On a physicist's dim view of probability theory, by Allyn Jackson
In this short piece, quantum information researcher David Deutsch discusses what he sees as misplaced faith in the use of probability theory in physics. He notes that probability theory can work as an approximation to reality, just as the theory that the Earth is flat can work when you are laying out your garden. But both theories have severe limitations. Deutsch writes: "The awful secret at the heart of probability theory is that physical events either happen or they don't: there's no such thing in nature as probably happening."
See "Probability is as useful to physics as flat-Earth theory," by David Deutsch. New Scientist, 30 September 2015. (Full access to the article requires a subscription.)
--- Allyn Jackson (Posted 10/13/15)
On quant investing, by Lisa DeKeukelaere
John Overdeck, a math genius who won a silver medal at the International Mathematics Olympiad at age 16, has parlayed his math skills into an estimated net worth of $2.8 billion from his Wall Street hedge fund, one of the biggest in America. Last year, Overstock bested other top mathematicians, including a Fields Medal winner, at an annual Museum of Mathematics soiree that included a competition to solve a series of complex equations. Overstock's firm, Two Sigma Investments, uses mathematics to predict stock prices—one of many successful trading operations driven by math experts known as quants. As a result of Two Sigma's success in achieving a high average return, the firm is able to charge higher fees than its competitors, further boosting its earning potential.
See "Rich Formula: Math And Computer Wizards Now Billionaires Thanks To Quant Trading Secrets," by Nathan Vardi, Forbes, online 29 September 2015 (print issue 19 October 2015).
--- Lisa DeKeukelaere
On patterns in art, by Lisa DeKeukelaere
Professor Frank Farris provides an elementary group theory lesson couched in pretty, patterned wallpaper. Starting with the concepts of translational symmetry and mirror symmetry in a "frieze pattern" that repeats indefinitely left and right, Farris explains how a frieze pattern with one vertical mirror axis must have another set of vertical mirror axes halfway between the others. The idea that a pattern with a given set of symmetries must have all the other symmetries that arise from combining those in the set is connected with the concept of a mathematical group. Farris notes that there are only seven different types of frieze groups, based on the possible combinations of rotational, vertical, and horizontal qualities, and he introduces a notation for expressing each group as well as examples. Positing that our fondness for patterns may be an evolutionary trait that helped humans compare edible and poisonous plants, or simply an attraction to something pretty, Farris notes that the popularity of patterns is timeless. See some of Farris's work on Mathematical Imagery.
See "Patterns are math we love to look at," by Frank A. Farris. The Conversation, 22 September 2015, and a subsequent article in Quanta magazine describing Farris's work and posing some questions about it: "How to Create Art With Mathematics," by Pradeep Mutalik.
--- Lisa DeKeukelaere
On encryption that's secure from attack by quantum computers, by Claudia Clark
"Quantum computers," writes Natalie Wolchover, "once seen as a remote theoretical possibility, are now widely expected to work within five to 30 years. By exploiting the probabilistic rules of quantum physics, the devices could decrypt most of the world's 'secure' data, from NSA secrets to bank records to email passwords. Aware of this looming threat, cryptographers have been racing to develop 'quantum-resistant' schemes efficient enough for widespread use." In this article, Wolchover introduces the reader to two of the most widely used public key encryption schemes--RSA encryption and Diffie-Hellman key exchange--and explains why these "can be broken by algorithms designed to run on future quantum computers." However, most of the article focuses on the work that has been done over the years to develop "quantum-secure" schemes based on the mathematics of lattices and on the challenge of finding the right balance between security and efficiency.
See "The Tricky Encryption that Could Stump Quantum Computers," by Natalie Wolchover. Wired (from Quanta magazine), 19 September 2015. Nature ran a related article: "Encryption faces quantum foe," by Chris Cesare. 10 September 2015, pages 167-168.
--- Claudia Clark
On Hamid Naderi Yeganeh's mathematical art, by Annette Emerson
"... [T]o 25-year-old Iranian student Hamid Naderi Yeganeh, using cosines are a part of daily life--what you would expect of a mathematics major and award-winning mathlete," writes CNN reporter Chung in this article accompanied by images. Yeganeh, a student in mathematics at the University of Qom, says in the article, "At first I was interested in beautiful, symmetrical shapes. So, I started to create mathematical figures using trigonometric functions to define the endpoints of line segments. After a while, I understood I could find interesting shapes that looked like real life things, such as animals." He is now creating animations of his circle and line images (the article includes a nice video animation), and 3D sculptures of fractals. See samples of his works in an album of his works, Mathematical Concepts Illustrated by Hamid Naderi Yeganeh, on AMS's Mathematical Imagery.
Read "Next da Vinci? Math genius using formulas to create fantastical works of art," by Stephy Chung, CNN, 17 September 2015.
--- Annette Emerson (Posted 9/18/15)
On mathemagician Art Benjamin, by Samantha Faria
Art Benjamin's love of math grew with every Martin Gardner puzzle he completed. Each problem was creative, stimulating and engaging. Unfortunately, he explains, "both the fun and the explanations are often missing from math instruction in today's schools." A proclaimed "mathemagician," Benjamin astonishes his audiences not just with his super-speedy calculations but with his ability to captivate their attention and explain topics in interesting ways. The Harvey Mudd professor admits that he "learned how to be a good teacher through [his] early experiences as a magician. My approach to teaching has always been, 'How do I make this material entertaining?'" In his new book, The Magic of Math, Benjamin hopes to do just that.
See "From Poof to Proof: Inside the Mind of a Mathemagician," by Liz Logan, Smithsonian Magazine, 8 September 2015.
---Samantha Faria
Photo courtesy of Harvey Mudd College.
On busy roads, by Claudia Clark
When there are two routes to your destination, why does it always seem that the route you choose has more traffic? According to the author, Pradeep Mutalik, the mathematical explanation for this phenomenon is called selection bias: "The busier road has more drivers on it, so if you sample a bunch of drivers randomly, more of them will be from the more crowded road." Mutalik asks readers to consider how large an impact this has by posing the first of two questions: "Let us say the drivers of 200 cars independently and randomly make their choice between…two roads with a 50 percent probability of choosing a given road. Assume that there are no other cars on the road. How many more of the 200 cars end up on the more crowded road?" But this is not the whole story, Mutalik notes. Doesn't the fact that some of the time you find yourself on the less busy road "decrease your tendency to overestimate, or engage in upward bias, somewhat?" Here, Mutalik poses the second question: "What happens to the driver's upward bias if there are an average of 100 cars on each road? How many cars do we think he or she sees?"
See "The Road Less Traveled," by Pradeep Mutalik. Quanta Magazine, 3 September 2015. In addition to the article, the website has readers' answers to the questions.
--- Claudia Clark
On building probabilistic reasoning into programming language, by Lisa DeKeukelaere
"Sum of three dice." The 216 possible outcomes of rolling three dice stack up to form an approximation to the normal bell curve. The snippet of computer code, written in a programming language called Church, gives the probability of each three-die sum from 3 to 18. Church is one of a new generation of languages designed to model probabilistic reasoning. Illustration by Brian Hayes.
Hayes's article provides a primer on computer programs that use pseudorandom numbers to estimate probability, and progress in enhancing these programs to increase their applicability. To illustrate how the program works, Hayes uses the example of computing the odds that the sum of the faces on three die is a certain number. Galileo calculated the answer by enumeration--writing out all the possibilities--but sampling, either by hand or computer modeling, offers the benefit of being closer to how nature works and being easier to do with a large number of dice. Probability programs can be inefficient in solving complex problems that require discarding the majority of the generated samples, such as if we require two of the die faces to be equal, but the article explains the Monte Carlo method, and a programming language called Church that uses it, to address this problem. Church doesn't work cleanly in all cases, however, and research is ongoing to improve it.
See "Programs and Probability," by Brian Hayes, American Scientist, September-October 2015.
--- Lisa DeKeukelaere
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