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Bourbaki Congress of 1938.

By many measures, Nicolas Bourbaki ranks among the greatest mathematicians of the 20th century.

Largely unknown today, Bourbaki is likely the last mathematician to master nearly all aspects of the field. A consummate collaborator, he made fundamental contributions to important mathematical fields such as set theory and functional analysis. He also revolutionized mathematics by emphasizing rigor in place of conjecture.

There’s just one problem: Nicolas Bourbaki never existed.

Never existed?

The cover of the first volume in Bourbaki’s textbook. Maitrier/Wikimedia, CC BY-SA

While it is now widely accepted that there never was a Nicolas Bourbaki, there is evidence to the contrary.

For example, there are wedding announcements for his daughter Betty, a baptismal certificate in his name and an impressive family lineage extending back to an ancestor Napoleon raised as his own son.

Even the professional mathematics community was misled for a time. When Ralph Boas, an editor of the journal Mathematical Reviews, wrote that Bourbaki was a pseudonym, he was promptly refuted by none other than Bourbaki himself. Bourbaki countered with a letter stating that B.O.A.S. actually just was an acronym of the last names of the editors of the Reviews.

These cases of confused identity were not all fun and games. For example, it is alleged that, while visiting Finland at the outset of World War II, French mathematician André Weil was investigated for spying. The authorities found suspicious papers in his possession: a fake identity, a set of business cards and even invitations from the Russian Academy of Science – all in Bourbaki’s name. Supposedly, Weil was freed only after an officer recognized him as a preeminent mathematician.

Who was Bourbaki?

If Bourbaki never existed, who – or what – was he?

The name Nicolas Bourbaki first appeared in a place rocked by turmoil at a volatile time in history: Paris in 1934.

World War I had wiped out a generation of French intellectuals. As a result, the standard university-level calculus textbook had been written more than two and half decades before and was out of date.

Newly minted professors André Weil and Henri Cartan wanted a rigorous method to teach Stokes’ theorem, a key result of calculus. After realizing that others had similar concerns, Weil organized a meeting. It took place December 10, 1934 at a Parisian café called Capoulade.

The nine mathematicians in attendance agreed to write a textbook “to define for 25 years the syllabus for the certificate in differential and integral calculus by writing, collectively, a treatise on analysis,” which they hoped to complete in just six months.

Cafe Capoulade in 1943. Langhaus, German Federal Archive/Wikimedia, CC BY-SA

 

As a joke, they named themselves after an old French general who had been duped in the Franco-Prussian war.

As they proceeded, their original goal of elucidating Stokes’ theorem expanded to laying out the foundations of all mathematics. Eventually, they began to hold regular Bourbaki “conferences” three times a year to discuss new chapters for the treatise.

Individual members were encouraged to engage with all aspects of the effort, to ensure that the treatise would be accessible to nonspecialists. According to one of the founders, spectators invariably came away with the impression that they were witnessing “a gathering of madmen.” They could not imagine how people, shouting – “sometimes three or four at the same time” – could ever come up with something “intelligent.”

Top mathematicians from across Europe, intrigued by the group’s work and style, joined to augment the group’s ranks. Over time, the name Bourbaki became a collective pseudonym for dozens of influential mathematicians spanning generations, including Weil, Dieudonne, Schwartz, Borel, Grothendieck and many others.

Since then, the group which has added new members over time, has proved to have a profound impact on mathematics, certainly rivaling any of its individual contributors.

Profound impact

Mathematicians have made a plethora of important contributions under Bourbaki’s name.

To name a few, the group introduced the null set symbol; the ubiquitous terms injective, surjective, bijective; and generalizations of many important theorems, including the Bourbaki-Witt theorem, the Jacobson-Bourbaki theorem and the Bourbaki-Banach-Alaoglu theorem.

Their text, “Elements of Mathematics,” has swelled to more than 6,000 pages. It provides a “solid foundation for the whole body of modern mathematics,” according to mathematician Barbara Pieronkiewicz.

Bourbaki’s influence is still alive and well. Now in “his” 80th year of research, in 2016 “he” published the 11th volume of the “Elements of Mathematics.” The Bourbaki group, with its ever-changing cast of members, still holds regular seminars at the University of Paris.

Partly thanks to the breadth and significance of “his” mathematical contributions, and also because – ageless, unchanging and operating in multiple places at once – “he” seems to defy the very laws of physics, Bourbaki’s mathematical prowess will likely never be equaled.

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

Grab a few balls and get calculating pi

Pi Day, which occurs every 14 March – or 3/14, in the US date format – celebrates the world’s favourite mathematical constant. This year, why not try an experiment to calculate its value? All you will need is a cardboard tube and a series of balls, each 100 times lighter than the next. You have those lying around the house, right?

This experiment was first formulated by mathematician Gregory Galperin in 2001. It works because of a mathematical trick involving the masses of a pair of balls and the law of conservation of energy.

First, take the tube and place one end up against a wall. Place two balls of equal mass in the tube. Let’s say that the ball closer to the wall is red, and the other is blue.

Next, bounce the blue ball off the red ball. If you have rolled the blue ball hard enough, there should be three collisions: the blue ball hits the red one, the red ball hits the wall, and the red ball bounces back to hit the blue ball once more. Not-so-coincidentally, three is also the first digit of pi.

To calculate pi a little bit more precisely, replace the red ball with one that is 100 times less massive than the blue ball – a ping pong ball might work, so we will call this the white ball.

When you perform the experiment again, you will find that the blue ball hits the white ball, the white ball hits the wall and then the white ball continues to bounce back and forth between the blue ball and the wall as it slows down. If you count the bounces, you’ll find that there are 31 collisions. That gives you the first two digits of pi: 3.1.

Galperin calculated that if you continue the same way, you will keep getting more digits of pi. If you replace the white ball with another one that is 10,000 times less massive than the blue ball, you will find that there are 314 collisions, and so on. If you have enough balls, you can count as many digits of pi as you like.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Leah Crane*

Machines are getting better at maths – artificial intelligence has learned to solve university-level calculus problems in seconds.

François Charton and Guillaume Lample at Facebook AI Research trained an AI on tens of millions of calculus problems randomly generated by a computer. The problems were mathematical expressions that involved integration, a common technique in calculus for finding the area under a curve.

To find solutions, the AI used natural language processing (NLP), a computational tool commonly used to analyse language. This works because the mathematics in each problem can be thought of as a sentence, with variables, normally denoted x, playing the role of nouns and operations, such as finding the square root, playing the role of verbs. The AI then “translates” the problem into a solution.

When the pair tested the AI on 500 calculus problems, it found a solution with an accuracy of 98 per cent. A comparable standard program for solving maths problems had only an accuracy of 85 per cent on the same problems.

The team also gave the AI differential equations to solve, which are other equations that require integration to solve as well as other techniques. For these equations, the AI wasn’t quite as good, solving them correctly 81 per cent for one type of differential equation and 40 per cent on a harder type.

Despite this, it could still correctly answer questions that confounded other maths programs.

Doing calculus on a computer isn’t especially useful in practice, but with further training AI might one day be able to tackle maths problems that are too hard for humans to crack, says Charton.

The efficiency of the AI could save humans time in other mathematical tasks, for example, when proving theorems, says Nikos Aletras at the University of Sheffield, UK.

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*Credit for article given to Gege Li*

AI can search through numbers and equations to find patterns

Artificial intelligence’s ability to sift through large amounts of data is helping us tackle one of the most difficult unsolved problems in mathematics.

Yang-Hui He at City, University of London, and his colleagues are using the help of machine learning to better understand the Birch and Swinnerton-Dyer conjecture. This is one of the seven fiendishly difficult Millennium Prize Problems, each of which has a $1 million reward on offer for the first correct solution.

The conjecture describes solutions to equations known as elliptic curves, equations in the form of y² = x³ + ax + b, where x and y are variables and a and b are fixed constants.

Elliptic curves were essential in cracking the long-standing Fermat’s last theorem, which was solved by mathematician Andrew Wiles in the 1990s, and are also used in cryptography.

To study the behaviour of elliptic curves, mathematicians also use an equation called the L-series. The conjecture, first stated by mathematicians Bryan Birch and Peter Swinnerton-Dyer in the 1960s, says that if an elliptic curve has an infinite number of solutions, its L-series should equal 0 at certain points.

“It turns out to be a very, very difficult problem to find a set of integer solutions on such equations,” says He, meaning solutions only involving whole numbers. “This is part of the biggest problem in number theory: how do you find integer solutions to polynomials?”

Finding integer solutions or showing that they cannot exist has been crucial. “For example, Fermat’s last theorem is reduced completely to the statement of whether you can find certain properties of elliptic curves,” says He.

He and his colleagues used an AI to analyse close to 2.5 million elliptic curves that had been compiled in a database by John Cremona at the University of Warwick, UK. The rationale was to search the equations to see if any statistical patterns emerged.

Plugging different values into the elliptic curve equation and plotting the results on a graph, the team found that the distribution takes the shape of a cross, which mathematicians hadn’t previously observed. “The distribution of elliptic curves seems to be symmetric from left to right, and up and down,” says He.

“If you spot any interesting patterns, then you can raise a conjecture which may later lead to an important result,” says He.  “We now have a new, really powerful thing, which is machine learning and AI, to do this.”

To see whether a theoretical explanation exists for the cross-shaped distribution, He consulted number theorists. “Apparently, nobody knows,” says He.

“Machine learning hasn’t yet been applied very much to problems in pure maths,” says Andrew Booker at the University of Bristol, UK. “Elliptic curves are a natural place to start.”

“Birch and Swinnerton-Dyer made their conjecture based on patterns that they observed in numerical data in the 1960s, and I could imagine applications of machine learning that tried to detect those patterns efficiently,” says Booker, but the approach so far is too simple to turn up any deep patterns, he says.

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*Credit for article given to Donna Lu*

Planning is a key ingredient for effective teaching. But why is it so important? What is ‘good’ planning? How do we make our planning purposeful and focused? Let’s delve further into planning, and what it encompasses.

Curriculum is central to planning. It guides what we teach. The other crucial factor is to know the needs of your students. Effective planning begins with finding where these two elements meet.

The Purposeful planning podcast has been developed to complement the Explicit Teaching in Maths professional learning modules. In this podcast, Dr Emily Ross from The University of Queensland uses the analogy of planning being like a road trip. Dr Ross explains that the curriculum (the knowledge and skills taught) is the destination of this road trip. It’s where you want your students to get to by the end of the lesson or topic.

The road trip itself includes the places you see and the stops on the way to your destination, and this is likened to the teaching and learning. It’s the steps in the lesson or unit plan that enable your students to reach their destination.

Using this road trip analogy, we can ask ourselves two big questions in terms of planning: where are we going? and how will we get there?

Let’s breakdown this analogy further.

• Some people like to be very well planned and outline the detailed steps required to reach the destination.
• Some people like to make a more general plan and they outline the main signposts required to travel past to get to the destination.
• Sometimes you may need to take some detours along the way depending on the needs of your learners. Listen to your learners: Are you moving too slowly? Are you going too fast? Or are you on the wrong road?There are different ways of getting to your end destination, and understanding your students and their needs will determine the path you take.

The process of planning

Planning involves interpreting the curriculum and working out how we can support our students to learn knowledge and develop skills. So, what might ‘good’ planning look like?

Start with the learner

What do your students know about the topic? What prior knowledge do they have? This isn’t always straightforward as students in your class will bring a range of skills and knowledge to each topic, and your planning needs to reflect this.

The curriculum

Use your curriculum knowledge and understandings and know exactly what you want your students to achieve.

The steps

Break the learning into small steps. Think about the chunks of knowledge and skills the students need to learn and build upon this throughout the lesson and topic.

Learning sequences

Build authentic teaching and learning sequences to support students to learn knowledge, develop skills, and understand and apply concepts.

Learning intentions and success criteria

Planning is enhanced by including purposeful learning intentions and success criteria.

A learning intention states the goal of the lesson. What will you learn?  The success criteria outlines how the students will know they have succeeded. How do you know you have learnt it? How do you know you can now do it?

Learning intention (LI) and success criteria (SC) checklist

• Sharethe LI and SC with your students.
• Make the LI and SC explicitso that students know exactly what is expected.
• Make the LI challenging,but not too difficult. Students need to be learning new knowledge and skills, and experience success in doing so.
• Make the SC measurableso students can easily see if they have been successful or not.
• Provide feedbackto students throughout the lesson, so they know what to do next to achieve the learning intention.

Good planning is essential for quality maths teaching and learning. If you’d like to know more about planning:

• sign-up to The Maths in schools: Explicit teaching in Maths learning modules. This self-paced, professional leaning course offers five modules that are designed around the seven components of explicit teaching. The modules are aligned to the Australian Institute and School Leadership (ASITSL) professional standards, and they include lessons and activities you can use to teach maths concepts from the Australian Curriculum.
• listen to the Maths hub podcast, Episode 1: Purposeful planning. This engaging and informative podcast is hosted by Allan Dougan, the CEO of the Australian Association of Mathematics Teachers (AAMT). Allan chats to expert Dr Emily Ross about planning. The podcast provides practical ideas you can readily use as you plan your maths lessons. A highlight is the discussion on how to incorporate learning intentions and success criteria in creative and flexible ways.
• watch the Explicit teaching in mathematics: purposeful planning webinar. Associate Professor Helen Chick, University of Tasmania, discusses purposeful planning when thinking through and constructing maths lessons using the explicit teaching model and how careful planning of lessons is just as important as teaching the lesson and can enable explicit teaching to be successful.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to The Mathematics Hub

Mathematicians don’t need to worry about AI taking over their jobs just yet

You don’t need a human brain to do maths — even artificial intelligence can write airtight proofs of mathematical theorems.

An AI created by a team at Google has proven more than 1200 mathematical theorems. Mathematicians already knew proofs for these particular theorems, but eventually the AI could start working on more difficult problems.

One of the core pillars of maths is the concept of proof. It is an argument based on known statements, assumptions, or rules, that a certain mathematical statement, such as a theorem, is true.

To train their AI, the Google team started with a database of more than 10,000 human-written mathematical proofs, along with the reasoning behind each step known as a tactic. Tactics could include using a known property about numbers, such as the fact that multiplying x by y is the same as multiplying y by x, or applying the chain rule.

Then, they tested the AI on 3225 theorems it hadn’t seen before and it successfully proved 1253 of them. Those that it couldn’t prove were because it had only 41 tactics at its disposal.

To prove each theorem, the AI split them into smaller and smaller components using the list of tactics. Eventually each of the smaller components could be proven with a single tactic, thus proving the larger theorem.

“Most of the proofs we used are relatively short, so they don’t require a lot of long complicated reasoning, but this is a start,” says Christian Szegedy at Google. “Where we want to get to is a system that can prove all the theorems that humans can prove, and maybe even more.”

Tackling harder problems

While this particular algorithm is focused on linear algebra and complex calculus, changing its training set could allow it to do any sort of mathematics, says Szgedy. For now, the AI’s main application is filling in the details of long and arduous proofs with extreme precision.

Mathematicians often make intellectual jumps in their proofs without spelling out the exact tactics used to get from one step to the next, and provers like this could walk through the intermediate work automatically, without requiring a human mathematician to fill in each exact tactic used.

“You get the maximum of precision and correctness all really spelled out, but you don’t have to do the work of filling in the details,” says Jeremy Avigad at Carnegie Mellon University in Pennsylvania. “Maybe offloading some things that we used to do by hand frees us up for looking for new concepts and asking new questions.”

AIs like this could one day even solve maths problems we don’t know how to solve or that are too long and complicated. But that will take a much larger training set, more tactics, and a simpler way to plug the theorems into the computer. “That’s far away, but I think it could happen in our lifetime,” says Szgedy.

“Pretty much anything that you can state and try to prove mathematically, you can put into this system,” says Avigad. “You can distill just about all of mathematics down to very basic rules and assumptions, and these systems implement those rules and assumptions.”

All of this happens in a matter of seconds per proof and the only source of error is the translation of the theorem into formal language the computer can understand. Szegedy says that the team is now working on the problem of automatic translation so that it’s easier for mathematicians to interact with the system.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Leah Crane*

 

P is not NP? That is the question

One of the biggest open problems in mathematics may be solved within the next decade, according to a poll of computer scientists. A solution to the so-called P versus NP problem is worth $1 million and could have a profound effect on computing, and perhaps even the entire world.

The problem is a question about how long algorithms take to run and whether some hard mathematical problems are actually easy to solve.

P and NP both represent groups of mathematical problems, but it isn’t known if these groups are actually identical.

P, which stands for polynomial time, consists of problems that can be solved by an algorithm in a relatively short time. NP, which stands for nondeterministic polynomial time, comprises the problems that are easy to check if you have the right answer given a potential candidate, although actually finding an answer in the first place might be difficult.

NP problems include a number of important real-world tasks, such as the travelling salesman problem, which involves finding a route between a list of cities that is shorter than a certain limit. Given such a route, you can easily check if it fits the limit, but finding one might be more difficult.

Equal or not

The P versus NP problem asks whether these two collections of problems are actually the same. If they are, and P = NP, the implications are potentially world-changing, because it could become much easier to solve these tasks. If they aren’t, and P doesn’t equal NP, or P ≠ NP, a proof would still answer fundamental questions about the nature of computation.

The problem was first stated in 1971 and has since become one of the most important open questions in mathematics – anyone who can find the answer either way will receive $1 million from the Clay Mathematics Institute in Cambridge, Massachusetts.

William Gasarch, a computer scientist at the University of Maryland in College Park, conducts polls of his fellow researchers to gauge the current state of the problem. His first poll, in 2002, found that just 61 per cent of respondents thought P ≠ NP. In 2012, that rose to 83 per cent, and now in 2019 it has slightly increased to 88 per cent. Support for P = NP has also risen, however, from 9 per cent in 2002 to 12 per cent in 2019, because the 2002 poll had a large number of “don’t knows”.

Confidence that we might soon have an answer is also rising. In 2002, just 5 per cent thought the problem would be resolved in the next decade, falling to 1 per cent in 2012, but now the figure sits at 22 per cent. “This is very surprising since there has not been any progress on it,” says Gasarch. “If anything, I think that as the problem remains open longer, it seems harder.” More broadly, 66 per cent believe it will be solved before the end of the century.

There was little agreement on the kind of mathematics that would ultimately be used to solve the problem, although a number of respondents suggested that artificial intelligence, not humans, could play a significant role.

“I can see this happening to some extent, but the new idea needed will, I think, come from a human,” says Gasarch. “I hope so, not for any reason of philosophy, but just because if a computer did it we might know that (say) P ≠ NP, but not really know why.”

Neil Immerman at the University of Massachusetts Amherst thinks that this kind of polling is interesting, but ultimately can’t tell us much about the P versus NP problem.

“As this poll demonstrates, there is no consensus on how this problem will be eventually solved,” he says. “For that reason, it is hard to measure the progress we have made since 1971 when the question was first stated.”

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*Credit for article given to Jacob Aron*

Lewis Caroll was the pen name for mathematician Charles Dodgson

Curiouser and curiouser! Particle physicists could have the author of Alice’s Adventures in Wonderland to thank for simplifying their calculations.

Lewis Carroll, the 19th century children’s author, was the pen name of mathematician Charles Lutwidge Dodgson. While his mathematical contributions mostly proved unremarkable, one particular innovation may have stood the test of time.

Marcel Golz at Humboldt University, Berlin has built on Dodgson’s work to help simplify the complex equations that arise when physicists try to calculate what happens when particles interact. The hope is that it could allow for speedier and more accurate computations, allowing experimentalists at places like the Large Hadron Collider in Geneva, Switzerland to better design their experiments.

Working out the probabilities of different particle interactions is commonly done using Feynman diagrams, named after the Nobel prize winning physicist Richard Feynman. These diagrams are a handy visual aid for encoding the complex processes at play, allowing them to be converted into mathematical notation.

One early way of representing these diagrams was known as the parametric representation, which has since lost favour among physicists owing to its apparent complexity. To mathematicians, however, patterns in the resulting equations suggest that it might be possible to dramatically simplify them in ways not possible for more popular representations. These simplifications could in turn enable new insights. “A lot of this part of physics is constrained by how much you can compute” says Karen Yeats, a mathematician at the university of Waterloo, Canada.

Golz’s work makes use of the Dodgson identity, a mathematical equivalence noted by Dodgson in an 1866 paper, to perform this exact sort of simplification. While much of the connecting mathematics was done by Francis Brown, one of Golz’s tutors at Oxford University, the intellectual lineage can be traced all the way back to Lewis Carroll. “It’s kind of a nice curiosity,” says Golz. “A nice conversation starter.”

In the past, parametric notation was only useful in calculating simplified forms of quantum theory. Thanks to work like Golz’s, these simplifications could be extended to particle behaviour of real interest to experimentalists. “I can say with confidence that these parametric techniques, applied to the right problems, are game-changing,” says Brown.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Gilead Amit*

Is there an error in there somewhere?

It’s the stuff of Hollywood. Somebody somewhere is surely selling the movie rights to what’s become the biggest spat in maths: a misunderstood genius, a 500-page proof almost nobody can understand and a supporting cast squabbling over what it all means. At stake: nothing less than the future of pure mathematics.

In 2012, Shinichi Mochizuki at Kyoto University in Japan produced a proof of a long-standing problem called the ABC conjecture. Six years later the jury is still out on whether it’s correct. But in a new twist, Peter Scholze at the University of Bonn – who was awarded the Fields Medal, the highest honour in maths, in August – and Jakob Stix at Goethe University Frankfurt – who is an expert in the type of maths used by Mochizuki – claim to have found an error at the heart of Mochizuki’s proof.

Roll credits? Not so fast. The pairs’ reputation means that their claim is a serious blow for Mochizuki. And a handful of other mathematicians claim to have lost the thread of the proof at the same point Scholze and Stix say there is an error. But there is still room for dispute.

a + b = c?

The ABC conjecture was first proposed in the 1980s and concerns a fundamental property of numbers, based around the simple equation a + b = c. For a long time, mathematicians believed that the conjecture was true but nobody had ever been able to prove it.

To tackle the problem, Mochizuki had to invent a fiendish type of maths called Inter-universal Teichmüller (IUT) theory. In an effort to understand IUT better, Scholze and Stix spent a week with Mochizuki in Tokyo in March. By the end of the week, they claim to have found an error.

The alleged flaw comes in Conjecture 3.12, which many see as the crux of the proof. This section involves measuring an equivalence between different mathematical objects. In effect, Scholze and Stix claim that Mochizuki changes the length of the measuring stick in the middle of the process.

No proof

“We came to the conclusion that there is no proof,” they write in their report, which was posted online on 20 September.

But Ivan Fesenko at the University of Nottingham, UK, who says he is one of only 15 people around the world who actually understand Mochizuki’s theory, thinks Scholze and Stix are jumping the gun. “They spent much less time than all of us who have been studying this for many years,” says Fesenko.

Mochizuki has tried to help others understand his work, taking part in seminars and answering questions. Mochizuki was even the one who posted Scholze and Stix’s critical report. “We have this paradoxical situation in which the victim has published the report of the villain,” says Fesenko with a laugh. “This is an unprecedented event in mathematics.”

So is the proof wrong or just badly explained? Fesenko thinks that the six-year dispute exposes something rotten at the heart of pure mathematics. These days mathematicians work in very narrow niches, he says. “People just do not understand what the mathematician in the next office to you is doing.”

This means that mathematicians will increasingly have to accept others’ proofs without actually understanding them – something Fesenko describes as a fundamental problem for the future development of mathematics.

This suggests the story of Mochizuki’s proof may forever lack a satisfactory ending – becoming a war between mathematicians that is doomed to spiral into infinity. “My honest answer is that we will never have consensus about it,” says Fesenko.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Douglas Heaven*

 

For millennia mathematicians have struggled to unify arithmetic and geometry. Now one young genius could have brought them in sight of the ultimate prize.

IF JOEY was Chloe’s age when he was twice as old as Zoe was, how many times older will Zoe be when Chloe is twice as old as Joey is now?

Or try this one for size. Two farmers inherit a square field containing a crop planted in a circle. Without knowing the exact size of the field or crop, or the crop’s position within the field, how can they draw a single line to divide both the crop and field equally?

You’ve either fallen into a cold sweat or you’re sharpening your pencil (if you can’t wait for the answer, you can check the bottom of this page). Either way, although both problems count as “maths” – or “math” if you insist – they are clearly very different. One is arithmetic, which deals with the properties of whole numbers: 1, 2, 3 and so on as far as you can count. It cares about how many separate things there are, but not what they look like or how they behave. The other is geometry, a discipline built on ideas of continuity: of lines, shapes and other objects that can be measured, and the spatial relationships between them.

Mathematicians have long sought to build bridges between these two ancient subjects, and construct something like a “grand unified theory” of their discipline. Just recently, one brilliant young researcher might have brought them decisively closer. His radical new geometrical insights might not only unite mathematics, but also help solve one of the deepest number problems of them all: the riddle of the primes. With the biggest prizes in mathematics, the Fields medals, to be awarded this August, he is beginning to look like a shoo-in.

The ancient Greek philosopher and mathematician Aristotle once wrote, “We cannot… prove geometrical truths by arithmetic.” He left little doubt he believed geometry couldn’t help with numbers, either. It was hardly a controversial thought for the time. The geometrical proofs of Aristotle’s near-contemporary Euclid, often called the father of geometry, relied not on numbers, but logical axioms extended into proofs by drawing lines and shapes. Numbers existed on an entirely different, more abstract plane, inaccessible to geometers’ tools.

And so it largely remained until, in the 1600s, the Frenchman René Descartes used the techniques of algebra – of equation-solving and the manipulation of abstract symbols – to put Euclid’s geometry on a completely new footing. By introducing the notion that geometrical points, lines and shapes could all be described by numerical coordinates on an underlying grid, he allowed geometers to make use of arithmetic’s toolkit, and solve problems numerically.

This was a moonshot that let us, eventually, do things like send rockets into space or pinpoint positions to needle-sharp accuracy on Earth. But to a pure mathematician it is only a halfway house. A circle, for instance, can be perfectly encapsulated by an algebraic equation. But a circle drawn on graph paper, produced by plotting out the equation’s solutions, would only ever capture a fragment of that truth. Change the system of numbers you use, for example – as a pure mathematician might do – and the equation remains valid, while the drawing may no longer be helpful.

Wind forward to 1940 and another Frenchman was deeply exercised by the divide between geometry and numbers. André Weil was being held as a conscientious objector in a prison just outside Rouen, having refused to enlist in the months preceding the German occupation of France – a lucky break, as it turned out. In a letter to his wife, he wrote: “If it’s only in prison that I work so well, will I have to arrange to spend two or three months locked up every year?”

Weil hoped to find a Rosetta stone between algebra and geometry, a reference work that would allow truths in one field to be translated into the other. While behind bars, he found a fragment.

It had to do with the Riemann hypothesis, a notorious problem concerning how those most fascinating numbers, the primes, are distributed (see below). There had already been hints that the hypothesis might have geometrical parallels. Back in the 1930s, a variant had been proved for objects known as elliptic curves. Instead of trying to work out how prime numbers are distributed, says mathematician Ana Caraiani at Imperial College London, “you can relate it to asking how many points a curve has”.

Weil proved that this Riemann-hypothesis equivalent applied for a range of more complicated curves too. The wall that had stood between the two disciplines since Ancient Greek times finally seemed to be crumbling. “Weil’s proof marks the beginning of the science with the most un-Aristotelian name of arithmetic geometry,” says Michael Harris of Columbia University in New York.

The Riemann Hypothesis: The million-dollar question

The prime numbers are the atoms of the number system, integers indivisible into smaller whole numbers other than one. There are an infinite number of them and there is no discernible pattern to their appearance along the number line. But their frequency can be measured – and the Riemann hypothesis, formulated by Bernhard Riemann in 1859, predicts that this frequency follows a simple rule set out by a mathematical expression now known as the Riemann zeta function.

Since then, the validity of Riemann’s hypothesis has been demonstrated for the first 10 trillion primes, but an absolute proof has yet to emerge. As a mark of the problem’s importance, it was included in the list of seven Millennium Problems set by the Clay Mathematics Institute in New Hampshire in 2000. Any mathematician who can tame it stands to win $1 million.

In the post-war years, in the more comfortable setting of the University of Chicago, Weil tried to apply his insight to the broader riddle of the primes, without success. The torch was taken up by Alexander Grothendieck, a mathematician ranked as one of the greatest of the 20th century. In the 1960s, he redefined arithmetic geometry.

Among other innovations, Grothendieck gave the set of whole numbers what he called a “spectrum”, for short Spec(Z). The points of this undrawable geometrical entity were intimately connected to the prime numbers. If you could ever work out its overall shape, you might gain insights into the prime numbers’ distribution. You would have built a bridge between arithmetic and geometry that ran straight through the Riemann hypothesis.

The shape Grothendieck was seeking for Spec(Z) was entirely different from any geometrical form we might be familiar with: Euclid’s circles and triangles, or Descartes’s parabolas and ellipses drawn on graph paper. In a Euclidean or Cartesian plane, a point is just a dot on a flat surface, says Harris, “but a Grothendieck point is more like a way of thinking about the plane”. It encompasses all the potential uses to which a plane could be put, such as the possibility of drawing a triangle or an ellipse on its surface, or even wrapping it map-like around a sphere.

If that leaves you lost, you are in good company. Even Grothendieck didn’t manage to work out the geometry of Spec(Z), let alone solve the Riemann hypothesis. That’s where Peter Scholze enters the story.

“Even the majority of mathematicians find most of the work unintelligible”

Born in Dresden in what was then East Germany in 1987, Scholze is currently, at the age of 30, a professor at the University of Bonn. He laid the first bricks for his bridge linking arithmetic and geometry in his PhD dissertation, published in 2012 when he was 24. In it, he introduced an extension of Grothendieck-style geometry, which he termed perfectoid geometry. His construction is built on a system of numbers known as the p-adics that are intimately connected with the prime numbers (see “The p-adics: A different way of doing numbers”). The key point is that in Scholze’s perfectoid geometry, a prime number, represented by its associated p-adics, can be made to behave like a variable in an equation, allowing geometrical methods to be applied in an arithmetical setting.

It’s not easy to explain much more. Scholze’s innovation represents “one of the most difficult notions ever introduced in arithmetic geometry, which has a long tradition of difficult notions”, says Harris. Even the majority of working mathematicians find most of it unintelligible, he adds.

Be that as it may, in the past few years, Scholze and a few initiates have used the approach to solve or clarify many problems in arithmetic geometry, to great acclaim. “He’s really unique as a mathematician,” says Caraiani, who has been collaborating with him. “It’s very exciting to be a mathematician working in the same field.”

This August, the world’s mathematicians are set to gather in Rio de Janeiro, Brazil, for their latest international congress, a jamboree held every four years. A centrepiece of the event is the awarding of the Fields medals. Up to four of these awards are given each time to mathematicians under the age of 40, and this time round there is one name everyone expects to be on the list. “I suspect the only way he can escape getting a Fields medal this year is if the committee decides he’s young enough to wait another four years,” says Marcus du Sautoy at the University of Oxford.

 

Peter Scholze, 30, looks like a shoo-in for mathematics’s highest accolade this summer

With so many grand vistas opening up, the question of Spec(Z) and the Riemann hypothesis almost becomes a sideshow. But Scholze’s new methods have allowed him to study the geometry, in the sense Grothendieck pioneered, that you would see if you examined the curve Spec(Z) under a microscope around the point corresponding to a prime number p. That is still a long way from understanding the curve as a whole, or proving the Riemann hypothesis, but his work has given mathematicians hope that this distant goal might yet be reached. “Even this is a huge breakthrough,” says Caraiani.

Scholze’s perfectoid spaces have enabled bridges to be built in entirely different directions, too. A half-century ago, in 1967, the then 30-year-old Princeton mathematician Robert Langlands wrote a tentative letter to Weil outlining a grand new idea. “If you are willing to read it as pure speculation I would appreciate that,” he wrote. “If not – I am sure you have a waste basket handy.”

In his letter, Langlands suggested that two entirely distinct branches of mathematics, number theory and harmonic analysis, might be related. It contained the seeds of what became known as the Langlands program, a vastly influential series of conjectures some mathematicians have taken to calling a grand unified theory capable of linking the three core mathematical disciplines: arithmetic, geometry and analysis, a broad field that we encounter in school in the form of calculus. Hundreds of mathematicians around the world, including Scholze, are committed to its completion.

The full slate of Langlands conjectures is no more likely than the original Riemann hypothesis to be proved soon. But spectacular discoveries could lie in store: Fermat’s last theorem, which took 350 years to prove before the British mathematician Andrew Wiles finally did so in 1994, represents just one particular consequence of its conjectures. Recently, the French mathematician Laurent Fargues proposed a way to build on Scholze’s work to understand aspects of the Langlands program concerned with p-adics. It is rumoured that a partial solution could appear in time for the Rio meeting.

In March, Langlands won the other great mathematical award, the Abel prize, for his lifetime’s work. “It took a long time for the importance of Langlands’s ideas to be recognised,” says Caraiani, “and they were overdue for a major award.” Scholze seems unlikely to have to wait so long.

The p-adics: A different way of doing numbers

Key to the latest work in unifying arithmetic and geometry are p-adic numbers.

These are an alternative way of representing numbers in terms of any given prime number p. To make a p-adic number from any positive integer, for example, you write that number in base p, and reverse it. So to write 20 in 2-adic form, say, you take its binary, or base-2, representation – 10100 – and write it backwards, 00101. Similarly 20’s 3-adic equivalent is 202, and as a 4-adic it is written 011.

The rules for manipulating p-adics are a little different, too. Most notably, numbers become closer as their difference grows more divisible by whatever p is. In the 5-adic numbers, for example, the equivalents of 11 and 36 are very close because their difference is divisible by 5, whereas the equivalents of 10 and 11 are further apart.

For decades after their invention in the 1890s, the p-adics were just a pretty mathematical toy: fun to play with, but of no practical use. But in 1920, the German mathematician Helmut Hasse came across the concept in a pamphlet in a second-hand bookshop, and became fascinated. He realised that the p-adics provided a way of harnessing the unfactorisability of the primes – the fact they can’t be divided by other numbers – that turned into a shortcut to solving complicated proofs.

Since then, p-adics have played a pivotal part in the branch of maths called number theory. When Andrew Wiles proved Fermat’s infamous last theorem (that the equation xⁿ + yⁿ = zⁿ has no solutions when x, y and z are positive integers and n is an integer greater than 2) in the early 1990s, practically every step in the proof involved p-adic numbers.

• Answers: Zoe will be three times as old as she is now. The farmers should draw a line across the field that connects the centre points of the field and the crop.

This article appeared in print under the headline “The shape of numbers”

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*Credit for article given to Gilead Amit*