We have before talked about math teaching in schools here (Why math is so difficult?). At that post a fact about Iranian teachers’ habits had been explained. I myself have had this experience that how much inspiration and motivation is important. But what I have experienced in Iran schools: you have got a good situation or you haven’t got a good situation. If your total situation (family background of academic education, family income and etc.) is good enough to support you, then educational system will work for you. If you haven’t got a good situation then all doors are closed to you. nobody will help you. Read the whole story, especially when he says: The 77 year-old professor who has been at Utica for nearly four decades said there’s no way to make a student like mathematics. Put simply, those that love the discipline will do best. He used crochet as an example.
Mathematics should be fun, according to Hossein Behforooz. Students shouldn’t be scared. If it’s not fun, what’s the point? “In more than half of every class we have, there is this kind of phobia. People think that math is difficult, that math is hard,” explained the mathematics professor at Utica College in New York. “Math is a way to prove something and to make it fun, not to scare students.”
Unlike some ICM colleagues, the Iranian doesn’t agree that it’s the teacher’s role to inspire or motivate students with their personal passion. The 77 year-old professor who has been at Utica for nearly four decades said there’s no way to make a student like mathematics. Put simply, those that love the discipline will do best. He used crochet as an example. “If you do not like it, you cannot make it. But when you like it, every knot is beautiful,” he said. “It’s art, not science.”
But, Behforooz’s chosen subject area fascinates both mathematicians and non-mathematicians. A large part of the scholarship he produced is in an area called magic squares, containing grids with special arrangements of numbers in them. Magic squares earned their name in ancient times, due to associations with magic and the supernatural. The earliest record of a magic square is from China (circa 2200 BC), when Emperor Yu saw a magic square in the shell of a divine tortoise. In the West, magic squares were first used in the work of Theon of Smyrna, and by Arab astrologers in 8 AD. They were later seen in the writings of Greek mathematician Moschopoulos in the 14th century.
Magic squares retain their popularality as tools to help students solve and practice addition problems. Their arrangements are special because every diagonal, row and column add up to the same number. Behforooz said Brazil is clearly on the right path. “I heard that there are more than 500 Math Olympiad medallists here,” he said. “That is great, that is fantastic.”
Hosting the International Congress of Mathematicians in the southern hemisphere will reinforce positive ideas around math. It gives the message that mathematics is both accessible and fun for any student, anywhere in the world, Behforooz said. He hopes to meet an younger cohort of mathematicians at the next ICM in Russia.
“This is my ninth time to ICM. I hope I will be alive to go to the tenth one in Russia!” he laughed. “It’s very prestigious here [in Rio]. You had the Olympics, and you had the World Cup. The IMC was another chance to show that high achievement isn’t not just for the upper hemisphere.”
Only at France, a mathematician could be a politician too: cederic vilani, the French mathematician who wins the 2010 fileds medal, Has Become a Crucial Political Figure in France, as https://www.bloomberg.com reported.
Read the complete article:
To hear Cedric Villani tell it, the French are better than everyone else at love, wine — and math.
A winner of the Fields Medal — the Nobel Prize equivalent for mathematics — Villani has in less than a year risen to become a key political figure in France with the ear of the tech-savvy President Emmanuel Macron. On Thursday, Villani takes center-stage when he unveils the country’s Artificial Intelligence strategy, aimed at putting his claim of France’s mathematical superiority to work in the global battle for emerging disruptive technologies.
“There is a deficit of contact between science and politics,” the 44-year-old said in an interview. “It’s part of my job to reinforce that link. It will be France’s role to lead the rest of Europe.”
Villani is an unlikely warrior in Europe’s AI battle, trying to take on China and the U.S. that are leaps ahead. The skinny scientist and lawmaker with his penchant for Gothic suits, giant frilly bow-ties favored in the late 19th century and bespoke spider-shaped brooches often draws more attention for the way he looks than for what he has to say.
Yet Macron is relying on Villani to help his modernization push by being one of the new — more optimistic — faces of France, a role the scientist has embraced with gusto. His 150-page AI report comes on top of the work he’s done on crafting a new and better way of teaching math in the country and as he prepares his next project that will involve reviewing France’s pedagogical techniques and reflects on data privacy.
Next month he’ll travel with Macron to the U.S., after having visited China with him in January. He was among the key speakers at a pre-Davos gig organized by Macron at the Versailles Palace in January to show foreign investors, including Google Chief Executive Officer Sundar Pichai, that science was now at the core of France’s ambitions.
“Villani brings France’s policies up to a whole new level of knowledge and thinking, and it seems fair that he is given even more help to do his scaling up,” said Andre Loesekrug-Pietri, an investor who launched the Joint European Disruption Initiative with major European scientific figures to accelerate investment in fundamental research. He and Villani studied together at the top Paris school of Louis-Le-Grand.
Ever since Villani won the Fields medal in 2010, the soft-spoken math whiz has endeavored to make math a part of the conversation in France and to bring more science to politics. Mathematics has taken him from Paris’s prestigious Ecole Normale Superieure, to stints at Berkeley University and Princeton University and to the helm of the French capital’s Institut Henri Poincare, the world-renowned mathematical center.
His mathematics research fellow, Giuseppe Toscani of the University of Pavia in Italy, recalls Villani’s phenomenal ability to synthesize everything. The two men published research together in the late 1990s.
“He has the (almost unique) characteristic to be the best at anything he takes on,” Toscani said in a written response to questions. “Mathematics is one, among others. From that point of view, I am sure he will make important contributions in his new political life.”
Villani is part of Macron’s effort to change France’s political landscape, drawing into parliament people who are not professional politicians. The scientist has attempted to be more than just a new face. A fan of Marvel Comics’s Amazing Fantasy, Villani abides by the superhero’s mantra that “with great power comes great responsibility.”
France doesn’t have a Science Advisory Committee like in the U.S. The French prime minister is supposed to have a similar body, but Villani notes, “it hasn’t been used in a long, long time.”
Villani, who carries a pocket watch at all times and a giant, full and often half-open backpack, is a busy man. His aides talk about their boss’s extreme multi-tasking: he writes with one hand, types with the other all while speaking on the phone.
“I must do everything at the same time, that’s the difficulty,” Villani said.
The scientist is also contributing to a much-debated government plan to revise the constitution, which has taken him into uncharted and controversial waters. For the most part, though, he’s sticking to his real passion — making France the place to be for math and science.
In a June 2016 TED Talk about why his field of study is “so sexy,” Villani joked about French people’s reputation and added more seriously that Paris has more mathematicians than any other city in the world.
“What is it that French people do better than all the others? If you take a poll, the top three answers might be: love, wine and whining… Maybe. But let me suggest a fourth one: mathematics.”
You know, when I was a middle school student, our math teachers at Iran were mostly like horror movie characters: they were bad-tempered; they gave us hard problems. They were very sensitive about the homework, the cleanness of the book, and the absolute silence all during the class. Whenever we addressed to go to the board, we were supposed to know the answer of any given problem. If any other cases, unless the standards of the class, happened the punishment was very dreadful. So much stress and physical punishments. However, generally teachers were also very talented and smart, motivated and knowledgeable. The common thing that students tried to do was being silent and organized to try stay far from the punishment time. In addition, there were few math geniuses at class. The teacher, up to the end of the 2nd session, discovered them. In addition, from the third session, it was like this: everybody was silent writing what was being written on the board by teacher or those 3 – 4 geniuses. Two separated part in the class, the genius and teacher part and the stupid part (as teachers call those students). So, yes! I was in the light part of the class. I was so much better than anybody else in the math class was. Nevertheless, it is not the complete story. I started math major at high school and I was lucky enough to face with some inspiring teachers whom were kind and very patients with students. Then I changed very fast. I have had this experience in my all teaches from the first time of teaching so far: the problem is not to be dumb! The problem is not to be successful in making communication with mathematics! Math is not a bunch of pure contents, but also it is a language, it is a way of thinking and it is like three-dimensional glass in a world that people can see only two-dimensional. Anyway, I have always tried to inspire my students and to teach them the logic of the math. This is the solution. In continue, you are about to read a post from beetleypete blog which is an honest confess. We teach same book at the same rate by using same words and to different students. Different people have got different educational backgrounds, different emotional experience about math and they have different rates in learning. So all together I am coming to this conclusion that it is our duty as math teachers to teach students how to learn math, how to understand the language and the logic of the math. I think you may feel familiar with this memory:
I have recently posted about the study of both History and Geography, so though I would continue that theme with something I was not at all good at, Maths. Short for Mathematics, and simply called ‘Math’ in the USA, most of us in Britain know this school subject as ‘Maths’.
When I started school at the age of five, I was taught simple counting. Using blocks, toys, or any other accessory, I soon learned how to count up to ten and more, along with my classmates. Then easy addition, nothing too complex for my developing mind. By the time I went to Junior School, aged seven, rote learning was still popular, and we were soon getting to grips with our ‘times tables’, to form the foundations of simple multiplication. This was 1959 of course, so no calculators, and not a thought of the computers to come. Just a teacher writing numbers on a board, and conducting our recital like a band leader.
“Once five is five.
Two fives are ten.
Three fives are fifteen,
Four fives are twenty”.
And so on.
We went as far as the number thirteen, stopping there for reasons best known to the teacher. Division was also introduced, often helped along by the use of counters or visual aids, as I learned that four into twenty makes five. Then around the age of nine, that ‘Eureka’ moment, when I suddenly got the connection between multiplication and division. We also tackled currency, as at that time we still used pounds, shillings and pence, with twelve pence to a shilling, and twenty shillings in a pound. Not that I ever had much cash, but it was good to know what change to expect when I bought something. We were also using rulers, and learning how to measure short distances.
When I was eleven, it was time to go to secondary school, and begin the exam syllabus. I had a list of things I would need just for Maths lessons; this included a set of compasses, a protractor, a triangle and a ‘proper’ ruler, with measurements down to 1/16th of an inch. The first real lesson was a double period, (why was Maths always a double?) and it hit me like a whirlwind. Algebra? Geometry? Even something called Trigonometry. I thought the teacher must be talking a foreign language, but she assured us that was all to come. Meanwhile, we were hit with some serious long division. That alone was enough to make my brain ache, and I watched my ‘working out’ get further and further down the page as I struggled with something like 295 divided by 16. By the time the first month of the new school was over, I had decided that I really didn’t like Maths, and was sure I would never be good at it.
And I was right.
Then came ‘Problems’. Things like, “If a two hundred gallon water tank has a leak of a quarter of a pint a day for ten days, then half a pint a day for twelve days, how much water will be left after twenty-two days?” I didn’t even know where to start, and my hand was soon up, informing the teacher that I didn’t have a clue. Even when she showed me how to work out the solution, I still got the answer wrong. It all got worse once we started with Algebra. “If X = ? and Y = ?, what is XY squared? ” I just laughed. There was no chance I got any of that at all. The teacher later explained that X and Y had a value and it could be anything I wanted on that occasion. X could be 2 and Y 6, for example. My reply was not well-received. “Please Miss, then why don’t you just write a 2 and 6?” I was told in no uncertain terms that I was being deliberately ‘stupid’.
But I wasn’t.
Later, we were given a complex book of numbers, called ‘Logarithms’. This baffling table introduced us to decimal points and such, but might just as well have been Sanskrit, for all my brain could take it in. I wasn’t getting any better, and had to face the next year, when it was all going to get harder. Double Maths changed to a Monday morning when I was twelve, and I began to dread the walk to school,, shuffling with the reluctance of a condemned man about to be hanged. I still had the same teacher, the formidable Mrs Widdowson, who could freeze me with one of her signature glares, and had given me a terrible entry on my end of term report the previous year. Inside, I considered I was doing alright. All the other subjects were going great. I was in the top set for English, Geography, French, History, and even Religious Education, something I had little interest in. So what if I didn’t really ‘get’ Maths? It wasn’t the end of the world, as far as I was concerned.
So, I muddled along. Bad reports, bottom section of the class, and never truly understanding anything new. I did well at everything except Maths, and that was enough for me. When it came to the final exams, I just scraped though the Maths one with a Grade Four, a ‘just passed’ result. But it wasn’t all bad. That early learning left me able to recall the times table instantly, work out money without hesitation, and even able to calculate foreign currency exchanges, on my trips abroad. These days, i see young peope reach for a mobile phone, when faced with the most basic sum to work out.
Maybe we need to go back to chanting the times tables, and using a ruler?
Mathematicians Caucher Birkar, Alessio Figalli, Peter Scholze, and Akshay Venkatesh were awarded Fields Medals at the 2018 International Congress of Mathematicians in Rio de Janeiro, Brazil. Read more about the mathematicians and their work.
Mathematicians Caucher Birkar, Alessio Figalli, Peter Scholze, and Akshay Venkatesh were awarded Fields Medals at the 2018 International Congress of Mathematicians in Rio de Janeiro, Brazil. Read more about the mathematicians and their work.
I just watched this video on YouTube and I really enjoyed the way he is explaining the solution step by step. More than the video by itself, the another thing that I really appreciate is how mathematicians and mathematics teachers use YouTube to communicate with a more vast world. actually mathematicians are among pioneers of bringing the new technologies into the class, although they can easily use the older methods and technologies. I hope you too enjoy the video.
“Researchers from Germany, India, Iran and Italy take home the 2018 Fields Medal”
the whole story:
“Four notable and promising researchers from four different countries – Germany, India, Iran, and Italy – are the winners of the most important international award in mathematics, the Fields Medal. Delivered for the first time in 1936, the medal is recognition for works of excellence and an incentive for new outstanding achievements.
Awarded every four years at the world’s largest mathematics event – the International Congress of Mathematicians (ICM) – the medal will be given this year to Peter Scholze, Akshay Venkatesh, Caucher Birkar, and Alessi Fegalli at ICM’s opening ceremony on August 1st, at Riocentro.
Founded by the Canadian mathematician John Charles Fields to celebrate outstanding achievements, the Fields Medal has already been awarded to 56 scholars of the most diverse nationalities, among them, Brazilian Fields laureate Artur Avila, an extraordinary researcher from IMPA, awarded in 2014 in South Korea. Due to its importance and prestige, the medal is often likened to a Nobel Prize of Mathematics.
The winners of the Fields medal are selected by a group of renowned specialists nominated by the Executive Committee of the International Mathematical Union (IMU), which organize the ICMs. Every four years, between two and four researchers under the age of 40 are chosen. Since 2006, a cash prize of 15 thousand Canadian dollars accompanies the medal.
Meet the winners of the Fields Medal 2018:
Conquering the greatest honor among the world’s mathematicians before the age of 40 is a notable accomplishment, although the life of Akshay Venkatesh is already marked with precocious feats. Born in New Delhi, India in 1981, and raised in Australia, at age 12 he became a medalist at the International Mathematical Olympiad. From there, he dived into world of mathematics, starting a promising career. When he began his bachelor’s degree in Mathematics and Physics at the University of Western Australia, he was a 13-year-old boy.
At 20, Venkatesh finished his PhD at Princeton University and soon became an instructor at C.L.E. Moore, at the Massachusetts Institute of Technology (MIT), a prestigious position offered to recent graduates in the area of Pure Mathematics, previously occupied by prominent figures such as the American John Nash (1928-1915). Upon leaving in 2004, he became a Clay Research Fellow and was appointed associate professor at the Courant Institute of Mathematical Sciences at New York University.
He became a professor at Stanford University at the age of 27, and as of this year is a faculty member at the Institute for Advanced Study (IAS).
Venkatesh has his feet in Number Theory – an area that deals with abstract issues and had no known application until the arrival of cryptography in the late 1970s – but roves with ease through related topics, such as Theory of Representation, Ergodic Theory, and Automorphic Forms. Armed with a meticulous, investigative and creative approach to research, detecting impressive connections between diverse areas, his contributions have been fundamental to several fields of research in Mathematics. It is no wonder that his work has been recognized by several distinguished awards such as Ostrowisk (2017), Infosys (2016), SASTRA Ramanujan (2008) and Salem (2007).
Previously a guest speaker at the 2010 ICM, Venkatesh has been invited back to speak in Rio this August.
Born in Naples, Italy on April 2, 1984, Alessio Figalli belatedly discovered an interest in science. Until high school, his only concern was playing football. The training for the International Mathematical Olympiad (IMO) awakened his interest in the subject and, upon joining the Scuola Normale Superiore di Pisa, chose Mathematics.
Figalli completed his PhD in 2007 at the École Normale Supérieure de Lyon in France, with the guidance of Fields Medal laureate Cédric Villani. He has worked at the French National Center for Scientific Research, École Polytechnique, the University of Texas in the USA and ETH Zürich in Switzerland. A specialist in calculating variations and partial differential equations, he was invited to speak at the 2014 ICM in Seoul. He has won several awards, including: Peccot-Vimont (2011), EMS (2012), Cours Peccot (2012), Stampacchia Medal (2015) and Feltrinelli (2017).
Caucher Birkar’s dedication to the winding and multidimensional world of algebraic geometry, with its ellipses, lemniscates, Cassini ovals, among so many other forms defined by equations, granted him the Philip Leverhulme prize in 2010 for exceptional scholars whose greatest achievement is yet to come. Given the substantial contributions of Birkar to the field, that prize was a prophecy: after eight years, the Cambridge University researcher joins the select group of Fields Medal winners at the age of 40.
Birkar, who just this year received recognition for his work as one of the London Mathematical Society Prize winners, was born in 1978 in Marivan, a Kurdish province in Iran bordering Iraq with about 200,000 inhabitants. His curiosity was awakened by algebraic geometry, the same interest that, in that same region, centuries earlier, had attracted the attention of Omar Khayyam (1048-1131) and Sharaf al-Din al-Tusi (1135-1213).
After graduating in Mathematics from Tehran University, Birkar went to live in the United Kingdom, where he became a British citizen. In 2004, he completed his PhD at the University of Nottingham with the thesis “Topics in modern algebraic geometry”. Throughout his trajectory, birational geometry has stood out as his main area of interest. He has devoted himself to the fundamental aspects of key problems in modern mathematics – such as minimal models, Fano varieties, and singularities. His theories have solved long-standing conjectures.
In 2010, the year in which he was awarded by the Foundation Sciences Mathématiques de Paris, Birkar wrote, alongside Paolo Cascini (Imperial College London), Christopher Hacon (University of Utah) and James McKernan (University of California, San Diego), an article called “Existence of minimal models for varieties of general log type” that revolutionized the field. The article earned the quartet the AMS Moore Prize in 2016.
Peter Scholze was born in Dresden, Germany on December 11, 1987. Only 30 years old, he is already considered by the scientific community as one of the most influential mathematicians in the world.
In 2012, at age 24, he became a full professor at the University of Bonn, Germany. Scholze impresses his colleagues with the intellectual ability he has shown since was a teenager, when he won four medals – three gold and one silver – at the International Mathematical Olympiad (IMO).
The mathematician completed his university graduate and masters in record time – five semesters – and gained notoriety at the age of 22, when he simplified a complex mathematical proof of numbers theory from 288 to 37 pages.
A specialist in arithmetic algebraic geometry, he stands out for his ability to understand the nature of mathematical phenomena and to simplify them during presentations.
At age 16, still a student at the Heinrich-Hertz-Gymnasium – a school with a strong scientific focus – Scholze decided to study Andrew Wiles’ solution to Fermat’s Last Theorem. Faced with the complexity of the result, he realized that he was on the right track in choosing Mathematics as a profession.
He was a guest speaker at ICM 2014 in Seoul, South Korea, and will be a plenary member this year at the Rio de Janeiro Congress.
Scholze has been repeatedly recognized for his contributions to arithmetic algebraic geometry. He collects major mathematics awards, such as EMS (2016), Leibniz (2016), Fermat (2015), Ostrowski (2015), Cole (2015), Clay Research 2014), SASTRA Ramanujan (2013), Prix and Cours Peccot (2012) and, finally, the Fields Medal (2018).”
You can also find some more information from the 1st issue of the Guardian:
Former refugee among winners of Fields medal – the ‘Nobel prize for maths’
An Kurdish man who came to Britain as a refugee after fleeing conflict two decades ago is one of four men who have been awarded the Fields medal, considered the equivalent of a Nobel prize for mathematics.
The winners of the prize, presented at the International Congress of the International Mathematical Union in Rio de Janeiro, have been announced as Prof Caucher Birkar, 40, from Cambridge University, Prof Akshay Venkatesh, 36, an Australian based at Princeton and Stanford in the US, Prof Alessio Figalli, 34, from ETH in Zurich and Prof Peter Scholze, 30, from Bonn University.
The Fields medal is perhaps the most famous mathematical award. It was first awarded in 1936 and since 1950 has been presented every four years to up to four mathematicians who are under 40. As well as the medal, each recipient receives prize money of 15,000 Canadian dollars (£8,750). With all the prizes this year going to men, the late Maryam Mirzakhani remains the only woman to have received the accolade.
Birkar was born in Marivan in Iran – a Kurdish city heavily affected by the Iran-Iraq war of the 1980s – and studied mathematics at the University of Tehran before coming to the UK in 2000. After a year, he was granted refugee status, became a British citizen and began a PhD.
“When I was in school it was a chaotic period, there was the war between Iran and Iraq and the economic situation was pretty bad,” said Birkar. “My parents are farmers, so I spent a huge amount of time actually doing farming. In many ways it was not the ideal place for a kid to get interested in something like mathematics.”
Birkar says it was his brother who at an early age introduced him to more advanced mathematical techniques.
Prof Ivan Fesenko of the University of Nottingham, one of Birkar’s PhD supervisors, told the Guardian how Birkar, who initially spoke very little English, came to study with him.
“The Home Office sent him to live in Nottingham while they were processing his application for asylum status,” said Fesenko. “He came to me because he was interested in research work related to my general areas.”
Birkar’s talent, says Fesenko, quickly became apparent as he began his PhD. “I thought I should give him some problem – if he solves it, then this will be his PhD. Typically a PhD lasts three or four years. I gave him a problem and he solved it in three months,” said Fesenko.
“He is very, very smart; you start to talk with him and you recognise that he can read your thoughts several steps ahead. But he never uses this to his advantage, he is very, very respectful and he gently helps people to develop further,” said Fesenko.
As with many of the winners of the Fields medal, Birkar’s research is in areas of mathematics that can seem incomprehensible to a lay audience. His citation for the award says he won the medal “for his proof of the boundedness of fano varieties and for contributions to the minimal model program.”
Prof Paolo Cascini of Imperial College London has worked with Birkar. He said that in simple terms Birkar’s work focused on classifying geometrical shapes and describing their building blocks.
Birkar said he hoped the news may “put a little smile on the lips” of the world’s 40 million Kurds.
Among his achievements, Scholze invented the theory of perfectoid spaces – which are noted in his citation for the Fields medal, and have been described as a class of fractal structures allowing problems to be moved from one number system to another, making them easier to solve.
“Geometry is the study of space and shape,” said Kevin Buzzard of Imperial College London. “One technique that geometers have introduced is the idea of studying a complicated space by mapping a simpler space onto it. For example, a line is a simpler object than a circle. But if you imagine wrapping a line up into a spring shape and compressing the spring, you have found a way of mapping a line into a circle. Geometers might use this technique to analyse questions about circles, by turning them into perhaps more complex questions about lines.”
Perfectoid spaces, he says, turns this logic on its head. “The counterintuitive idea introduced by Scholze is that to study a geometric object, you might instead want to find a mapping from a space which is so grotesque and twisted that in some sense it cannot be twisted up any more. The result is that instead of ending up having to solve complicated questions about simple objects, you have to solve simple questions about extremely complicated objects.”
The Italian winner, Figalli, works in the field of optimal transport, which has its roots in the research of 18th-century mathematician Gaspard Monge, who studied where to send material dug from the ground for use in construction so that the transport costs are as low as possible.
Venkatesh becomes only the second Australian to win the prestigious medal, after Terence Tao in 2006.Venkatesh was recognised for his use of dynamics theory, which studies the equations of moving objects to solve problems in number theory, which is the study of whole numbers, integers and prime numbers.
Venkatesh grew up in Perth and at age 13 became the youngest person to study at the University of Western Australia. He earned first class honours in pure mathematics aged 16 before studying at Princeton.
At UWA, he went straight into second-year maths courses after he proved he could write the exam papers for all the first year subjects he had never taken.
His work also uses representation theory, which represents abstract algebra in terms of more easily-understood linear algebra, and topology theory, which studies the properties of structures that are deformed through stretching or twisting, like a Mobius strip.
Receiving his award on Wednesday, he said: “A lot of the time when you do math, you’re stuck, but at the same time there are all these moments where you feel privileged that you get to work with it.
“You have this sensation of transcendence, you feel like you’ve been part of something really meaningful.”
One of his early mentors, Prof Cheryl Praeger, who has known Venkatesh since he was 12, and supervised his honours thesis when he was 15, said he was always “extraordinary”.
“At our first meeting, I was speaking with Akshay’s mother Svetha, while Akshay was sitting at a table in my office reading my blackboard which contained fragments from a supervision of one of my PhD students.
“At Akshay’s request I explained what the problem was. He coped with quite a lot of detail and I found that he could easily grasp the essence of the research.”