
Pages

Categories
 algebra
 applications of maths
 Basic Set Theory and Logic
 calculus
 careers in mathematics
 Cnennai Math Institute Entrance Exam
 coordinate geometry
 combinatorics or permutations and combinations
 Complex Numbers
 Fun with Mathematics
 geometry
 IITJEE Advanced
 IITJEE Advanced Mathematics
 IITJEE Foundation Math IITJEE Main and Advanced Math and RMO/INMO of (TIFR and Homibhabha)
 IITJEE Foundation mathematics
 IITJEE Mains
 Inequalities
 Information about IITJEE Examinations
 INMO
 ISI Kolkatta Entrance Exam
 KVPY
 mathematicians
 memory power concentration retention
 miscellaneous
 motivational stuff
 physicisrs
 PreRMO
 probability theory
 pure mathematics
 RMO
 RMO Number Theory
 Statistics
 time management
 Trigonometry

Archives
 March 2019
 February 2019
 January 2019
 November 2018
 October 2018
 September 2018
 August 2018
 July 2018
 June 2018
 May 2018
 April 2018
 March 2018
 February 2018
 January 2018
 December 2017
 November 2017
 October 2017
 September 2017
 August 2017
 July 2017
 June 2017
 May 2017
 April 2017
 March 2017
 February 2017
 January 2017
 November 2016
 October 2016
 September 2016
 August 2016
 July 2016
 June 2016
 May 2016
 April 2016
 March 2016
 February 2016
 January 2016
 December 2015
 November 2015
 October 2015
 September 2015
 August 2015
 July 2015
 June 2015
 May 2015
 April 2015
 March 2015
 February 2015
 January 2015
 December 2014
 November 2014
 October 2014
 September 2014
 August 2014
 July 2014
 June 2014
Category Archives: mathematicians
Women in Science 2019 UNESCO Awards to Ingrid Daubechies and Claire Voisin
Prof Ali Nesin, Eeelavati Prize 2018, and his Nesin Mathematics Village in Turkey
The only way I can pay an honour to an outstanding generous mathematician, Prof Ali Nesin, is to share information about him and his Mathematics Village is to share information collected from the internet:
http://www.nesinkoyleri.org/eng/
https://www.ams.org/notices/201506/rnotip652.pdf
https://en.wikipedia.org/wiki/Nesin_Mathematics_Village
https://interestingengineering.com/nesinmathematicsvillagelearnenjoymath
http://mathematicsineurope.eu/?p=1568
http://www.hurriyetdailynews.com/abriefintroductiontoturkeysmathematicsvillage70193
I can’t help myself noticing similarities between Prof Anand Kumar of India’s Super 30 http://www.super30.org and Prof Ali Nesin 🙂
Regards,
Nalin Pithwa
Maths and the Bomb: Sir Michael Atiyah at 80
Just paying yet another tribute to Sir Michael Atiyah (resharing one of the articles I have collected about him):
*************************************************************************************88
Maths and the bomb Sir Michael Atiyah at 80
The Times
April 21 2009
When, five years ago, he shared the £480,000 Abel Prize, the equivalent of a Nobel prize in the world of mathematics, Sir Michael Atiyah might have listened to his wife’s urgings to put his feet up and settle into a comfortable life. But that would not have been his style. “Some mathematicians retire,” he concedes with a smile. “I don’t think I have.”
This week, Sir Michael’s 80th birthday and a life dedicated to science and political activism is celebrated in a series of events. A threeday conference celebrating his contribution to geometry and physics, at the University of Edinburgh Informatics Forum, ends today, his birthday. Tomorrow and on Friday his Sir Michael’s role in promoting disarmament is recognised with readings and lectures dedicated to exposing the folly of nuclear weapons.
Much has been achieved at an age when contemporaries might have settled for a quiet life. In 1995, as president of the Royal Society and aged 67, Sir Michael made a stinging attack on Britain’s nuclear weapons policy.
Subsequently he accepted the presidency of the influential Pugwash disarmament conferences, which unite scientists in opposition to the arms race.
He still believes passionately in the cause, which, he says, is more important to the world than maths, “because if we blow ourselves up, there will be no mathematics anyway”.
Sir Michael discovered his aptitude for mathematics during his boyhood in the Sudan. His Lebanese father was an Oxford graduate and a civil servant, his mother was Scottish and he grew up regarding himself as British, studying at Manchester Grammar School and Cambridge University.
The key professional encounters in his life came in the United States in the 1950s, when he joined the Institute for Advanced Study, at Princeton University, a gathering place for the world’s most brilliant mathematical minds. Here he forged relationships which have endured, and much of his greatest work has come from what he calls the “dialogues of ideas” established there.
His greatest achievement has been the AtiyahSinger theorem, which secured his fame and prize money, shared with his collaborator, Isadore Singer, of the US. At the time, he said he couldn’t think what to do with his share; the sporty red Lexus parked outside the Informatics building suggests he has since given it more thought.
In simple terms, the theorem provided a kind of analytical bridge which could be shifted between disciplines. “The theorem technique enables you to get to an answer bypassing all the intervening calculations,” he says. The idea “was something where you could calculate numbers of solutions by very indirect methods which applied in a very wide range of situations: geometry, algebra, physics…”
Maths, he says, is something he plays out in his mind as he walks around his flat and his garden, and he jots things down – “the dull stuff” – only when he has to check something.
“Walking helps the physiological process. You have to maintain a very high pitch of concentration when you do mathematics. It’s illumination – shining the mind’s eye on a problem and really seeing through it.
“The old clichés about the beauty of maths are true. It has beauty within it, but not all parts are equally beautiful. Beauty in mathematics is the thing that helps you in the search for truth.”
Some people, he believes, are born with mathematical brains, although they might choose other careers. One former student won the Nobel Prize for Economics, another is the bestpaid hedge fund manager in the US. So was Sir Michael never tempted to use his mathematical skill in a wider world? Could he have solved the global financial crisis?
“Economics is a combination of gambling, psychology and who knows what,” he says. “The current crisis? I think people made a bloody mess. You can foretell that the bubble will burst – the question is when. If you gambled on it you might win or lose a lot of money. I just didn’t gamble.”
**************************************************************************************
Regards,
Nalin Pithwa.
Mathematics versus Physics
The object of pure Physics is the unfolding of the laws of the intelligible world; the object of pure Mathematic that of unfolding the laws of human intelligence. — J. J. Sylvester.
In my opinion, for example, Boole’s Laws of (Human) Thought.
An observation about Sir Isaac Newton
Newton’s patience was limitless. Truth, he said much later, was the offspring of silence and meditation. And, he said, I keep the subject constantly before me and wait till the first dawnings open slowly, by little and little into a full and clear light.
A leaf out of Paul Erdos’ biography: My Brain is Open: by Bruce Schechter
Reference:
My Brain is Open: The Mathematical Journeys of Paul Erdos by Bruce Schechter, a TouchStone Book, Published by Simon and Schuster, New York.
Amazon India link:
Chapter One: Traveling.
The call might come at midnight or an hour before dawn — mathematicians are oddly unable to handle the arithmetic of time zones. Typically, a thickly accented voice on the other end of the line would abruptly begin:
“I am calling from Berlin. I want to speak to Erdos.”
“He’s not here, yet.”
“Where is he?”
“I don’t know.”
“Why don’t you know!” Click!
Neither are mathematicians always observant of social graces.
For more than sixty years mathematicians around the world have been roused from their abstract dreams by such calls, the first of the many disruptions that constituted a visit from Paul Erdos. The frequency of the calls would increase over the next several days and would culminate with a summons to the airport, where Erdos himself would appear, a short, frail man in a shapeless old suit, clutching two small suitcases that contained all of his worldly possessions. Stepping off the plane he would announce to the welcoming group of mathematicians, “My brain is open!”
Paul Erdos’ brain, when open, was one of the wonders of the world, an Ali Baba’s cave, glittering with mathematical treasures, gems of the most intricate cut and surpassing beauty. Unlike Ali Baba’s cave, which was hidden behind a huge stone in a remote desert, Erdos and his brain were in perpetual motion. He moved between mathematical meetings, universities, and corporate think tanks, logging hundreds of thousands of miles. “Another roof, another proof,” as he liked to say. “Want to meet Erdos?” mathematicians would ask. “Just stay here and wait. He’ll show up.” Along the way, in borrowed offices, guest bedrooms, and airplane cabins, Erdos wrote in excess of 1600 papers, books and articles, more than any other mathematician who ever lived. Among them are some of the great classics of the twentieth century, papers that opened up entire new fields and became the obsession and inspiration of generations of mathematicians.
The meaning of life, Erdos often said, was to prove and conjecture. Proof and conjecture are the tools with which mathematicians explore the Platonic universe of pure form, a universe that to many of them is as real as the universe in which they must reluctantly make their homes and livings, and far more beautiful. “If numbers aren’t beautiful, I don’t know what is,” Erdos frequently remarked. And although, like all mathematicians, he was forced to make his home in the temporal world, he rejected worldly encumbrances. He had no place on earth called home, nothing resembling a conventional yearround, ninetofive job, and no family in the usual sense of the word. He arranged his life with only one purpose, to spend areas many hours a day as possible engaged in the essential, lifeaffirming business of proof and conjecture.
For Erdos, the mathematics that consumed most of his waking hours was not a solitary pursuit but a social activity, a movable feast. One of the greatest mathematical discoveries of the twentieth century was the simple equation that two heads are better than one. Ever since Archimedes traced his circles in sand, mathematicians, for the most part, have laboured alone — that is, until some forgotten soul realized that mathematics could be done anywhere. Only paper and pencil were needed, and those were not strictly essential. A tablecloth would do in a pinch, or the mathematician could carry his equations in his head, like a chessmaster playing blindfolded. Strong coffee, and in Erdos’ case even more powerful stimulants, helped too. Mathematicians began to frequent the coffeehouses of Budapest, Prague, and Paris, which led to the quip often attributed to Erdos:”A mathematician is a machine for turning coffee into theorems.” Increasingly, mathematical papers became the work of two, three, or more collaborators. That radical transformation of how mathematics is created is the result of many factors, not the least of which was the infectious example set by Erdos.
Erdos had more collaborators than most people have acquaintances. He wrote papers with more than 450 collaborators — the exact number is still not known, since Erdos participated in the creation of new mathematics until the last day of his life, and his collaborators are expected to continue writing and publishing for years. The briefest encounter could lead to a publication — for scores of young mathematicians a publication that could become the cornerstone of their life’s work. He would work with anyone who could keep up with him, the famous or the unknown. Having been a child prodigy himself, he was particularly interested in meeting and helping to develop the talents of young mathematicians. Many of the world’s leading mathematicians owe their careers to an early meeting with Erdos.
Krishna Alladi, who is now a mathematician at the University of Florida, Gainesville, is one of the many young mathematicians whom Erdos helped. In 1974, when Alladi was an undergraduate in Madras, India, he began an independent investigation of a certain number theoretic function. His teachers could not help Alladi with his problem, nor could his father, who was a theoretical physicist and head of Madras Institute of Mathematics. Alladi’s father told some of his knowledgeable friends about his son’s difficulty, and they suggested that he write to Erdos.
Because Erdos was constantly on the move, Alladii sent a letter to the Hungarian Academy of Sciences. In an astonishingly short time, Alladi heard from Erdos, who said he would soon be lecturing in Calcutta. Could Alladi come there to meet him? Unfortunately, Alladi had examinations and could not attend, so he sent his father in his place to present the results of his research. After his father’s talk, Alladi recounts, “Erdos walked up to him and told him in very polite terms that he was not interested in the father but in the son.” Determined to meet with the promising young mathematician, Erdos, who was bound for Australia, rerouted his trip to stop briefly in Madras, which lies about 860 miles south of Calcutta.
Alladi was astonished that a great mathematician should change his plans to visit a student. He was nervous when he met Erdos at the airport, but that soon passed. “He talked to me as if he had known me since childhood,” Alladi recalls. The first thing Erdos asked was, “Do you know my poem about Madras?” And then he recited:
This the city of Madras
The home of the curry and the dhal,
Where Iyers speak only to Iyengars
And Iyengars speak only to God.
The Iyers and Iyengars are two Brahmin sects. The Iyers worship Shiva the Destroyer but will also worship in the temples of the Iyengars, who worship only Lord Vishnu, the Protector. Erdos explained that this was his variation on the poem about Boston and the pecking order among the Lowells, the Cabots, and God. Having put Alladi at ease, Erdos launched into a discussion of mathematics. Erdos was so impressed with Alladi, who was applying to graduate schools in the United States, that he wrote a letter on his behalf. Within a month, Alladi received the Chancellor’s Fellowship at the University of California, Los Angeles.
A celebrated magazine article about Erdos was called, “The Man Who Loved Only Numbers.” While it is true that Erdos loved numbers, he loved much more. He loved to talk about history, politics, and almost any other subject. He loved to take long walks and to climb towers, no matter how dismal the prospective view, he loved to play pingpong, chess, and Go, he loved to perform silly tricks to amuse children and to make sly jokes and thumb his nose at authority. But, most of all, Erdos loved those who loved numbers, mathematicians. He showed that love by opening his pocket as well as his mind. Having no permanent job, Erdos also had little money, but whatever he had was at the service of others. If he heard of a graduate student who needed money to continue his studies, he would sent a cheque. Whenever he lectured in Madras, he would send his fee to the needy widow of the great Indian mathematician Srinivasa Ramanujan; he had never met Ramanujan or his wife, but the beauty of Ramanujan’s equations had inspired Erdos as a young mathematician. In 1984, he won the prestigious Wolf prize, which came with a cash reward of $at 50000, easily the most money Erdos had ever received at one time. He gave $30000 to endow a postdoctoral fellowship in the name of his parents at the Technion in Haifa, Israel, and used the remainder to help relatives, graduate students, and colleagues:”I kept only $720,” Erdos recalled.
In the years before the internet, there was Paul Erdos. He carried a shopping bag crammed with latest papers, and his brain was stuffed with the latest gossip as well as an amazing database of the world of mathematics. He knew everybody: what they were interested in; what they had conjectured, proved, or were in the midst of proving; their phone numbers; the names and ages of their wives, children, pets; and, much more. He could tell off the top of his head on which page in which obscure Russian journal a theorem similar to the one you were working on was proved in 1922. When he met a mathematician in Warsaw, say, he would immediately take up the conversation where they had left it two years earlier. During the iciest years of the Cold War Erdos’s fame allowed him freely to cross the Iron Curtain, so that he became vital link between the East and the West.
In 1938, with Europe on the brink of war, Erdos fled to the United States and embarked on his mathematical journeys. This book is the story of those adventures. Because they took Erdos everywhere mathematics is done, this is also the story of the world of mathematics, a world virtually unknown to outsiders.. Today perhaps the only mathematician most people can name is Theodore Kacznyski. The names of Karl Friedrich Gauss, Bernhard Riemann, Georg Cantor and Leonhard Euler, who are to Mathematics what Shakespeare is to literature and Mozart to music, are virtually unknown outside of the worlds of math and science.
For all frequent flier miles Erdos collected, his true voyages were journeys of the mind. Erdos carefully constructed his life to allow himself as much time as possible for those inward journeys, so a true biography of Erdos should spend almost as much time in the Platonic realm of mathematics as in the real world. For a layman this may seem to be a forbidding prospect. Fortunately, many of the ideas that fascinated Erdos can be easily grasped by anyone with a modest recollection of high school mathematics. The proofs and conjectures that made Erdos famous are, of course, far more difficult to follow, but that should not be of much concern to the reader. As Ralph Boas wrote, “Only professional mathematicians learn anything from proofs. Other people learn from explanations.” Just as it is not necessary to understand how Glenn Gould fingers a difficult passage to be dazzled by his performance of thee “Goldberg Variations,” one does not have to understand the details of Erdos’s elegant proofs to appreciate the beauty of mathematics. And, it is the nature of Erdos’s work that while his proofs are difficult, the questions he asks can be quite easy to understand. Erdos often offered money for the solution to problems he proposed. Some of those problems are enough for readers of this book to understand — and, perhaps, even solve. Those who decide to try should be warned that, as Erdos has pointed out, when the number of hours it takes to solve one of his problems is taken into account, the cash prizes rarely exceed minimum wage. The true prize is to share in the joy that Erdos knew so well, joy in understanding a page of the eternal book of mathematics.
— shared by Nalin Pithwa (to motivate his students and readers.)
Diophantus of Alexandria: some trivia, some tidbits
The name/word Diophantine equation honours the mathematician Diophantus, who initiated the study of such equations. Practically, nothing is known of Diophantus as an individual, save that he lived in Alexandria sometime around 250 A.D. The only positive evidence as to the date of his activity is that the Bishop of Laodicea, who began his episcopate in 270, dedicated a book on Egyptian computation to his friend Diophantus. Although Diophantus’ works were written in Greek and he displayed the Greek genius for theoretical abstraction, he was most likely a Hellenized Babynolian. The only personal particulars we have of his career come from the wording of an epigramproblem (apparently dating from the 4th century). His boyhood lasted 1/6 of his life; his beard grew after 1/12 more; after 1/7 more he married; and his son was born 5 years later, the son lived to half his father’s age and the father died 4 years after his son. If x was the age at which Diophantus died, these data lead to the equation:
with solution . Thus, he must have reached an age of 84, but in what year or even in what century is not certain.
The great work upon which the reputation of Diophantus rests is his Arithmetica, which may be described as the earliest treatise on algebra. Only six Books of the original thirteen have been preserved. It is in the Arithmetica that we find the first systematic use of mathematical notation, although the signs employed are of the nature of abbreviations for words rather than algebraic symbols in the sense with which we use them today. Special symbols are introduced to represent frequently occurring concepts, such as the unknown quantity in an equation and the different powers of the unknown up to the sixth power. Diophantus also had a symbol to express subtraction, and another for equality.
It is customary to apply the term Diophantine equation to any equation in one or more unknowns that is to be solved in the integers. The simplest type of Diophantine equation is the linear Diophantine equation in two variables:
where a, b, c are given integers and a, b are not both zero. A solution of this equation is a pair of integers that, when substituted in to the equation, satisfy it; that is, we ask that . Curiously enough, the linear equation does not appear in the extant works of Diophantus (the theory required for its solution is to be found in Euclid’s Elements), because he viewed it as trivial, most of his problems deal with finding squares or cubes with certain properties.
Reference:
Elementary Number Theory, David M. Burton, 6th edition, Tata McGraw Hill Edition.
More such interesting information about famous mathematical personalities is found in the classic, “Men of Mathematics by E. T. Bell”.
— Nalin Pithwa
B.S. in Mathematics: IIT Bombay program:
http://www.math.iitb.ac.in/Academics/bs_programme.php
Note that the admission is through IITJEE Advanced only.
–Nalin Pithwa.