Edited by Mike Breen and Annette Emerson, AMS Public Awareness Officers
Contributors: Mike Breen (AMS), Claudia Clark (writer and editor), Rachel Crowell (2015 AMS Media Fellow), Annette Emerson (AMS), Samantha Faria (AMS), and Allyn Jackson (Deputy Editor, Notices of the AMS)
Read a summary of reviews of the new film, The Man Who Knew Infinity. Photo: Ken Ono, associate producer and math consultant on the film; Jeremy Irons, who plays G.H. Hardy; Devika Bhise, who plays Ramanujan's wife, Janaki; Dev Patel, who plays Ramanujan; and Manjul Bhargava, associate producer and math consultant on the film. Photo courtesy of Manjul Bhargava.
"The news should start with mathematics, then poetry, and move down from there," from The Humans, by Matt Haig.
Math Digest Archives ||
1997 || 1996 || 1995
Click here for a list of links to web pages of publications covered in the Digest.
See also: The AMS Blog on Math Blogs: Mathematicians tour the mathematical blogosphere. PhD mathematicians Evelyn Lamb and Anna Haensch blog on blogs that have posts related to mathematics research, applied mathematics, mathematicians, math in the news, mathematics education, math and the arts, and more. Recent posts: "Math and Verbal Gymnastics," by Anna Haensch, and "Fold Your Way to Glory," by Evelyn Lamb.
Named after mathematician and computing pioneer Alan Turing, a "Turing machine" is a theoretical model of a computer that carries out a specific task. The machine has a certain number of "states" it can be in. Given an input, the machine goes through various states, and the final state renders the output--that is, if a final state is ever reached. It is possible for a Turing machine to run forever without stopping. The epochal work of Kurt Gödel implies that there exist Turing machines whose behavior cannot be predicted using the standard axioms of mathematics. Those axioms are known as Zermelo-Fraenkel set theory with the Axiom of Choice, or ZFC for short. Now Scott Aaronson and Adam Yedidia of MIT have set out to test how complicated such an unpredictable Turing machine would have to be. They created a Turing machine called Z that has 7,918 states and has two significant properties: 1) Z will run forever, and 2) it is not possible to prove, within ZFC, that Z will run forever. If Z ever stopped running, that would show that ZFC is inconsistent. The purpose is not to find an inconsistency in ZFC; indeed, mathematicians believe no such inconsistency exists. Rather, the purpose is to explore the question of how big Z must be--that is, how many states it must have--in order to elude proof in ZFC that Z will ever stop. How big Z must be says something interesting about the limitations on the foundations of mathematics, the New Scientist article says--and this is something Gödel and Turing would liked to have known. The article goes on to quote Aaronson: "[Gödel and Turing] might have said 'that's nice, but can you get 800 states? What about 80 states?' I would like to know if there is a 10-state machine whose behavior is independent of ZFC." He and Yedidia have also produced a 4,888-state Turing machine that halts if and only if there is a counterexample to Goldbach's Conjecture, and a 5,372-state machine that halts if and only if there is a counterexample to the Riemann Hypothesis. The fact that those two machines require fewer states than Z suggests that the question of consistency of ZFC is the most complex of the three problems. "That would match most people's intuitions about these sorts of things," Aaronson is quoted as saying.
See "This Turing machine should run forever unless maths is wrong," by Jacob Aron. New Scientist, 11 May 2016, and for more information, see Aaronson's blog entry on this topic.
--- Allyn Jackson (Posted 5/19/16)
In this article the L-functions and Modular Forms Database (LMFDB) is compared to the first periodic table of the elements by John Voight, an associate professor at Dartmouth College. Anna Haensch, assistant professor of mathematics at Duquesne University said (in a PR Newswire press release), "Like the public DNA database, the LMFDB lets us peek, for the first time, into the relationships of different mathematical items and trace their common ancestry...Another way to think about this is a huge Facebook for mathematical items. We can see what forms are friendly with other items--and can investigate these possibilities." [Anna is a former Digest writer and current blogger on the AMS Blog on Math Blogs.]
Haensch and Voight were on the international team that constructed the database containing data supplied by more than 80 mathematicians from 12 countries. The database catalogs millions of mathematical objects and relationships between them. The LMFDB allows mathematicians around the world to conserve valuable resources--time and brain power--that would be spent if they had to do calculations that are available in the database. Researchers believe the database will be helpful to mathematicians who are working to solve problems in pure mathematics, such as the Riemann hypothesis. They also think scientists could use the database to learn about relationships between mathematical objects and apply this knowledge to design better data encryption systems, including those used in cloud storage. Image: Graph of zeros of Riemann-zeta function along its critical line, from lmfdb.org.
See "A new way to explore the mathematical universe," by Viviane Richter, Cosmos Magazine, 11 May 2016.
--- Rachel Crowell (Posted 5/18/16)
The National Science Foundation (NSF) "has unveiled a research agenda intended to shape the agency's next few decades and win over the next U.S. president and Congress. The nine big ideas illustrate how increased support for the type of basic research that NSF funds could help answer pressing societal problems, she says, ranging from how humans interact with technology to how climate change in the polar regions will impact the global economy, environment, and culture." The emphasis is on transdisciplinary research to find solutions and innovations. NSF Director France Córdova "is counting on rank-and-file scientists to help sell the initiative by submitting more grant proposals that don't fit traditional categories or are especially ambitious." The research areas are: Harnessing data for 21st century science and engineering; Shaping the human-technology frontier; Understanding the rules of life (i.e., predicting phenotypes from genotypes); The next quantum revolution (physics); Navigating the new Arctic (including a fixed and mobile observing network); and Windows on the universe: multimessenger astrophysics--in which mathematical sciences can play a role. See "NSF Ideas for Future Investment".
See "NSF director unveils big ideas, with an eye on the next president and Congress," by Jeffrey Mervis, Science, 10 May 2016.
--- Annette Emerson (Posted 5/13/16)
"Cheng already made clear her conviction that in mathematics, rules are like eggs: meant to be broken, stirred, flipped over and taste-tested," writes Angier as she joined mathematician Eugenia Cheng to work on her mathematically inspired desserts. One dessert is a "Bach pie," named so because "BAnana added to CHocolate gives you Bach," Cheng tells the reporter. "The braiding illustrated the structure of a Bach prelude and the sorts of patterns that knot theorists study 'to see how looped up the braids are.'" Cheng is receiving a lot of media attention for her popularizing mathematics through cooking, covered in her book, How to Bake π: An Edible Exploration of the Mathematics of Mathematics, which devotes chapters to illuminate mathematical concepts. (See "On math and food," by Claudia Clark, coverage of when Cheng appeared on The Late Show with Stephen Colbert in November 2015.) In addition to her work in category theory, Cheng blogs and has online tutorials. John Baez (who writes the AMS Visual Insight Blog) described Cheng's outreach, "She's trying to explain math to everybody, with or without pre-existing expertise, and I think she’s doing wonderfully."
Cheng "insists that the public has it all wrong about math being difficult, something that only the gifted mathletes among us can do. To the contrary, she says, math exists to make life smoother, to solve those problems that can be solved by applying math's most powerful tool: logic." Angier has written a good profile of Cheng, quoting other mathematicians, and explaining some of the mathematical concepts and connections with cooking. That The New York Times has covered Cheng and her outreach in the Science section is notable.
See "Eugenia Cheng Makes Math a Piece of Cake," by Natalie Angier, The New York Times, 2 May 2016.
--- Annette Emerson (Posted 5/12/16)
This article discusses Claude Shannon, who has been called the father of the information age. April 30th was the 100th anniversary of Shannon's birth.
In the 1948 paper "A Mathematical Theory of Communication" Shannon introduced the concept of using zeroes and ones to measure data. He called these measurements "bits," a term he credited to John Turkey, his colleague at Bell Telephone Laboratories.
Shannon defined another important concept: channel capacity. Consider different materials that information can travel through, such as telephone wires and fiber optic cables. There is an upper bound to the rate at which information can travel through these channels and remain intact. This maximum rate is called the channel capacity or the Shannon limit.
James Gleik, who wrote "The Information," told the author of this article, "It's Shannon whose fingerprints are on every electronic device we own, every computer screen we gaze into, every means of digital communication. He's one of these people who so transform the world that, after the transformation, the old world is forgotten."
One of Shannon's hobbies was creating off-beat inventions, from rocket-powered frisbees to flame-throwing trumpets. Shannon died on February 24th, 2001 ("MIT Professor Claude Shannon dies; was founder of digital communications," MIT News, February 27, 2001). Each year since 1973, the IT Society has presented one Claude E. Shannon Award, "to honor consistent and profound contributions to the field of information theory." Shannon was the first recipient of the award. The 2016 recipient was Alexander S. Holevo of the Steklov Mathematical Institute of the Russian Academy of Sciences in Moscow. Photo: Oberwolfach Photo Collection (CC BY-SA 2.0 DE).
See "Claude Shannon, the Father of the Information Age, Turns 1100100," by Siobhan Roberts, The New Yorker, 30 April 2016.
--- Rachel Crowell (Posted 5/6/16)
Tricia Serio, professor and head of molecular and cellular biology at the University of Arizona in Tucson, has personally experienced "microaggression" -- indirect, subtle or unintentional comments -- in the workplace. Her lead example is when she was early in her academic career she told her department head that she was pregnant and his response was "Was it planned?" Another was when she inquired about research opportunities at the institution and the response was "Why? Jeff [my significant other at the time] is doing a postdoc in another city." Serio believes that there are many untold stories of unconscious gender bias in academic science, and so she has set up Speak Your Story Survey, a website where women in mathematics and all sciences are invited to post their stories anonymously. "The purpose of this invitation is not to identify individual offenders. Rather, I hope to shine a light on the perception gap that I suspect leads to many microaggressions (and their subsequent impact), and to begin to quantify its scope by field, type of institution and location. My goal is to narrow or, ideally, to eliminate this gap. Let's inspire change by moving from unspoken anecdotes to awareness." She plans to share the stories periodically, though she doesn't indicate how she will do that.
See "Speak up about subtle sexism in science," by Tricia Serio, Nature, 26 April 2016 (print 28 April 2016).
--- Annette Emerson (Posted 5/9/16)
In this article Jo Boaler explains why it is essential for children to use their fingers for counting in math class. Apparently, there is "a specific region in our brain that is dedicated to perception and representation of fingers known as the somatosensory finger area." When students learned ways to use their fingers to help their mathematical understanding it lead to greater math successes. Neuroscientists recommend that students learn 'finger discrimination' or distinguishing between their fingers. Unfortunately, conventional methods have warned students away from counting on their fingers but Boaler explains that new brain research shows that, "Stopping students from using their fingers when they count could be akin to halting their mathematical development." Teachers should encourage their students to use their fingers to "strengthen this brain capacity" and ask the students to visualize mathematical ideas and draw what they see.
See "Why Kids Should Use Their Fingers in Math Class," by Jo Boaler and Lang Chen, The Atlantic, April 13, 2016.
--- Samantha Faria
"Scientists have long debated whether the basis of high-level mathematical thought is tied to the brain's language-processing centers—that thinking at such a level of abstraction requires linguistic representation and an understanding of syntax—or to independent regions associated with number and spatial reasoning," writes the author of this Scientific American article. According to a study that was recently published in Proceedings of the National Academy of Sciences by INSERM–CEA Cognitive Neuroimaging Unit director Stanislas Dehaene and graduate student Marie Amalric, that question has been answered: "Our results show that high-level mathematical reflection recycles brain regions associated with an evolutionarily ancient knowledge of number and space," Amalric states. In the study, Cepelewicz explains, functional magnetic resonance imaging (fMRI) was used "to scan the brains of 15 professional mathematicians and 15 non-mathematicians of the same academic standing. While in the scanner the subjects listened to a series of 72 high-level mathematical statements, divided evenly among algebra, analysis, geometry and topology, as well as 18 high-level nonmathematical (mostly historical) statements. They had four seconds to reflect on each proposition and determine whether it was true, false or meaningless." Previously, Dehaene, an experimental psychologist, "has studied how humans (and even some animal species) are born with an intuitive sense of numbers—of quantity and arithmetic manipulation—closely related to spatial representation," Cepelewicz notes. "How the connection between a hardwired 'number sense' and higher-level math is formed, however, remains unknown."
See "How Does a Mathematician's Brain Differ from That of a Mere Mortal?," by Jordana Cepelewicz, Scientific American, 12 April 2016, and "Math brains do differ from the rest of us," by Geoff Johnson, Times Colonist, 26 April 2016.
--- Claudia Clark
The only way to get to the meaty, higher-level science, technology, engineering and math college courses is by successfully passing the introductory course. Unfortunately, these courses can be "stultifying bores," reported a presidential council. This may leave students "with the impression that all STEM fields are dull and unimaginative." Transforming Post-Secondary Education in Mathematics, referred to as Tipsy, is addressing this issue. With support from large foundations, organizations, well-known mathematicians, colleges and universities, this group has already worked out four areas in which to focus. The first goal is to make the math relevant to the students by demonstrating how it is used in the subjects that already interest them. For example, political science students could take a course in mathematical modeling. Next, Tipsy recommends more experimentation in the classroom by trying out options that have already been shown to work, such as flipped classrooms. This method has the students watch lectures outside of class and then spend class time working on problems with the instructor. Often, mathematicians are trained in graduate school to research rather than learning how to effectively teach future students. One can be a brilliant mathematician yet an unsuccessful teacher. Tipsy urges that "future professors train in effective teaching methods." Lastly, the only way to accomplish these goals, Tipsy admits, is to create a higher-education network. Through partnerships with various organizations, such as the Association of Public and Land-Grant Universities and the National Association of System Heads, the network can reach out to large numbers of provosts, presidents, deans and department chairs.
See "4-Part Plan Seeks to Fix Mathematics Education," by Dan Berret, The Chronicle of Higher Education, April 10, 2016 (requires subscription).
--- Samantha Faria
According to this article, some scientists, such as Seth Shostak, director of the Search for Extra Terrestrial Intelligence (SETI), anticipate successfully contacting aliens within our lifetime. Without knowing their language, how will we communicate with extraterrestrials once we have achieved contact? Long before the invention of telephones and computers, scientists began pondering this question. Their answers have ranged from wacky and environmentally hazardous to specialized and tedious. In the early 1800’s, Joseph Johann Von Littrow, an Austrian astronomer, had a five step plan for alerting aliens to our presence: 1. Trek to the Sahara Desert. 2. Dig deep trenches. 3. Fill trenches with water. 4. Top water with kerosene. 5. Light kerosene on fire, catching extraterrestrial attention.
Littrow’s blazing idea was never tested. Hans Freudenthal, a German mathematician, proposed the next major idea in the field of extraterrestrial communication. If utilized, Freudenthal’s idea--a spoken, mathematically-based language called Lincos--wouldn’t produce quite the same spectacle as burning desert trenches. Lincos is a combination of lingua and cosmica. Freudenthal described Lincos in his 1960 book Lincos: Design of a Language for Cosmic Intercourse. Lincos shows promise as a basis for communicating with aliens who understand logic, mathematics and science. Scientists at SETI say these are the ETs we are likely to achieve communication with anyway. The reason? In order for an alien civilization to build receivers capable of understanding our messages, they would need to understand the mathematics and science behind constructing such devices.
In 1999, astrophysicists Yvan Dutil and Stéphane Dumas used a Ukrainian radio telescope to send messages intended for an extraterrestrial audience. Their transmissions, known as the Evaptoria Messages, were the third set of messages ever directed at potential ET civilizations. Unlike the two previous sets of messages sent out by other scientists, the Evaptoria Messages were created based on protocols outlined in Freudenthal’s Lincos. Dutil and Dumas’s 1999 messages were sent to cosmic addresses between 50 and 70 light years away. The first message is expected to reach cosmic address Hip4872 in Cassiopeia in approximately 19 years.
Researchers continue to refine their methods for communicating with possible ETs. CosmicOS, a computer program that aliens could run if they receive it, was developed by Paul Fitzpatrick, a former postdoctoral lecturer at MIT and co-founder of Robot Rebuilt, a U.S. robotics company. Alexander Ollongren, a Dutch mathematician, created a second-generation lingua cosmica. Messages in Ollongren’s language are created using constructive logic. Scientists have even considered combining Fitzpatrick and Ollongren’s approaches.
Scientists agree that mathematics is central to our pursuit of meaningful communication with ETs. (Image: Allan Telescope Array, Colby Gutierrez-Kraybill/Wikimedia.)
See "Building a Language to Communicate With Extraterrestrials," by Daniel Oberhaus, The Atlantic, 6 April 2016.
--- Rachel Crowell (Posted 4/21/16)
Looking for some worthwhile cinematic entertainment? Check out the film The Man Who Knew Infinity, based on a biography of Indian mathematician Srinivasa Ramanujan, written by Robert Kanigel. Director Matthew Brown "dramatizes the purest of mathematics for a general audience, and explores the strange personal life of Ramanujan, who died at 32, at the height of his powers, probably from tuberculosis," writes Robinson in a review in Nature.
Ramanujan is played by actor Dev Patel while Jeremy Irons plays Ramanujan's colleague and champion, G. H. Hardy. Their performances benefited from the careful advising of mathematician and Ramanujan scholar, Ken Ono (who has just published an autobiography entitled My Search for Ramanujan) and Fileds Medalist Manjul Bhargava. Much of the film—and the mathematics—takes place at Trinity College in Cambridge, England, where Ramanujan and Hardy collaborated intensely between 1914 and 1919. While praising the film, Robinson notes that it "struggles to shed light on the origins of Ramanujan's prodigious gift. Biographers have had the same problem with Gauss and many other mathematicians. As India's great film director Satyajit Ray put it: 'This whole business of creation, of the ideas that come in a flash, cannot be explained by science.'" Hardy was awed and mystified as well, writing of Ramanujan "All his results, new or old, right or wrong, had been arrived at by a process of mingled argument, intuition and induction, of which he was entirely unable to give any coherent account."
(Photo: Ken Ono, associate producer and math consultant on the film; Jeremy Irons, who plays G.H. Hardy; Devika Bhise, who plays Ramanujan's wife, Janaki; Dev Patel, who plays Ramanujan; and Manjul Bhargava, associate producer and math consultant on the film.) Photo courtesy of Manjul Bhargava. Ono has recently co-authored a book with Amir Aczel, My Search for Ramanujan: How I Learned to Count.)
See "In search of Ramanujan," by Andrew Robinson, Nature, 31 March 2016, "Ramanujan biopic "The Man Who Knew Infinity" tells mathematician's journey," Reuters/Daily Mail, 7 April 2016, and "Genius by numbers: why Hollywood maths movies don't add up," by Stuart Jeffries, The Guardian, 6 April 2016; "The Man Who Knew Infinity fails to break the mathematical mould," by Jacob Aron, New Scientist, 8 April 2016; "A Math Biopic, With Dev Patel, Applies a Different Calculus," by Kathryn Shattuck, New York Times, 20 April 2016; "Jeremy Irons on how prejudice blinds us to genius," Q interview with Shadrach Kabango, CBC Radio, 19 May 2016.
--- Claudia Clark
Math Digest Archives ||
1997 || 1996 || 1995
Click here for a list of links to web pages of publications covered in the Digest.