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Monday, December 21, 2015

Ramanujan The great Mathematician

He started his journey on 17th March

April 14th is important in many ways. One of which that goes unreported is the great son of India whose journey into recognition as one of the greatest genius happened with his arrival in London on this Day.

We need to be reminded that top mathematicians have mined many theorems from his works and worked up their minds to find out how so many mind boggling theorems have popped out of a single mind.

Two great people were born on 22nd December one is this great genius and another is great soul, my mother.

My God, I could not stop reading the link given below at a stretch yesterday night. I am always amazed at this great genius, ordinary human being with an extraordinary brain. He is a reminder of people who can be labeled as a category of 'the greater the thinking the simpler the living'. The world is witness to many unschooled geniuses. But seldom on subjects as scientific and complicated as this, especially a wide range of contribution in mathematical analysis, number theory, infinite series, continued fractions etc.

I am very poor even in the basic Mathematics but I know its immense importance in terms of its application in almost all spheres of life. I have read many interesting books on mathematics and its relevance. I have also listened to many lectures about how India has been a great pioneer in some of the most advanced Mathematics which were even embedded in many ancient scriptures etc. Equally I have also read some interesting books on advanced number theories by Arabic scholars. 

This PDF gives brief accounts of this great genius whose life continues to be greater puzzle than the most complicated theorems he came up with, the complex mathematical problems he solved so easily, the great solutions he provided with ease to many unsolvable problems.

He was a man immersed in number who understood and interpreted everything connected with life through numbers.

French historian of Mathematics and author of the book, The Universal History of Numbers Georges Ifrah says,
"The Indian mind has always had for calculations and the handling of numbers an extraordinary inclination, ease and power, such as no other civilization in history ever possessed to the same degree. So much so that Indian culture regarded the science of numbers as the noblest of its arts...A thousand years ahead of Europeans, Indian savants knew that the zero and infinity were mutually inverse notions."
(source: Histoire Universelle des Chiffres - By Georges Ifrah  Paris - Robert Laffont, 1994, volume 2.  p. 3 ).

Claiming India to be the true birthplace of our numerals, Ifrah salutes the Indian researchers saying that the "...real inventors of this fundamental discovery, which is no less important than such feats as the mastery of fire, the development of agriculture, or the invention of the wheel, writing or the steam engine, were the mathematicians and astronomers of the Indian civilization: scholars who, unlike the Greeks, were concerned with practical applications and who were motivated by a kind of passion for both numbers and numerical calculations."

He refers to 24 evidences from scriptures from India, whose dates range from 1150 BC until 458 BC. Of particular interest is the work by Indian mathematician Bhaskaracharya known as Bhaskara (1150 BC) where he makes a reference to zero and the place-value system were invented by the god Brahma. In other words,  these notions were so well established in Indian thought and tradition that at this time they were considered to have always been used by humans, and thus to have constituted a "revelation" of the divinities.

"It was only after the eighth century BC, and doubtless due to the influence of the Indian Buddhist missionaries, that Chinese mathematicians introduced the use of zero in the form of a little circle or dot (signs that originated in India),...".
The early passion which Indian civilization had for high numbers was a significant factor contributing to the discovery of the place-value system, and not only offered the Indians the incentive to go beyond the "calculable" physical world, but also led to an understanding (much earlier than in our civilization) of the notion of mathematical infinity itself.

Sanskrit notation had an excellent conceptual quality. It was easy to use and moreover it facilitated the conception of the highest imaginable numbers. This is why it was so well suited to the most exuberant numerical or arithmetical-cosmogonic speculations of Indian culture."

"The Indian people were the only civilization to take the decisive step towards the perfection of numerical notation. We owe the discovery of modern numeration and the elaboration of the very foundations of written calculations to India alone."

"It is clear how much we owe to this brilliant civilization, and not only in the field of arithmetic; by opening the way to the generalization of the concept of the number, the Indian scholars enabled the rapid development of mathematics and exact sciences. The discoveries of these men doubtless required much time and imagination, and above all a great ability for abstract thinking. These major discoveries took place within an environment which was at once mystical, philosophical, religious, cosmological, mythological and metaphysical."

"In India, an aptitude for the study of numbers and arithmetical research was often combined with a surprising tendency towards metaphysical abstractions; in fact, the latter is so deeply ingrained in Indian thought and tradition that one meets it in all fields of study, from the most advanced mathematical ideas to disciplines completely unrelated to 'exact sciences. 

In short, Indian science was born out of a mystical and religious culture and the etymology of the Sanskrit words used to describe numbers and the science of numbers bears witness to this fact. "

"Sanskrit means “complete”, “perfect” and “definitive”. In fact, this language is extremely elaborate, almost artificial, and is capable of describing multiple levels of meditation, states of consciousness and psychic, spiritual and even intellectual processes. As for vocabulary, its richness is considerable and highly diversified. Sanskrit has for centuries lent itself admirably to the diverse rules of prosody and versification. Thus we can see why poetry has played such a preponderant role in all of Indian culture and Sanskrit literature. "

1729 = 13 + 123 = 93 + 103.
Generalizations of this idea have created the notion of "taxicab numbers". Coincidentally, 1729 is also a Carmichael number.
1729 = 13 + 123 = 93 + 103.

Robert Schneider is the lead singer of The Apples in stereo, a record producer (Neutral Milk Hotel, Olivia Tremor Control), and cofounder of the Elephant 6 collective of musicians and artists. He is currently pursuing a PhD in number theory at Emory University, in Atlanta, Georgia, where he lives with his wife and son.

Benjamin Phelan is a writer and musician who lives in Louisville, Kentucky, and is a multi-instrumentalist in Apples in Stereo..
There is a form of Buddhism so potent, adherents say, that to hear its name spoken is to receive a promise of premature enlightenment, of early freedom from the wheel of incarnations. Something similar is true of Srinivasa Ramanujan, the super-genius who was born into deep poverty in an obscure part of southern India, who taught himself mathematics from a standard textbook, and in total isolation became a mathematician of such power that a hundred years after his death, at the age of thirty-two, the meaning of much of his work is still a mystery. In the middle of what I thought would be my life’s work, writing and producing music, I heard his story; now I find myself in graduate school studying number theory.

That even as he approached the infinite, Ramanujan found a wormhole through, and beyond. Even on his deathbed, mathematics was an act of worship. Worship of a single infinity, in infinite forms, all of them knowable.

Ramanujan was unfashionable. His body of work consisted of notebooks filled with short formulae, so there was no overarching theory to study, and formula writing had been out of style in serious mathematics for more than a century.2 The formulists had had their time. They were the sorcerers of math’s prehistory who had discovered the deep connections among the key concepts and encoded them in mathematical haiku. Modern mathematicians-in-training studied modern theorists, technicians who labored over proofs of narrowly defined conjectures, mastered this or that technique, and polished the gleaming apparatus free of fingerprints.
When Ono began to dig a little more deeply into Ramanujan’s formulae, he was surprised at the tangle of roots he encountered below the surface. Ramanujan’s crazy tricks linked up with some of the deepest concepts in math. They could not exist unless they concealed massive theoretical edifices.
Take the tau function, an oddity that Ramanujan discovered and studied during his five years at Cambridge. A function is a mathematical expression that, when fed with a number, produces another number. It’s a machine that takes some raw material and then stretches, compresses, reshapes, or transforms it into something else. Functions embody the relationships between numbers; they are central objects of study in number theory. Ramanujan found the tau function important enough to spend upward of thirty pages in his notebook exploring it, but it was hard for other mathematicians to see why he’d been so interested. On its face, there was nothing special about the tau function. Hardy, Ramanujan’s chief collaborator at Cambridge, worried that the tau function’s homeliness might lead future mathematicians to see it as a mathematical “backwater.” For decades after Ramanujan’s death, it was treated as one.
Then, in the 1960s, a French mathematician named Jean-Pierre Serre realized that the tau function was an unassuming front for a powerful force. Its existence could be explained only if there was a brand-new theory of functions encoded in it. Serre called this theory, suspected but not proven, the Galois representations. Not long after, the Belgian researcher Pierre Deligne proved that the Galois representations actually existed, and in the process clarified that the tau function was deeply connected to algebraic geometry and algebraic number theory. For proving the Galois representations, Deligne won a Fields Medal, the ne plus ultra of mathematical achievement, awarded every four years to a mathematician under the age of forty. In 1995, the Galois representations appeared as the key component of Andrew Wiles’s epochal proof of Fermat’s Last Theorem, the largest, most notorious open problem in mathematics, which had gone unproved for over three hundred years and was suspected of being unprovable. Wiles, forty-one when he published the final version of his proof, was ineligible for a Fields, which only seems unjust: no prize, not even a Fields medal, could be adequate to the mastery in his proof. When the International Mathematical Union convened to hand out Fields Medals that year, it created a special award for Wiles and, for the ceremony, built two stages: one for the Fields Medalists and one above it, where Wiles stood alone.
“All that, from Serre to the Fields medal to Wiles, is from only about ten or fifteen pages from Ramanujan’s notebooks, out of the hundreds that he wrote,” Ono says. “Which is typical! And in fact, studying the tau function, the British mathematician Louis Mordell proved some properties that were later developed into Hecke algebras and the Langlands program, among the two or three most important developments in twentieth-century math. And that’s from a different five pages of Ramanujan’s work on tau that have no intersection with the previous fifteen. In fact, it might be as short as a page. One page from Ramanujan’s work may have given birth to all that.”
There’s a subtlety here that needs to be made explicit. It’s not remarkable that Ramanujan’s work on the tau function led to interesting new mathematics. That kind of thing happens all the time; it’s how the subject advances.
With Ramanujan there is a seeming reversal of cause and effect. No one can write down a formula with deep, hidden properties unless they first know what the deep properties are that they are trying to encode. This is the way mathematicians understand math to work; it is the only way they—we—know to approach the subject. But the significance of the tau function—the reason to write it down—wasn’t discovered until Ramanujan had been dead for sixty years.
“There’s no way Ramanujan knew all these intermediate things,” says Ono. “The concepts [encoded in the tau function] didn’t exist when he was alive. That’s the mind-boggling part: Ramanujan anticipated the work of people who would live long after him. He had visions that said there were going to be some theories in the future. Somehow. He didn’t need any intermediate steps for him to anticipate that there would be all these subjects, and that he would find the first examples of them, and that they would go on to be the prototypes that we desperately needed to build our subjects. Whether he’s in fashion or out of fashion has more to do with us, with where we are in coming to grips with him.”
When Ono started looking into the mock-theta functions, there were a few hints as to what they might mean. They seemed to help describe the spread of cancer tumors, and physicists had begun to find them useful in understanding how black holes unravel space and time and how string theory knits them together. This was peculiar, since the concept of string theory didn’t exist in 1920, when Ramanujan wrote his letter, and black holes were brand-new objects of speculation among a handful of physicists. But still—when modern astrophysicists peer inside their black-hole models, they find they are looking at mock-theta functions.
Despite a few research applications, the mathematical understanding of mock-theta functions was in a bizarre state. Dozens of papers had been written on them, but no one could explain in the most basic sense what a mock-theta functionwas. When Ramanujan died, there were no clues anywhere in the mathematical literature to explain why he found the mock-theta functions interesting. It’s probably not going too far to say that, in fact, they weren’t interesting. All they did, Ramanujan wrote, was imitate a class of functions called the theta functions, which had been around for a century or so. In that time, the theta functions had been working perfectly on their own. No one had needed to imitate them. Ramanujan had produced a solution to a nonexistent problem. Who cares?would not have been an unreasonable response.
In the summer of 2012, Ono found that the only way he could understand the mock-theta functions was via Serre, Deligne, and others’ work on the tau function. This made no sense. It meant that it was not Ramanujan’s own work on tau that had led Ramanujan from tau to the mock thetas, but the work of others, of Serre and Deligne, that would not be carried out until he’d been dead for decades.
Ono had the sensation of Ramanujan walking in his footsteps, but from the wrong direction in time.
“Whatever Ramanujan was thinking about between the tau function and the deathbed letter somehow must have been parallel to what I was doing, without him knowing I was doing it, ninety years later,” he says.
With Ramanujan looking over his shoulder like a “chubby guardian angel,” Ono found that, as the numbers being spit out by the theta functions started to grow at an unimaginable speed, approaching and then far exceeding the number of atoms in the universe, the mock-theta functions began to imitate them with eerie precision. In the lower reaches of the number line, the behavior of the function and its doppelgänger was unlovely and chaotic. But out here, in the immense realms that had driven Cantor insane and enraged the European mathematics establishment, their relationship became clear. You could take the ludicrous, unmanageable output of a theta function, then subtract the ludicrous, unmanageable output of a mock-theta function, and the answer was shocking in its simplicity. The answer was 4.
With pencil and paper and pages of calculations in front of you, to see these titanic quantities consume each other so precisely bends the mind.
“It doesn’t take any imagination,” says Ono, “to recognize that four is a beautiful number.”
As Ramanujan lies dying, racing toward infinity, a dot of light appears in the great wall. The gleaming apparatus is about to crash, but the mock theta function does its crazy trick, and the infinite dissolves, just a little. A portal the size of an atom appears. The apparatus threads the hole. And then it keeps going, and going, and going…”

and this blog has lot of interesting stuff on Maths

some useful sites for mathematics

 Also read a very interesting research work 'Alex's Adventures in Numberland' by Alex Bellos

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