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Credit: The mathematician Ken Ono in his office at Emory University in Atlanta. Raymond McCrea Jones for Quanta Magazine

The mathematician Ken Ono believes that the story of Srinivasa Ramanujan—mathematical savant and two-time college dropout—holds valuable lessons for how we find and reward hidden genius

For the first 27 years of his life, the mathematician Ken Ono was a screw-up, a disappointment and a failure. At least, that’s how he saw himself. The youngest son of first-generation Japanese immigrants to the United States, Ono grew up under relentless pressure to achieve academically. His parents set an unusually high bar. Ono’s father, an eminent mathematician who accepted an invitation from J. Robert Oppenheimer to join the Institute for Advanced Study in Princeton, N.J., expected his son to follow in his footsteps. Ono’s mother, meanwhile, was a quintessential “tiger parent,” discouraging any interests unrelated to the steady accumulation of scholarly credentials.

This intellectual crucible produced the desired results—Ono studied mathematics and launched a promising academic career—but at great emotional cost. As a teenager, Ono became so desperate to escape his parents’ expectations that he dropped out of high school. He later earned admission to the University of Chicago but had an apathetic attitude toward his studies, preferring to party with his fraternity brothers. He eventually discovered a genuine enthusiasm for mathematics, became a professor, and started a family, but fear of failure still weighed so heavily on Ono that he attempted suicide while attending an academic conference. Only after he joined the Institute for Advanced Study himself did Ono begin to make peace with his upbringing.

Through it all, Ono found inspiration in the story of Srinivasa Ramanujan, a mathematical prodigy born into poverty in late-19th-century colonial India. Ramanujan received very little formal schooling, yet he still produced thousands of independent mathematical results, some of which—like the Ramanujan theta function, which has found applications in string theory—are still intensely studied. But despite his genius, Ramanujan’s achievements didn’t come easily. He struggled to gain acceptance from Western mathematicians and dropped out of university twice before dying of illness at the age of 32.

While Ono, now 48, doesn’t compare himself to Ramanujan in terms of ability, he has built his career in part from Ramanujan’s insights. In 2014, Ono and his collaborators Michael Griffin and Ole Warnaar published a breakthrough result in algebraic number theory that generalized one of Ramanujan’s own results. Ono’s work, which is based on a pair of equations called the Rogers-Ramanujan identities, can be used to easily produce algebraic numbers (such as phi, better known as the “golden ratio”).

More recently, Ono served as an associate producer and mathematical consultant for The Man Who Knew Infinity, a recently released feature film about Ramanujan’s life. And his new memoir, My Search for Ramanujan: How I Learned to Count (co-authored with Amir D. Aczel), draws connections between Ramanujan’s life and Ono’s own circuitous path to mathematical and emotional fulfillment. “I wrote this book to show off my weaknesses, to show off my struggles,” Ono said. “People who are successful in their careers were not always successful from day one.”

Like Ramanujan, who benefited from years of mentorship by the British mathematician G.H. Hardy, Ono credits his own success to serendipitous encounters with teachers who helped his talents flourish. He now spends a great deal of time mentoring his own students at Emory University. Ono has also helped launch the Spirit of Ramanujan Math Talent Initiative, a venture that “strives to find undiscovered mathematicians around the world and match them with advancement opportunities in the field.”

Quanta Magazine spoke with Ono about finding his way as a mathematician and a mentor, and about Ramanujan’s inspiring brand of creativity. An edited and condensed version of the interview follows.

QUANTA MAGAZINE: What was so special about Ramanujan’s approach to doing mathematics?
KEN ONO: First, he was really a poet, not a problem solver. Most professional mathematicians, whether they’re in academia or industry, have problems that they’re aiming to solve. Somebody wants to prove the Riemann hypothesis, and sets out to do it. That’s how we think science should proceed, and in fact almost every scientist should work that way, because in reality science develops through the work of thousands of individuals slowly adding to a body of knowledge. But what you find in Ramanujan’s original notebooks is just formula after formula, and it’s not apparent where he’s going with his ideas. He was someone who could set down the paths of beginnings of important theories without knowing for sure why we would care about them as mathematicians of the future.

He’s credited with compiling thousands of identities—that is, equations that are true regardless of what values the variables take. Why is that important?
It is true that the vast majority of the contents of his notebooks are what you would call identities. Identities that relate continued fractions to other functions, expressions for integrals, expressions for hypergeometric functions, and expressions for objects that we call q-series.

But that would be a literal interpretation of his notebooks. In my opinion, that would be like taking a cookbook by Julia Child, reading the recipes and saying that it’s about assembling chemical compounds into something more complicated. Strictly speaking that would be true, but you would be missing out on what makes delicious recipes so important to us.

Ramanujan’s work came through flights of fancy. If he had been asked to explain why he did his work, he would probably say that he recorded formulas that he found beautiful, and they were beautiful because they revealed some unexpected phenomenon. And they’re important to us today because these special phenomena that Ramanujan identified, over and over again, have ended up becoming prototypes for big mathematical theories in the 20th and 21st centuries.

Here’s an example. In one of his published manuscripts, Ramanujan recorded a lot of elementary-looking results called congruences. In the 1960s, Jean-Pierre Serre, himself a Fields medalist, revisited some of these results, and in them he found glimpses of a theory that he named the theory of Galois representations. This theory of Galois representations is the language that Andrew Wiles used in the 1990s to prove Fermat’s last theorem.

There’s no “theory of Ramanujan,” but he anticipated mathematical structures that would be important to all of these other more contemporary works. He lived 80 years before his time.

How do you approach your own mathematical work—more as an artist, like Ramanujan, or with the aim of solving specific problems, like a scientist?
I’m definitely much more of a scientist. Science proceeds at a much faster rate than when I started in my career in the early 1990s, and I have to stop often to recognize the beauty in it and try not to be so caught up in the more professionalized side of how one does science. The grant getting, the publications, and all of that—I have to admit, I don’t like it.

What compelled you to juxtapose your own story with his?
Well, I almost didn’t write it. There are a lot of very personal things that I’ve never told anyone before. It wasn’t until I started writing this book that I was mature enough as a parent myself to try to understand the circumstances that led my parents to raise us the way they did. And as a professor at Emory, I see all these kids under tremendous pressure—rarely pressure that they understand the origin of. So many of these super-talented kids are just going through the motions, and aren’t passionate about their studies at all, and that’s terrible. I was like that too. I’d given up on ever trying to live up to my parents’ expectations, but somehow because I’ve had Ramanujan as a guardian angel, things have worked out well for me. It makes you a better teacher when you just tell people how hard it was for you.

This book and your story don’t fit the typical “great man of science” narrative.
I think you’ll find that’s much more common than people are willing to admit. I didn’t discover my passion for mathematics until my early 20s—that’s when [my doctoral adviser Basil] Gordon turned me on to mathematics at a time when I didn’t think anything was beautiful. I thought it was all about test scores, grades and trying to do as well as possible without putting in effort. Colleges are full of kids who think that way. How do you beat the system, right? I wasn’t beating the system. The system was beating me, and Gordon turned me around. When I’ve told people the story I’ve discovered that I’m really not alone.

That’s what I see in Ramanujan. He was a two-time college dropout whom my father looked up to as a hero—which made no sense to me when I was 16, because I was told I had to be a child prodigy. I was supposed to do my geometry problems during the summer sitting next to my dad while he did his research. I wasn’t even really allowed to go out and play, and then to just have my father tell me about Ramanujan out of the blue—it was beyond earth-shattering.

If you’d been interested in something conventionally “artistic,” like music, this kind of painful journey toward success would not seem so surprising. Why does it surprise us to hear about a mathematician having the same struggles?
For whatever reason, we live in a culture where we think that the abilities of our best scientists and our best mathematicians are somehow just God-given. That either you have this gift or you don’t, and it’s not related to help, to hard work, to luck. I think that’s part of the reason why, when we try to talk about mathematics to the public, so many people just immediately respond by saying, “Well, I was never very good at math. So I’m not really supposed to understand it or identify with it.” I might have had some mathematical talent passed through my father genetically, but that was by no means enough. You have to be passionate about a subject.

At the same time, I want it to be known that it’s totally OK to fail. In fact, you learn from your mistakes. We learn early on if that you want to be good at playing the violin, you’ve got to practice. If you want to be good at sports, you practice. But for some crazy reason, our culture assumes that if you’re good at math, you were just born with it, and that’s it. But you can be so good at math in so many different ways. I failed my [graduate-school] algebra qualifications! That doesn’t mean I can’t end up being a successful mathematician. But when I tell people I failed at this, nobody believes me.

But Ramanujan seems to be just that: a unique genius who appeared out of nowhere. What does that have to do with a regular person’s life?
You think no one can be like Ramanujan? Well, I disagree. I think we can search the world looking for a mathematical talent, just not by the usual metrics. I want teachers and parents to recognize that when you do see unusual talent, instead of demanding that these people have certain test scores, let’s find a way to help nurture them. Because I think humanity needs it. I think these are the lessons we learn from Ramanujan.

You’re leading the Spirit of Ramanujan Math Talent Initiative. What is this spirit? How do we recognize it?
First of all, it’s the idea that talent is often found in the most unforgiving and unpromising of circumstances. It’s the responsibility of mentors, teachers and parents first to recognize that talent, which is not always easy to do, and then to offer opportunities that nurture that talent.

There are no age limits, and I don’t want this to be a competition where you’re recognized for high test scores. I have no trouble finding people who can get an 800 on the math SAT. That’s easy. Those people don’t need to be identified. They’ve already self-identified. I’m searching for creativity.

That said, the spirit of Ramanujan does not require finding the next Ramanujan. We would be super lucky to do that, but if we make opportunities for 30 talented people around the world who are presently working in an intellectual desert, or are subjected to inelastic educational systems where they’re not allowed to flourish—or if we can provide an opportunity for someone to work with a scientist who could be their G.H. Hardy—then this initiative will be successful.

Do you wish you had been nurtured differently? Do you resent your parents?
I love my parents. We discussed the draft of the book for months last summer. They were very upset with me at first, because it was difficult for them to get past the first 30 pages, but now they embrace it. One reviewer actually saw the book as a love letter to my parents and to my mentors, because they taught me skills I needed.

If you had never joined the Institute for Advanced Study, would you still be struggling to reconcile your own path with your parents’ expectations?
I think I would still be searching for that recognition today if I hadn’t gotten there.

Both my parents will tell you that you only get to live once, so you might as well be the very, very best that you can be at whatever you choose. Which I don’t necessarily agree with, because if everyone lived that way, there would be nothing but a whole bunch of unhappy people in the world. But that’s how they brought us up. They taught me to be competitive. They taught me not to falsely believe I had done well when I hadn’t. They taught me standards, and those are important. But it’s true that if I hadn’t had the opportunity to work at the Institute, I’m not sure I would have been able to write this book. I might still be struggling with these things.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to John Pavlus & Quanta Magazine

Over the past few days, the mathematics world has been abuzz over the news that Sir Michael Atiyah, the famous Fields Medalist and Abel Prize winner, claims to have solved the Riemann hypothesis.

If his proof turns out to be correct, this would be one of the most important mathematical achievements in many years. In fact, this would be one of the biggest results in mathematics, comparable to the proof of Fermat’s Last Theorem from 1994 and the proof of the Poincare Conjecture from 2002.

Besides being one of the great unsolved problems in mathematics and therefore garnishing glory for the person who solves it, the Riemann hypothesis is one of the Clay Mathematics Institute’s “Million Dollar Problems.” A solution would certainly yield a pretty profitable haul: one million dollars.

The Riemann hypothesis has to do with the distribution of the prime numbers, those integers that can be divided only by themselves and one, like 3, 5, 7, 11 and so on. We know from the Greeks that there are infinitely many primes. What we don’t know is how they are distributed within the integers.

The problem originated in estimating the so-called “prime pi” function, an equation to find the number of primes less than a given number. But its modern reformulation, by German mathematician Bernhard Riemann in 1858, has to do with the location of the zeros of what is now known as the Riemann zeta function.

The technical statement of the Riemann hypothesis is “the zeros of the Riemann zeta function which lie in the critical strip must lie on the critical line.” Even understanding that statement involves graduate-level mathematics courses in complex analysis.

Most mathematicians believe that the Riemann hypothesis is indeed true. Calculations so far have not yielded any misbehaving zeros that do not lie in the critical line. However, there are infinitely many of these zeros to check, and so a computer calculation will not verify all that much. Only an abstract proof will do.

If, in fact, the Riemann hypothesis were not true, then mathematicians’ current thinking about the distribution of the prime numbers would be way off, and we would need to seriously rethink the primes.

The Riemann hypothesis has been examined for over a century and a half by some of the greatest names in mathematics and is not the sort of problem that an inexperienced math student can play around with in his or her spare time. Attempts at verifying it involve many very deep tools from complex analysis and are usually very serious ones done by some of the best names in mathematics.

Atiyah gave a lecture in Germany on Sept. 25 in which he presented an outline of his approach to verify the Riemann hypothesis. This outline is often the first announcement of the solution but should not be taken that the problem has been solved – far from it. For mathematicians like me, the “proof is in the pudding,” and there are many steps that need to be taken before the community will pronounce Atiyah’s solution as correct. First, he will have to circulate a manuscript detailing his solution. Then, there is the painstaking task of verifying his proof. This could take quite a lot of time, maybe months or even years.

Is Atiyah’s attempt at the Riemann hypothesis serious? Perhaps. His reputation is stellar, and he is certainly capable enough to pull it off. On the other hand, there have been several other serious attempts at this problem that did not pan out. At some point, Atiyah will need to circulate a manuscript that experts can check with a fine-tooth comb.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to William Ross

The humble honeybee can use symbols to perform basic maths including addition and subtraction, shows new research published today in the journal Science Advances.

Despite having a brain containing less than one million neurons, the honeybee has recently shown it can manage complex problems – like understanding the concept of zero.

Honeybees are a high value model for exploring questions about neuroscience. In our latest study we decided to test if they could learn to perform simple arithmetical operations such as addition and subtraction.

Addition and subtraction operations

As children, we learn that a plus symbol (+) means we have to add two or more quantities, while a minus symbol (-) means we have to subtract quantities from each other.

To solve these problems, we need both long-term and short-term memory. We use working (short-term) memory to manage the numerical values while performing the operation, and we store the rules for adding or subtracting in long-term memory.

Although the ability to perform arithmetic like adding and subtracting is not simple, it is vital in human societies. The Egyptians and Babylonians show evidence of using arithmetic around 2000BCE, which would have been useful – for example – to count live stock and calculate new numbers when cattle were sold off.

This scene depicts a cattle count (copied by the Egyptologist Lepsius). In the middle register we see 835 horned cattle on the left, right behind them are some 220 animals and on the right 2,235 goats. In the bottom register we see 760 donkeys on the left and 974 goats on the right. Wikimedia commons, CC BY

But does the development of arithmetical thinking require a large primate brain, or do other animals face similar problems that enable them to process arithmetic operations? We explored this using the honeybee.

How to train a bee

Honeybees are central place foragers – which means that a forager bee will return to a place if the location provides a good source of food.

We provide bees with a high concentration of sugar water during experiments, so individual bees (all female) continue to return to the experiment to collect nutrition for the hive.

In our setup, when a bee chooses a correct number (see below) she receives a reward of sugar water. If she makes an incorrect choice, she will receive a bitter tasting quinine solution.

We use this method to teach individual bees to learn the task of addition or subtraction over four to seven hours. Each time the bee became full she returned to the hive, then came back to the experiment to continue learning.

Addition and subtraction in bees

Honeybees were individually trained to visit a Y-maze shaped apparatus.

The bee would fly into the entrance of the Y-maze and view an array of elements consisting of between one to five shapes. The shapes (for example: square shapes, but many shape options were employed in actual experiments) would be one of two colours. Blue meant the bee had to perform an addition operation (+ 1). If the shapes were yellow, the bee would have to perform a subtraction operation (- 1).

For the task of either plus or minus one, one side would contain an incorrect answer and the other side would contain the correct answer. The side of stimuli was changed randomly throughout the experiment, so that the bee would not learn to only visit one side of the Y-maze.

After viewing the initial number, each bee would fly through a hole into a decision chamber where it could either choose to fly to the left or right side of the Y-maze depending on operation to which she had been trained for.

The Y-maze apparatus used for training honeybees. Scarlett Howard

At the beginning of the experiment, bees made random choices until they could work out how to solve the problem. Eventually, over 100 learning trials, bees learnt that blue meant +1 while yellow meant -1. Bees could then apply the rules to new numbers.

During testing with a novel number, bees were correct in addition and subtraction of one element 64-72% of the time. The bee’s performance on tests was significantly different than what we would expect if bees were choosing randomly, called chance level performance (50% correct/incorrect)

Thus, our “bee school” within the Y-maze allowed the bees to learn how to use arithmetic operators to add or subtract.

Why is this a complex question for bees?

Numerical operations such as addition and subtraction are complex questions because they require two levels of processing. The first level requires a bee to comprehend the value of numerical attributes. The second level requires the bee to mentally manipulate numerical attributes in working memory.

In addition to these two processes, bees also had to perform the arithmetic operations in working memory – the number “one” to be added or subtracted was not visually present. Rather, the idea of plus one or minus “one” was an abstract concept which bees had to resolve over the course of the training.

Showing that a bee can combine simple arithmetic and symbolic learning has identified numerous areas of research to expand into, such as whether other animals can add and subtract.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Scarlett Howard, Adrian Dyer, Jair Garcia

Credit: If you know your height, can you predict how big your arm span is? What about the length of your femur? Or the circumference of your head? Try this simple activity and find out how you knowing these simple ratios can even make you a better artist!  George Retseck

A project that measures up

Introduction
Our bodies are amazing! They are full of mysteries and surprising facts such as this one: Did you know that you are about a centimeter taller in the morning, when you have just woken up after hours of lying down, than you are in the evening? You might never have noticed it. These interesting facts only reveal themselves when you look closely, measure and compare. That is what this activity is about: recording, comparing and discovering how the human body measures up!

Background
Did you know that human bodies come in all sizes and forms? When you start measuring them, however, you will find our bodies show surprising similarities—and even more surprisingly, we can express these with mathematical concepts.

For one thing, our bodies are quite symmetrical. When you draw a vertical line down the center of a body, the left and right sides are almost mirror images of each other. Human bodies also show interesting ratios. Ratios compare two quantities, like the size of one part of the body to the size of another part, or to the size of the whole. An example of a human body ratio is a person’s arm span—the distance from the middle fingertip of the left hand to that of the right hand when stretching out both arms horizontally—to their height. This ratio is approximately a one to one ratio, meaning that a person’s arm span is about equal to their height. There are many more human body ratios; some are independent of age, and others change as we grow from a baby to an adult.

Wondering who would be interested in these ratios? Artists are avid users of human body ratios, because it helps them draw realistic-looking figures. They are also used in the medical world; a sizable deviation from a human body ratio can indicate a body that does not develop according to expectations. In this science activity we will examine some human body ratios and, if you like, we can explore how they can help you draw more realistic-looking figures.

Materials

• Yarn
• Scissors
• A hardcover book
• A helper
• Pen and paper (optional)
• Measuring tape (optional)

Preparation

• To compare the length of different parts of your body with your height, we will first create a string the length of your height. Take off your shoes. The easiest way is to lie on the ground with your heels pressing against a wall. Look straight up and have your helper place a hardcover book flat against the top of your head, resting on the ground. Get out from under the book and, together, span the yarn across the floor from the wall to the book, cutting the yarn just where it reaches it. Now you have a piece of yarn that is as long as you are tall. (If lying on the ground is not possible, you can also stand flat on the floor against the wall and have the book rest on top of your head and against the wall.)

Procedure

• First, we examine your arm span to height ratio. Your arm span is the distance between the middle fingertips on each hand when you stretch your arms out as far as they can reach. How do you think your height compares with your arm span? Would it be similar, way longer or way shorter?
• Now stretch your arms out as far as they can reach. Your arms will be parallel to the ground. Hold one end of the piece of yarn you just cut off with the fingertips of your left hand. Let your helper span the yarn toward the tip of your right hand’s middle finger. Is piece long enough, way longer or way too short? What does this tell you about how your arm span compares to your height?
• For most people, their arm span is about equal to their height. Mathematicians say the arm span to height ratio is one to one: your arm span goes once into your height.
• Now let’s explore another ratio: the length of your femur bone to your height. The femur bone is the only bone in your thigh. To measure its length, sit down and span a new piece of yarn over your thigh from the hip joint to the edge of your knee and cut the yarn there.
• Make an estimate. How many times would this piece of yarn go into the piece that is as long as you are tall? Can you find a way to test your estimate?
• There are several ways to compare the length of the two pieces of yarn: You might cut several pieces of the length of your shorter string, lay them end to end next to your longer piece, and count how many you need. Another way is to fold the longer string into equal parts so the length of the folded string equals the length of the shorter string. The number of folds needed is exactly the number of times your shorter string goes into your longer string.
• Did you see that the length of your femur bone goes about four times into your height? You can also say that if you divide your height in four equal pieces, you have the length of your femur bone, or the length of your femur bone is one fourth of your height. Mathematicians call this a one to four ratio.
• Now let’s move on to a ratio that might help you make more realistic drawings: the head to body ratio. How many times would the length of your head fit into your height? Maybe four, six or eight times? To test six times, fold the yarn with length equaling your height into six equal pieces. Have your helper place a book flat on your head and hang the folded string from the side of the book. If the other end of the string is about level with your chin, your height would be about six times the length of your head, or your head to body ratio would be one to six. Which number of folds fits best for you?
• There are many more bodily ratios you can explore: the circumference of your head compared with your height, or the length ratios of your forearm and foot or thumb and hand. Use pieces of yarn to measure, compare and detect these and/or your other bodily ratios.
• Extra: You have explored some ratios in your body and might wonder if these hold for other people as well. Do you think they hold for most people of your age? What about adults or babies? Do you think these ratios hold for them or would some be different? Make a hypothesis, find some volunteers, measure and compare. Was your hypothesis correct?
• Extra: This activity uses pieces of yarn to compare lengths. You can also measure your height, arm span, femur bone, etcetera with measuring tape, round the values and write the ratios as fractions. Can you find a way to simplify these fractions?
• Extra: Draw some stick figures on a sheet of paper. Can you apply some of the bodily ratios you explored (like the arm span to height or the head to body ratio) to the figures?Which ones look most realistic to you?
• Extra: Ratios are all around us. Can you find other places where ratios play an important role? To get you started, think about a recipe and the ratio of the quantity of one ingredient to another. For avid bikers, can you find the ratios that correspond to the different gears on a bike?

Observations and results
You probably found your arm span to height ratio approximately to be one to one whereas the femur to height was approximately one to four. This is expected because on average and over a large age range the human body has an arm span that is roughly equal to its height and a femur bone roughly a quarter of its height.

The head to body ratio is a little more complex as it changes from a ratio of about one to four for a small child to about one to eight for an adult. A five-year-old is likely to have a head to body ratio of about one to six.

It is good to remember these ratios are averages over a large group of people. Individual variations occur; some might even be used to one’s advantage—for example, having exceptionally long arms can be advantageous when playing basketball.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Science Buddies & Sabine De Brabandere

In mathematics, no researcher works in true isolation. Even those who work alone use the theorems and methods of their colleagues and predecessors to develop new ideas.

But when a known technique is too difficult to use in practice, mathematicians may neglect important – and otherwise solvable – problems.

Recently, I joined several mathematicians on a project to make one such technique easier to use. We produced a computer package to solve a problem called the “S-unit equation,” with the hope that number theorists of all stripes can more easily attack a wide variety of unsolved problems in mathematics.

Diophantine equations

In his text “Arithmetica,” the mathematician Diophantus looked at algebraic equations whose solutions are required to be whole numbers. As it happens, these problems have a great deal to do with both number theory and geometry, and mathematicians have been studying them ever since.

Why add this restriction of only whole-number solutions? Sometimes, the reasons are practical; it doesn’t make sense to raise 13.7 sheep or buy -1.66 cars. Additionally, mathematicians are drawn to these problems, now called Diophantine equations. The allure comes from their surprising difficulty, and their ability to reveal fundamental truths about the nature of mathematics.

In fact, mathematicians are often uninterested in the specific solutions to any particular Diophantine problem. But when mathematicians develop new techniques, their power can be demonstrated by settling previously unsolved Diophantine equations.

Andrew Wiles, right, receives the Wolflskehl award for his solution of Fermat’s Last Theorem. Peter Mueller/REUTERS

Andrew Wiles’ proof of Fermat’s Last Theorem is a famous example. Pierre de Fermat claimed in 1637 – in the margin of a copy of “Arithmetica,” no less – to have solved the Diophantine equation xⁿ + yⁿ = zⁿ, but offered no justification. When Wiles proved it over 300 years later, mathematicians immediately took notice. If Wiles had developed a new idea that could solve Fermat, then what else could that idea do? Number theorists raced to understand Wiles’ methods, generalizing them and finding new consequences.

No single method exists that can solve all Diophantine equations. Instead, mathematicians cultivate various techniques, each suited for certain types of Diophantine problems but not others. So mathematicians classify these problems by their features or complexity, much like biologists might classify species by taxonomy.

Finer classification

This classification produces specialists, as different number theorists specialize in the techniques related to different families of Diophantine problems, such as elliptic curves, binary forms or Thue-Mahler equations.

Within each family, the finer classification gets customized. Mathematicians develop invariants – certain combinations of the coefficients appearing in the equation – that distinguish different equations in the same family. Computing these invariants for a specific equation is easy. However, the deeper connections to other areas of mathematics involve more ambitious questions, such as: “Are there any elliptic curves with invariant 13?” or “How many binary forms have invariant 27?”

The S-unit equation can be used to solve many of these bigger questions. The S refers to a list of primes, like {2, 3, 7}, related to the particular question. An S-unit is a fraction whose numerator and denominator are formed by multiplying only numbers from the list. So in this case, 3/7 and 14/9 are S-units, but 6/5 is not.

The S-unit equation is deceptively simple to state: Find all pairs of S-units which add to 1. Finding some solutions, like (3/7, 4/7), can be done with pen and paper. But the key word is “all,” and that is what makes the problem difficult, both theoretically and computationally. How can you ever be sure every solution has been found?

In principle, mathematicians have known how to solve the S-unit equation for several years. However, the process is so convoluted that no one could ever actually solve the equation by hand, and few cases have been solved. This is frustrating, because many interesting problems have already been reduced to “just” solving some particular S-unit equation.

The process of solving the S-unit equation is so convoluted that few have attempted to do it by hand. Jat306/shutterstock.com

How the solver works

Circumstances are changing, however. Since 2017, six number theorists across North America, myself included, have been building an S-unit equation solver for the open-source mathematics software SageMath. On March 3, we announced the completion of the project. To illustrate its application, we used the software to solve several open Diophantine problems.

The primary difficulty of the S-unit equation is that while only a handful of solutions will exist, there are infinitely many S-units that could be part of a solution. By combining a celebrated theorem of Alan Baker and a delicate algorithmic technique of Benne de Weger, the solver eliminates most S-units from consideration. Even at this point, there may be billions of S-units – or more – left to check; the program now tries to make the final search as efficient as possible.

This approach to the S-unit equation has been known for over 20 years, but has been used only sparingly, because the computations involved are complicated and time-consuming. Previously, if a mathematician encountered an S-unit equation that she wanted to solve, there was no automated way to solve it. She would have to carefully step through the work of Baker, de Weger and others, then write her own computer program to do the computations. Running the program could take hours, days or even weeks for the computations to finish.

Our hope is that the software will help mathematicians solve important problems in number theory and enhance their understanding of the nature, beauty and effectiveness of mathematics.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Christopher Rasmussen

Have you ever taken a walk through the rain on a warm spring day and seen that perfect puddle? You know, the one where the raindrops seem to touch down at just the right pace, causing a dance of vanishing circles?

Even before I entered the field of fluid flow research nearly 15 years ago, I was fascinated by the waves that appear after a raindrop hits a puddle.

As I became focused on the study of unstable waves in liquid sheets – geared toward mitigating undesirable waves in industrial coating and atomization processes – my fascination with puddle waves turned into an obsession. What is going on? Where does the pattern come from? Why does the impact of rain in a puddle look different than when rain falls elsewhere, like in a lake or the ocean?

It turns out that it all has to do with something called dispersion.

In the context of water waves, dispersion is the ability of waves of different wavelengths to each move at their own individual speeds. Looking down on a puddle, we see a collection of such waves moving together as one ripple in the water.

When a raindrop touches down, imagine it as a “ding” to the water surface. This ding can be idealized as a packet of waves of all different sizes. After the raindrop falls, the packet’s waves are ready to begin their new life in the puddle.

However, whether we see those waves as ripples depends on the body of water that the raindrop lands on. The number and spacing of rings that you see depends on the height of the puddle. This has been verified in some very cool ripple tank experiments, where a drop of the same velocity falls into a container with water at different depths.

Shallow puddles enable ripples, because they are much thinner than they are wide. The balance between the surface force – between the water puddle and the air above it – and the gravitational force tips in favor of surface force. This is key, since the surface force depends on the curvature of the water surface, whereas the gravitational force does not.

An initially still shallow puddle becomes curved at the surface after the raindrop hits. The surface force is different for long waves than for short ones, causing waves of different sizes to separate into ripples. For shallow puddles, the long waves move slowly away from the point of impact, while the short waves move fast, and the really short waves move really fast, becoming tightly packed at the perimeter. This creates the enchanting pattern that we see.

Raindrops may react differently in other situations. Imagine that rain is hitting a lake or ocean – or those deep pothole puddles that require galoshes. Here, the raindrop hits the water, but the force due to gravity becomes more important. It moves waves of all sizes at the same speed which may overpower the rippling effect due to the surface force.

The combination of teaching undergraduate partial differential equations while simultaneously continuing to research liquid sheets led to what I’ve been calling the “puddle equation.” When solved, the equation creates an animated simulation of what happens after a raindrop hits a puddle. It’s a simplified version of an equation in one of our group’s more recent research endeavors, but it’s also consistent with the classical description of ripples.

I use this approximate description of puddle waves as one way to get students excited about math by relating it to the world around them.

A model of waves in a dispersive puddle, after a raindrop hits. The top three figures show what happens after a drop hits the puddle, with arrows indicating the passage of time. The bottom figure shows the cross-sectional view through the puddle, highlighting that the initial wave bundle caused by the raindrop splits into waves of different sizes. Large waves in the center move more slowly than small waves at the perimeter. Nate Barlow

The study of surface-force-driven waves is important for applications such as coating processes involved in making batteries and solar cells.

Such waves also appear as a result of the leg stroke of a water strider insect, but research has found that the water strider isn’t specifically looking to make those waves to enable travel.

The beauty of puddle waves is no small thing by itself. By connecting nature with its primal language – mathematics – we gain access to its control panel, allowing us to observe every little detail, uncovering all the secrets.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Nate Barlow

Alice in Wonderland enthusiasts recently celebrated the story’s anniversary with creative events like playing with puzzles and time — and future Alice exhibits are in the works. The original 1865 children’s book Alice’s Adventures in Wonderland, sprung from a mathematician’s imagination, continues to inspire exploration and fun.

But is a connection between math and creativity captured in schools? Much discussion across the western world from both experts and the public has emphasized the need to revitalize high school mathematics: critics say the experience is boring or not meaningful to most students. Experts concerned with the public interest and decision-making say students need skills in critical thinking, creativity, communication and collaboration.

Mathematicians, philosophers and educators are also concerned with the excitement and energy of creative expression, with invention, with wonder and even with what might be called the romance of learning.

Mathematics has all the attributes of the paragraph above, and so it seems to me that what’s missing from high school math is mathematics itself.

I am now working with colleagues at Queen’s University and the University of Ottawa to develop RabbitMath, a senior level high-school math curriculum designed to enable students to work together creatively with a high level of personal engagement. My preparation for this has been 40 years of working with teachers in high-school classrooms.

In partnership with grades 11 and 12 math teachers, we will be piloting this curriculum over the next few years.

Mathematical novels

When students study literature, drama or the creative arts in high school, the curriculum centres on what can be called sophisticated works of art, created in response to life’s struggles and triumphs.

But currently in school mathematics, this is rarely the case: students are not connected to the larger imaginative projects through which professional mathematicians confront the world’s problems or explore the world’s mysteries.

The author, Peter Taylor, right, at a Lisgar Collegiate Institute Grade 11 math classroom in Ottawa, 2018. (Ann Arden), Author provided

Mathematician Jo Boaler from the Stanford Graduate School of Education says that a “wide gulf between real mathematics and school mathematics is at the heart of the math problems we face in school education.”

Of the subject of mathematics, Boaler notes that:

“Students will typically say it is a subject of calculations, procedures, or rules. But when we ask mathematicians what math is, they will say it is the study of patterns that is an aesthetic, creative, and beautiful subject. Why are these descriptions so different?”

She points out the same gulf isn’t seen if people ask students and English-literature professors what literature is about.

In the process of constructing the RabbitMath curriculum, problems or activities are included when team members find them engaging and a challenge to their intellect and imagination. Following the analogy with literature, we call the models we are working with mathematical novels.

For example, one project invites students to work with ocean tides. It would hard to find a dramatic cycle as majestic as the effect of that sublime distant moon on the powerful tidal action in the Bay of Fundy.

Student engagement

In the 1970s, the extraordinary mathematician and computer scientist at Massachusetts Institute of Technology, Seymour Papert, noticed that in art class, students, just as mature artists, are involved in personally meaningful work. Papert’s objective was to be able to say the same of a mathematics student.

I had a parallel experience in 2013 when I was the internal reviewer for the Drama program at Queen’s. I marvelled at students’ creative passion as they prepared to stage a performance. And they weren’t all actors: they were singers, musicians, writers, composers, directors and technicians.

In Papert’s curriculum model, students with diverse abilities and interests work together on projects, whereby they collaborate on problems, strategies and outcomes.

As a pioneering computer scientist, Papert understood that students could directly access the processes of design and construction through digital technology. Papert used his computer system LOGO for this technical interface. LOGO was limited in its scope, but Papert’s idea was way ahead of its time.

Students in the RabbitMath classroom will work together using the programming language Python to construct diagrams and animations to better understand their experiments with springs and tires, mirrors and music. They will produce videos that can explain to their classmates the workings of a sophisticated structure.

Today, technology, the internet, computer algebra systems and mathematical programming provide possibilities for immediate engagement in processes of design and construction — exactly what Papert wanted. The platform for RabbitMath is the Jupyter Notebook, a direct descendant of LOGO.

Technical skill

For too many years, real progress in school mathematics education has been hamstrung by a ridiculous confrontation between so-called “traditional” and “discovery” math. The former is concerned with technical facility and the latter is about skills of inquiry and investigation.

There is no conflict between the two; in fact they support each other rather well. Every sophisticated human endeavour, from conducting a symphony orchestra to putting a satellite into orbit, understands the complementary nature of technical facility and creative investigation.

Stanford University Graduate School of Education mathematician Keith Devlin advises parents to ensure their child has mastery of what he calls number sense, “fluidity and flexibility with numbers, a sense of what numbers mean, and an ability to use mental mathematics to negotiate the world and make comparisons.” But for students embarking on careers in science, technology or engineering, that is not enough, he says. They need a deep understanding of both those procedures and the concepts they rely on — the capacity to analyze and work with complex systems.

A high-school math class is a rich ecosystem of differing abilities, capacities, objectives and temperaments.

The educator’s goal must be to enable a diverse mix of students to work together in a math class as creatively and intensely as students in the drama program, or to bring the same personal passion as they might to writing fiction.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Peter Taylor

Bias in, bias out: many algorithms have inherent design problems. Vintage Tone/Shutterstock

Could this be the first line of a “Hippocratic Oath” for mathematicians and data scientists? Hannah Fry, Associate Professor in the mathematics of cities at University College London, argues that mathematicians and data scientists need such an oath, just like medical doctors who swear to act only in their patients’ best interests.

“In medicine, you learn about ethics from day one. In mathematics, it’s a bolt-on at best. It has to be there from day one and at the forefront of your mind in every step you take,” Fry argued.

But is a tech version of the Hippocratic Oath really required? In medicine, these oaths vary between institutions, and have evolved greatly in the nearly 2,500 years of their history. Indeed, there is some debate around whether the oath remains relevant to practising doctors, particularly as it is the law, rather than a set of ancient Greek principles, by which they must ultimately abide.

How has data science reached the point at which an ethical pledge is deemed necessary? There are certainly numerous examples of algorithms doing harm – criminal sentencing algorithms, for instance, have been shown to disproportionately recommend that low-income and minority people are sent to jail.

Similar crises have led to proposals for ethical pledges before. In the aftermath of the 2008 global financial crisis, a manifesto by financial engineers Emanuel Derman and Paul Wilmott beseeched economic modellers to swear not to “give the people who use my model false comfort about its accuracy. Instead, I will make explicit its assumptions and oversights.”

Just as prejudices can be learned as a child, the biases of these algorithms are a result of their training. A common feature of these algorithms is the use of black-box (often proprietary) algorithms, many of which are trained using statistically biased data.

In the case of criminal justice, the algorithm’s unjust outcome stems from the fact that historically, minorities are overrepresented in prison populations (most likely as a result of long-held human biases). This bias is therefore replicated and likely exacerbated by the algorithm.

Machine learning algorithms are trained on data, and can only be expected to produce predictions that are limited to those data. Bias in, bias out.

Promises, promises

Would taking an ethical pledge have helped the designers of these algorithms? Perhaps, but greater awareness of statistical biases might have been enough. Issues of unbiased representation in sampling have long been a cornerstone of statistics, and training in these topics may have led the designers to step back and question the validity of their predictions.

Fry herself has commented on this issue in the past, saying it’s necessary for people to be “paying attention to how biases you have in data can end up feeding through to the analyses you’re doing”.

But while issues of unbiased representation are not new in statistics, the growing use of high-powered algorithms in contentious areas make “data literacy” more relevant than ever.

Part of the issue is the ease with which machine learning algorithms can be applied, making data literacy no longer particular to mathematical and computer scientists, but to the public at large. Widespread basic statistical and data literacy would aid awareness of the issues with statistical biases, and are a first step towards guarding against inappropriate use of algorithms.

Nobody is perfect, and while improved data literacy will help, unintended biases can still be overlooked. Algorithms might also have errors. One easy (to describe) way to guard against such issues is to make them publicly available. Such open source code can allow joint responsibility for bias and error checking.

Efforts of this sort are beginning to emerge, for example the Web Transparency and Accountability Project at Princeton University. Of course, many proprietary algorithms are commercial in confidence, which makes transparency difficult. Regulatory frameworks are hence likely to become important and necessary in this area. But a precondition is for practitioners, politicians, lawyers, and others to understand the issues around the widespread applicability of models, and their inherent statistical biases.

Ethics is undoubtedly important, and in a perfect world would form part of any education. But university degrees are finite. We argue that data and statistical literacy is an even more pressing concern, and could help guard against the appearance of more “unethical algorithms” in the future.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Lewis Mitchell, Joshua Ross

The first modern electronic digital computer was called the Atanasoff–Berry computer, or ABC.

It was built by physics Professor John Vincent Atanasoff and his graduate student, Clifford Berry, in 1942 at Iowa State College, now known as Iowa State University.

That’s where I have been teaching computer engineering for over 30 years, and I’m also a collector of old computers. I got to meet Atanasoff when he visited Iowa State and got a signed copy of his book.

Before ABC, there were mechanical computing devices that could perform simple calculations. The first mechanical computer, The Babbage Difference Engine, was designed by Charles Babbage in 1822. The ABC was the basis for the modern computer we all use today.

The ABC’s drums. Courtesy of Iowa State University Library Special Collections and University Archives, CC BY-ND

The ABC weighed over 700 pounds and used vacuum tubes. It had a rotating drum, a little bigger than a paint can, that had small capacitors on it. A capacitor is device that can store an electric charge, like a battery.

The ABC was designed to solve problems with up to 29 different variables. You might be familiar with equations with one variable, like 2y = 14. Now imagine 29 different variables. These are common problems in physics and other sciences, but were difficult and time-consuming to solve by hand.

Atanasoff was credited with several breakthrough ideas that are still present in modern computers. The most important idea was using binary digits, just ones and zeroes, to represent all numbers and data. This allowed the calculations to be performed using electronics.

Another idea was the separation of the program (the computer instructions) and memory (places to store numbers).

The ABC completed one operation about every 15 seconds. Compared to the millions of operations per second of today’s computer, that probably seems very slow.

Unlike today’s computers, the ABC did not have a changeable stored program. This meant the program was fixed and designed to do a single task. This also meant that, to solve these problems, an operator had to write down the intermediate answer and then feed that back into the ABC. Atanasoff left Iowa State before he perfected a storage method that would have eliminated the need for the operator to reenter the intermediate results.

Part of the ABC. Courtesy of Iowa State University Library Special Collections and University Archives, CC BY-ND

Shortly after Atanasoff left Iowa State, the ABC was dismantled. Atanasoff never filed a patent for his invention. That means that, for a long time, many people weren’t aware of the ABC.

In 1947, the creators of the Electronic Numerical Integrator And Computer, or ENIAC, filed a patent. This allowed them to claim they were the inventors of the digital computer. For several decades, most people thought that the ENIAC was the first modern computer.

But one of the inventors of the ENIAC had visited Atanasoff in 1941. The courts later ruled that this visit influenced the design of the ENIAC. The ENIAC patent was thrown out by a judge in 1973.

The holders of the ENIAC patent argued that the ABC never really worked. Since all that remained was one of the drum memory units, it was hard to prove otherwise.

In 1997 a team of faculty, researchers and students at Iowa State University finished building a replica of the ABC. They were able to demonstrate that the ABC did function. You can see the replica today at the Computer History Museum in Mountain View, California.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Doug Jacobson

Bees are pretty good at maths – as far as insects go, at least. We already know, for example, that they can count up to four and even understand the concept of zero.

But in a new study, published today in the Journal of Experimental Biology, we show honeybees can also understand numbers higher than four – as long as we provide feedback for both correct and incorrect responses as they learn.

Even our own brains are less adept at dealing with numbers greater than four. While we can effortlessly estimate up to four items, processing larger numbers requires more mental effort. Hence why when asked to count, a young child will sometimes answer with “1, 2, 3, 4, more”!

If you don’t believe me, try the test below. The various colour groupings representing 1-4 stars are easy to count quickly and accurately. However, if we try estimating the number of all stars at once by ignoring colours, it requires more concentration, and even then our accuracy tends to be poorer.

For numbers of elements ranging from 1-4, as represented here in different colours, we very efficiently process the exact number. However, if we try estimating the number of all stars at once by ignoring colour, it requires a lot more cognitive effort.

This effect isn’t unique to humans. Fish, for example, also show a threshold for accurate quantity discrimination at four.

One theory to explain this is that counting up to four isn’t really counting at all. It may be that many animals’ brains can innately recognise groups of up to four items, whereas proper counting (the process of sequentially counting the number of objects present) is needed for numbers beyond that.

By comparing the performance of different animal species in various number processing tasks we can better understand how differences in brain size and structure enable number processing. For example, honeybees have previously been shown to be able to count and discriminate numbers up to four, but not beyond. We wanted to know why there was a limit at four – and whether they can go further.

Best bee-haviour

Bees are surprisingly good at maths. We recently discovered that bees can learn to associate particular symbols with particular quantities, much like the way we use numerals to represent numbers.

Bees learn to do this type of difficult task if given a sugary reward for choosing the correct association, and a bitter liquid for choosing incorrectly. So if we were to push bees beyond the four threshold, we knew success would depend on us asking the right question, in the right way, and providing useful feedback to the bees.

We trained two different groups of bees to perform a task in which they were presented with a choice of two different patterns, each containing a different number of shapes. They could earn a reward for choosing the group of four shapes, as opposed to other numbers up to ten.

We used two different training strategies. One group of ten bees received only a reward for a correct choice (choosing a quantity of four), and nothing for an incorrect choice. A second group of 12 bees received a sugary reward for picking four, or a bitter-tasting substance if they made a mistake.

In the test, bees flew into a Y-shaped maze to make a choice, before returning to their hive to share their collected sweet rewards.

Each experiment conducted with a single bee lasted about four hours, by which time each bee had made 50 choices.

Bees were individually trained and tested in a Y-shaped maze where a sugar reward was presented on the pole directly in front of the correct stimulus. Author provided

The group that only received sweet rewards could not successfully learn to discriminate between four and higher numbers. But the second group reliably discriminated the group of four items from other groups containing higher numbers.

Thus, bees’ ability to learn higher number discrimination depends not just on their innate abilities, but also on the risks and rewards on offer for doing so.

Bee’s-eye view of either four or five element displays that could be discriminated. Inserts show how we normally see these images.

Our results have important implications for understanding how animals’ brains may have evolved to process numbers. Despite being separated by 600 million years of evolution, invertebrates such as bees and vertebrates such as humans and fish all seem to share a common threshold for accurately and quickly processing small numbers. This suggests there may be common principles behind how our brains tackle the question of quantity.

The evidence from our new study shows bees can learn to process higher numbers if the question and training are presented in the right way. These results suggest an incredible flexibility in animal brains, of all sizes, for learning to become maths stars.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Adrian Dyer, Jair Garcia, Scarlett Howard