how the upcoming appearance of a paper on extra sensory perception (ESP) in the Journal of Personality and Social Psychology has "sparked a lively discussion on blogs and in the mainstream media," and "rekindled a long-running debate about whether the statistical tools commonly used in psychology--and most other areas of science--too often lead researchers astray." By applying standard statistical methods like the t-test to the results of several experiments, the paper's author, Daryl Bem, a social psychologist and professor emeritus at Cornell University, "found statistically significant evidence suggesting his subjects had unconscious knowledge of future events." But other statisticians, including University of Washington, Seattle statistician Adrian Raftery, argue that there are problems with such standard statistical methods. "Scientists generally want to know…the probability that a given hypothesis is true, given the data they've observed. But that's not what a p-value tells them," Miller writes. Instead, Raftery and others argue for an approach based on Bayesian statistics, a more "intuitive" approach "designed to determine the probability that a hypothesis is true given the data a researcher has observed." Miller goes on to describe this approach, as well as some of the analyses that have already been undertaken of Bem's data, in the remainder of this article.
--- Claudia Clark
"Bending and Stretching Classroom Lessons to Make Math Inspire," by Kenneth Chang. The New York Times, 18 January 2011.
Victoria "Vi" Hart (pictured at left), the daughter of mathematician and artist George Hart, is the one helping math inspire. She wants to make math cool, and one way she is doing that is by posting videos on You Tube. Her first video is about doodling during a class on exponential functions (read more in the summary of Hart's blog at the top of the page). She uses binary trees and Sierpinski's triangle to make the topic more interesting than it might first appear to students. Hart has written papers with MIT's Erik D. Demaine and according to Chang hopes to become "a Martin Gardner for the Web 2.0 era." (Photo of Hart looking through an icosahedron made of six balloons, by Erik D. Demaine.)
--- Mike Breen
"Santa Cruz police first in nation to try Santa Clara University model to predict crime," by Stephen Baxter. Mercury News, 14 January 2011;
"The Santa Cruz Experiment," by Kalee Thompson. Popular Science, November 2011, pages 38-49 and 97.
Police in Santa Cruz, California are going to use a mathematical model to try to predict and prevent crime. The model was developed by George Mohler at nearby Santa Clara University and is based on eight years of crime data from Santa Cruz. The police still plan to rely on traditional methods to battle crime but will use the model to create likely places for extra patrol checks. The Popular Science article is an in-depth look at the approach and early results of its implementation.
--- Mike Breen
"An equation for friendship," by Eryn Brown. Los Angeles Times, 14 January 2011.
Structural balance theory describes how relationships in networks evolve. Many networks evolve into either two unfriendly camps--intractable conflict--or a network in which everyone is friendly--global harmony but until recently it wasn't known if other situations could arise. Researchers at Cornell University have shown that with generic initial conditions, those are the only two possibilities. It was also unknown exactly how shifts in individual relationships--such as a triangle of three enemies--result in larger alliances in the network. The researchers also filled in that gap in knowledge by showing that shifts in positive and negative relationships occur incrementally and both affect and are affected by other network relationships. They used their insight to predict the alliances in World War II (correct on all nations except Portugal and Denmark). Brown writes that the theory could be applied to online networks, but one member of the research team, Jon Kleinberg, advises caution: "This is abstracted reality; it's not reality. Realistic scenarios are messier." The research "Continuous-time model of structural balance," was published in the Proceedings of the National Academy of Science in January (read the abstract and link to the full article) .
--- Mike Breen
"Mathematics-Inspired Dance Work Makes World Premiere at Staller," by Christine Sampson. ThreeVillagePatch, 11 January 2011.
Choreographers Kyla Barkin and Aaron Selisson probably did not see much advanced mathematics in their future when they decided to pursue a career in modern dancing. Yet they found themselves attending a class on the mathematical theory of differential cohomology to prepare for a work assignment. They were commissioned to develop an original dance work based on the mathematical theory of differential cohomology by the Simons Center at Stony Brook University. The performance was accompanied by music from three composers and performed by New York City-based Sirius String Quartet. Hexagons played a large role in the performance as the structure of a hexagon is a key component of the theory of differential cohomology. See a video excerpt and photos of the performance. (Photo of Yin Yue, Loni Landon, and Frances Chiaverini by Christopher Duggan, courtesy of the Barkin/Selissen Project.)
--- Baldur Hedinsson
"The Statistical Turn in Literary Studies," by Jeffrey J. Williams. The Chronicle of Higher Education, 7 January 2011, pages B14-15.
Novelists and poets aren’t known for being very good with numbers, however mathematics is playing an increasingly important role in examining their works. Millions of books are now accessible in digital form, which makes flicking through them easier than ever. Mining the content of all these books allows scholars to find trends and detect changes in how books are written which would have been impossible before. Literary scholars are now able to apply statistics to rigorously study how the use of language changes over time, what topics are most written about and even how ideas spread from one place to another, just to name a few. This mathematical approach of studying written works is called "distant reading" as apposed to "close reading," which is based on more subjective interpretations of individual literary works.
--- Baldur Hedinsson
"Make way for mathematical matter," by Michael Brooks. New Scientist, 5 January 2011.
Norwegian mathematician Nils Baas is working to uncover new forms of matter— beyond your typical solid, liquid, and gas—using topology, the study of the properties objects share because of their shape. The foundation for his work is a structure of linked objects that can all be separated if only one is cut, also known as “Brunnian rings.” The construction of a shape with these properties containing three rings is well-known, and researchers have shown that it occurs in nature, in cesium atoms and atomic nuclei. Baas has taken the idea a step further by proving it is possible to create a such structure with more than three objects, or for sets of objects with these properties. The proof enables the possibility that these more complex Brunnian rings also appear in nature, as new forms of matter, and Baas has teamed up with a researcher at New York University to create them.
--- Lisa DeKeukelaere
"Animal Instincts," by John Allen Paulos. Scientific American, January 2011, page 18.
Paulos is reacting to various articles that claim that animals understand some math better than humans do. For example, when pigeons were presented with doors and rewards as in the Monty Hall Problem, the pigeons made the correct "decision" and switched more often than humans did. Although some concluded that pigeons understand conditional probability better than people do, Paulos writes that pigeons didn't do any calculations, they simply observed, whereas people presented with the problem "overanalyze and get confused." He debunks other claims, such as dogs doing calculus, and notes that the results from the experiments are of scientific interest, but don't indicated a deep understanding of math. In the February issue of Scientific American, Paulos writes about the misuse of averages.
--- Mike Breen
"Year in Science 2010," Discover, January/February 2011.
Whether you are playing a game, translating a language, looking for patterns in shark behavior, or fighting crime, you may need some mathematical expertise. Four of the 100 Top Stories of 2010 in Discover feature mathematics.
"Rubik's Cube Decoded," by Bruno Maddox. In July of this year, a team including a university professor, an engineer, a programmer, and a math teacher provided a computer-assisted proof that any initial configuration of the Rubik's cube could be solved in no more than 20 moves. The success of their proof relied on previous work that showed that there were some configurations requiring 20 moves.
"Fighting Crime with Mathematics," by Daniel Lametti. Also this year, a multi-disciplinary group of researchers from UCLA studied the spatio-temporal behavior of crime in their area so as to determine optimal responses to crime waves. Their research combined statistical mechanics and dynamics, studying crime waves with some of the same tools used to study earthquakes.
"Computer Rosetta Stone," by Elizabeth Svoboda. Meanwhile, a team of computer scientists devised a deciphering algorithm that probabilistically maps the alphabet of a lost language onto the alphabet of a known and somewhat related language. They checked their algorithm by applying it to Ugaritic (which had already been deciphered without computers over the course of several years) and Hebrew. The program correctly identified over 60 percent of cognates and may be used in the future to improve computer translation programs.
"Sharks Use Math to Hunt," by Stephen Ornes. Computer models of search patterns of predators in low-prey areas have suggested that the fractal behavior described as Levy flight, which combines short sporadic movements with much longer trajectories, would optimize chances of a shark running into its prey. However, this June a study in Nature analyzing 13 million movements of 55 different animals has provided the largest data set to date, and confirms the computer models' predictions.
--- Brie Finegold
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