# Math Digest

## On Media Coverage of Math

Edited by Mike Breen and Annette Emerson, AMS Public Awareness Officers
Contributors:
Mike Breen (AMS), Claudia Clark (freelance science writer), Rachel Crowell (2015 AMS Media Fellow), Annette Emerson (AMS), Samantha Faria (AMS), and Allyn Jackson (Deputy Editor, Notices of the AMS)

### May 2016

On the game Set and combinatorics problems, by Rachel Crowell

According to this article, in the game Set, if a player can find three cards out of a group of 12 that have certain attributes, then the player has a set. Each of the 81 cards in a deck has an image that has a color (red, green, or purple), shape (oval, diamond, or squiggle), shade (solid, striped, or outlined) and number of objects (one, two, or three). If a player finds three cards for which all of the images are identical or three cards for which the images don't have any attributes in common, then the player has a set. If a player can't find a set within the 12 cards that are in play, then three more cards are added. This pattern continues until a player finds at least one set. If you are wondering just how many cards players need in order for them to be guaranteed at least one set in the game, a 1971 result from mathematician Giuseppe Pellegrino provides the answer: 21.

But what if there were a different version of the game that included more cards in the deck, with more options for the properties on each card? What if each card had more than four attributes, or there were more than three options for each attribute? In May, three mathematicians published a proof that can be used to solve the same problem for a collection of cards where there are four options for each of the four attributes on a card. Mathematicians Ernie Croot of the Georgia Institute of Technology, Vsevolod Lev of the University of Haifa, Oranim, in Israel and Péter Pál Pach of the Budapest University of Technology and Economics in Hungary used a method based on finding polynomials that evaluate to zero at certain points in a collection.

Jordan Ellenberg (pictured), a mathematician at the University of Wisconsin, Madison and Dion Gijswijt, a mathematician at the Delft University of Technology in the Netherlands, independently published papers that generalized Croot, Lev, and Pach's result. The proofs that Ellenberg and Gijswijt published can be used to show that for a collection of cards each with n attributes, the largest collection of cards which will not contain a set will be at most (2.756/3)n as large as the entire deck of cards. Ellenberg and Gijswijt also based their proofs on the polynomial method. Terence Tao, a mathematician at the University of California, Los Angeles, and a winner of the Fields Medal, told Erica Klarreich of Quanta Magazine that the polynomial method creates, "beautiful short proofs," and the method is, "sort of magical." (Photo: Mats Rudels.)

See "Simple Set Game Proof Stuns Mathematicians," by Erica Klarreich. Quanta, 31 May 2016.

--- Rachel Crowell (Posted 6/13/16)

On scheduling the NCAA tennis championships, by Claudia Clark

On the occasion of the NCAA tennis championships held at the University of Tulsa's Michael D. Case Tennis Center, sports writer Kelly Hines introduces the reader to a key player at the event: Brian Garman, a University of Tampa professor of mathematics who has been managing the scheduling for the NCAA tennis championships for the past 30 years. "Known as a living legend in the sport," Hines writes, "Brian Garman is a former tennis umpire who saw a need for better scheduling at juniors events. He came up with a formula called the Garman System that takes into account the number of courts that are available and the average time it takes to conclude a match." Garman explains that "the way the matches are distributed is linear…Because of that, I could then make a prediction on any number of courts." This system was adopted by the NCAA in 1984, eventually spread worldwide, and is used at all levels except professional.

See "NCAA Tennis: Meet the 71-year-old mathematician who keeps the championships running on schedule," by Kelly Hines. Tulsa World, 26 May 2016.

--- Claudia Clark

Proof shows Ramsey's theorem for pairs is finitistically reducible, by Rachel Crowell

According to this article by Natalie Wolchover, researchers Keita Yokoyama (left) and Ludovic Patey (right) recently found a new proof of Ramsey’s theorem for pairs (RT22) which shows that the theorem is "finitistically reducible." Yokoyama is a mathematician at the Japan Institute of Science and Technology and Patey is a computer scientist at Paris Diderot University.

RT22 works in the following way: Imagine you have any infinite set of numbers you’d like. For simplicity, let’s choose the set of all integers. Next, each object is paired with all other objects in the set. Next, each object will become red or blue depending on a rule you choose. For example, a rule could be, for a pair of numbers A < B, if B < 3A, then the pair of objects will be colored blue. Otherwise, you will color the objects red. When you finish this process, you will have an infinite monochromatic subset of numbers. Yokoyama and Patey’s finding means that mathematicians can use the theorem to prove statements in finite mathematics, even though RT22 describes a property of infinite sets. Wolchover writes, "Yokoyama and Patey's proof shows that mathematicians are free to use this infinite apparatus to prove statements in finitistic mathematics--including the rules of numbers and arithmetic, which arguably underlie all the math that is required in science--without fear that the resulting theorems rest upon the logically shaky notion of infinity." Of all statements about infinity that are known to be finitistically reducible, RT22 is thought to be the most complex. Ramsey’s theorem for triples (RT23), which works in a similar way to RT22 but for triples of numbers instead of pairs, cannot be finitistically reduced. Wolchover calls RT22, "the longest bridge yet between the finite and the infinite." (Photo courtesy of Keita Yokoyama.)

See "Mathematicians Bridge Finite-Infinite Divide," by Natalie Wolchover. Quanta, 24 May 2016.

--- Rachel Crowell (Posted, palindromically, 6/1/16)

On math and the brain, by Claudia Clark

In this article, Chrystopher Nehaniv, a professor of mathematical and evolutionary computer science at the University of Hertfordshire (UK), describes the international effort with which he has been involved: exploring whether simple non-abelian groups (SNAGs) can be used to model complex processes in living cells. "We have for the first time shown that there are SNAGs hidden in common biological networks," he reports. "To do this, we analysed the internal workings of cells (their gene regulation and metabolism) using mathematics, computers and models from systems biology. We found that SNAG symmetries accurately describe potential activities in the genetic regulatory network that controls a cell's response to certain kinds of stress--such as radiation and DNA damage." It turns out that this genetic network involves the smallest simple non-abelian group, A5, which describes the symmetries of the icosahedron and the dodecahedron. "The 60 symmetries in this case are the result of particular sequences of manipulations by the cell's genetic regulatory network to transform ensembles of proteins into other forms," Nehaniv explains. "For example, when a set of five concentration levels of proteins is manipulated, it can be transformed to another set. When this is done many times, it can break some of the proteins down, join some together or synthesise new types of proteins. But after a specific number of manipulations the original five concentration levels of proteins will eventually return." Nehaniv is hopeful about the potential use of SNAG-based computations in non-living organisms as well: "In the future, new kinds of computers and software systems may deploy resources the way some living organisms do, in robust adaptive responses."

Image: Simulating the human brain is proving tricky. But could mathematics based on symmetries help? youtube, CC BY-SA

See "How the hidden mathematics of living cells could help us decipher the brain ," by Chrystopher Nehaniv. The Conversation, 20 May 2016 and the research article, "Symmetry structure in discrete models of biochemical systems: natural subsystems and the weak control hierarchy in a new model of computation driven by interactions," Philosophical Transactions A. For more information about math and the brain, listen to the Mathematical Moment podcast of Van Weeden talking about his use of math in researching the brain's communications pathways.

--- Claudia Clark

On calculators in the classroom, by Samantha Faria

In a recent position paper, the National Council of Teachers of Mathematics (NCTM) wrote that "calculators promote the higher-order thinking and reasoning needed for problem solving and help students learn arithmetic operations, algorithms and numerical relationships." Yet, a divide exists over whether calculators help or hinder learning. This may be because "people haven't figured out what math is. Is it calculations or is it the thinking that goes in to producing calculations," explained Barbara Reys, an expert on math education. Experts agree that teachers must be trained in using calculators in the classroom and "know when and how to incorporate them into lessons." Despite the fact that the NCTM encourages the use of calculators by every student in every grade, research shows that their use varies wildly.

See "Calculators in Class: Use Them or Lose Them?" by Jo Craven McGinty, The Wall Street Journal, May 23, 2016.

---Samantha Faria

On a book by Ken Ono about Ramanujan's influence on him, by Claudia Clark

In this episode of the weekly radio program Science Friday, host Ira Flatow interviews mathematician Ken Ono, professor of math and computer science at Emory University, about the great Indian mathematician, Srinivasa Ramanujan. Ono was an advisor to the just-released film about Ramanujan, The Man Who Knew Infinity, as well as the author of a memoir, My Search for Ramanujan: How I Learned to Count. During the interview, Ono discusses how Ramanujan taught himself mathematics and did mathematical work, as well as the important and unique relationship Ramanujan had with G. H. Hardy. Ono also describes the tremendous impact that Ramanujan's life and work had on him personally, inspiring him with his struggles as well as his creativity and passion for mathematics, and serving as a bridge between himself and his eminent mathematician father. When asked how he would compare Ramanujan to other mathematicians, Ono describes him as "an incomplete prophet. He died young. He left behind three notebooks that we've been mining 100 years after his death…and we are still learning about the full potential of his ideas. So as an anticipator of the future, as someone whose ideas mean a lot for the future of science, I'm not sure Ramanujan has an equal." Ono would like to see Ramanujan become a household name: "Of course, to mathematicians like me he matters. But what he symbolizes is much greater. What he symbolizes is that greatness can be found in the most unforgiving of circumstances and it's the responsibility of teachers and mentors alike to first recognize that talent, and then find a way to nurture it…Science usually proceeds by the work of thousands, slowly adding to a body of work. But, …every once in a while, some people come along and they propel knowledge further. They are rare. And that's what we have in Ramanujan."

See "Finding Ramanujan: Interview with Ken Ono," by Ira Flatow. Science Friday, 13 May 2016.

--- Claudia Clark

On Bayesian math and mental disorders, by Claudia Clark

Our experience of reality is built upon on information taken in by our senses woven together with our expectations based on past experiences. In the words of computational neuroscientist Peggy Seriès, "The brain is a guessing machine, trying at each moment of time to guess what is out there." She and other neuroscientists suspect that "guessing gone seriously awry may play a part in mental illnesses such as schizophrenia, autism and even anxiety disorders," writes article author Laura Sanders. Practitioners in the young field of computational psychiatry apply mathematical theories to such mental disorders in order to provide new insights. "Scientists hope that a deeper description of mental illnesses may lead to clearer ways to identify a disorder, chart how well treatments work and even improve therapies." In this article, Sanders describes some of the work researchers are doing applying Bayes's theorem--which describes the probability of an event based on related probabilities--to schizophrenia, autism, and anxiety disorders. "Experiments guided by Bayesian math reveal that the guessing process differs in people with some disorders," Sanders notes. "People with schizophrenia, for instance, can have trouble tying together their expectations with what their senses detect. And people with autism and high anxiety don't flexibly update their expectations about the world, some lab experiments suggest. That missed step can muddy their decision-making abilities."

See "Bayesian reasoning implicated in some mental disorders," by Laura Sanders, Science News, 13 May 2016.

--- Claudia Clark

On using a Turing machine to test some mathematics, by Allyn Jackson

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

On the periodic table of math: the L-functions and Modular Forms Database, by Rachel Crowell

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

On NSF's Ideas for Future Investment, by Annette Emerson

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

On Eugenia Cheng, Math and Cooking, by Annette Emerson

"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