Srinivasa Ramanujan: the greatest Savant of all time

[Razor]

It is surprising that the general public don't know or have forgotten this very special and very unique specimen of the human species.
He was called Srinivasa Ramanujan.

So what is special about him?
Numbers.
He was a genius. No, wait I don't think it is apt to call him a genius because what he did came naturally to him, like breathing.
He had a magical gift of being able to perceive numbers in a way us "normal" humans, can not.

He did not have any formal training in pure mathematics, yet he made enormous contributions to mathematical analysis, number theory, infinite series, and continued fractions. Ask the average citizen of any country about Ramanujan and they will blink.

It is most likely that Ramanujan was what is today called a "savant." Savants see the world in a very different way. @Mad Indian might be interested.)
They do not care or are unable to form much social connections and are obsessed with what interests them (in Ramanujan's case: numbers.)
Also Savants in some cases see things very differently. For example they may see numbers as pictures or they may see things around them as numbers and so on. Or in the case of Ramanujan he said that his mathematics was divinely inspired and that a goddess helped him with this mathematics. So Ramanujan was probably hallucinating "numbers."
One Savant of our times may have been Shakuntala Devi, the human computer, though I'm not very sure of this.


Anyway some background about Ramanujan from wiki

Srinivasa Ramanujan was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. Ramanujan initially developed his own mathematical research in isolation, which was quickly recognized by Indian mathematicians. When his skills became apparent to the wider mathematical community, centered in Europe at the time, he began a famous partnership with the English mathematician G. H. Hardy. He rediscovered previously known theorems in addition to producing new work. Ramanujan was said to be a natural genius, in the same league as mathematicians such as Euler and Gauss.

Note here that our text books and our students will learn about Euler and Gauss and their theorems but our textbooks don't really have anything on Ramanujan. That's a shame.
But thinking about it, this may be due to Ramanujan's theorems being so high level that they are not included in text books, at least I don't remember coming across anything in textbooks about Ramanujan.

@pmaitra Have you come across any of Ramanujan's stuff in your work?

This thread is dedicated to this mind blowing natural mathematician, most of his stuff I don't even understand.

@Mad Indian @Keshav Murali @Free Karma @Peter @sorcerer and others who are interested may add stuff if you are interested.

 

[Razor]

People need to be aware.

In the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to British mathematicians. One mathematician, M. J. M. Hill of University College London, commented that Ramanujan's papers were riddled with holes.[51] He said that although Ramanujan had "a taste for mathematics, and some ability", he lacked the educational background and foundation needed to be accepted by mathematicians.[52] Although Hill did not offer to take Ramanujan on as a student, he did give thorough and serious professional advice on his work. With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.[53]

The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers without comment.[54] On 16 January 1913, Ramanujan wrote to G. H. Hardy. Coming from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible "fraud".[55] Hardy recognised some of Ramanujan's formulae but others "seemed scarcely possible to believe".
Hardy was also impressed by some of Ramanujan's other work relating to infinite series. The first result had already been determined by a mathematician named Bauer. The second one was new to Hardy, and was derived from a class of functions called a hypergeometric series which had first been researched by Leonhard Euler and Carl Friedrich Gauss. Compared to Ramanujan's work on integrals, Hardy found these results "much more intriguing".[57] After he saw Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy commented that "they [theorems] defeated me completely; I had never seen anything in the least like them before".[58] He figured that Ramanujan's theorems "must be true, because, if they were not true, no one would have the imagination to invent them".

On 8 February 1913, Hardy wrote a letter to Ramanujan, expressing his interest for his work. Hardy also added that it was "essential that I should see proofs of some of your assertions".[61] Before his letter arrived in Madras during the third week of February, Hardy contacted the Indian Office to plan for Ramanujan's trip to Cambridge. Secretary Arthur Davies of the Advisory Committee for Indian Students met with Ramanujan to discuss the overseas trip.[62] In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land".[63] Meanwhile, Ramanujan sent a letter packed with theorems to Hardy, writing, "I have found a friend in you who views my labour sympathetically."[64]

To supplement Hardy's endorsement, a former mathematical lecturer at Trinity College, Cambridge, Gilbert Walker, looked at Ramanujan's work and expressed amazement, urging him to spend time at Cambridge.[65] As a result of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an engineering college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Board of Studies in Mathematics to discuss "what we can do for S. Ramanujan".[66] The board agreed to grant Ramanujan a research scholarship of 75 rupees per month for the next two years at the University of Madras.[67] While he was engaged as a research student, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society. In one instance, Narayana Iyer submitted some theorems of Ramanujan on summation of series to the above mathematical journal adding "The following theorem is due to S. Ramanujan, the mathematics student of Madras University". Later in November, British Professor Edward B. Ross of Madras Christian College, whom Ramanujan had met a few years before, stormed into his class one day with his eyes glowing, asking his students, "Does Ramanujan know Polish?" The reason was that in one paper, Ramanujan had anticipated the work of a Polish mathematician whose paper had just arrived by the day's mail.[68] In his quarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to evaluate formerly unyielding integrals.

 

[Razor]

In mathematics, there is a distinction between having an insight and having a proof. Ramanujan's talent suggested a plethora of formulae that could then be investigated in depth later. It is said by G. H. Hardy that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a by-product, new directions of research were opened up. Ramanujan's series for π converges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently used to calculate π.

One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P. C. Mahalanobis who had a problem, "Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to the left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are n and x?" This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. "It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind", Ramanujan replied.[84][85]

In 1918, Hardy and Ramanujan studied the partition function P(n) extensively and gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. Hans Rademacher, in 1937, was able to refine their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae, called the circle method.[86]

He discovered mock theta functions in the last year of his life.[87] For many years these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms.

The number 1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a visit to the hospital to see Ramanujan. In Hardy's words:[90]
" I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No", he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

For more on his mathematical achievements Srinivasa Ramanujan - Wikipedia, the free encyclopedia

Ramanujan claimed that his mathematical work was divinely inspired

Ramanujan credited his acumen to his family Goddess, Namagiri of Namakkal. He looked to her for inspiration in his work, and claimed to dream of blood drops that symbolised her male consort, Narasimha, after which he would receive visions of scrolls of complex mathematical content unfolding before his eyes. He often said, "An equation for me has no meaning, unless it represents a thought of God."

 

[Razor]

Hardy said : "He combined a power of generalization, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day. The limitations of his knowledge were as startling as its profundity.

Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly periodic function or of Cauchy's theorem, and had indeed but the vaguest idea of what a function of a complex variable was...". (note: this is what I meant by natural talent and @Mad Indian this is what interests me the most. This is what makes Ramu unique/rare. Another person that I can think is Mozart. Beethoven was not naturally talented, he worked very hard through pain and tough times and produced masterpieces. But Mozart was a natural genius (like Ramu). Mozart produced music effortlessly as though it was like breathing or drinking water.)

When asked about the methods employed by Ramanujan to arrive at his solutions, Hardy said that they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account." He also stated that he had "never met his equal, and can compare him only with Euler or Jacobi."

Quoting K. Srinivasa Rao,[93] "As for his place in the world of Mathematics, we quote Bruce C. Berndt: 'Paul Erdős has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, J.E. Littlewood 30, David Hilbert 80 and Ramanujan 100.'"

Professor Bruce C. Berndt of the University of Illinois, during a lecture at IIT Madras in May 2011, stated that over the last 40 years, as nearly all of Ramanujan's theorems have been proven right, there had been a greater appreciation of Ramanujan's work and brilliance. Further, he stated Ramanujan's work was now pervading many areas of modern mathematics and physics. (More on this later.)

.............

 

[Razor]

Maths Genius Ramanujan's Deathbed Dream Predictions Proved 90 Years Later

While on his death bed, the brilliant Indian mathematician savant Srinivasa Ramanujan wrote down cryptic mathematical functions he said came to him in dreams from the Hindu goddess Namagiri. Now, amazingly, 90 years later, researchers say they've proved he was right.

It was on his deathbed that he described the mysterious functions that mimicked theta functions, or modular forms, in a letter to Hardy. Like trigonometric functions such as sine and cosine, theta functions have a repeating pattern, but the pattern is more complicated than a sine curve.

Theta functions are also "super-symmetric" meaning that if a specific type of mathematical function called a Moebius transformation is applied to the functions, they turn into themselves. Because they are so symmetric, these theta functions are useful in many types of mathematics and physics.

Ramanujan died before he could prove these theories but finally more than 90 years later, Ono and his team have proved that these functions do indeed actually mimic modular forms.

Ono says. "We found the formula explaining one of the visions that he believed came from his goddess." Ramanujan could not possibly have known this formula, which arises from a bed-rock of modern mathematics built after his death. Ono said. "It is inconceivable he had this intuition, but he must have." The proof deepens further the intrigue surrounding the workings of Ramanujan's enigmatic mind.

The team were also astonished to realise the functions are useful today when studying advanced ideas barely imagined when Ramanujan was alive, such as String Theory and the study of Black Holes. "No one was talking about black holes back in the 1920s when Ramanujan first came up with mock modular forms, and yet, his work may unlock secrets about them," Ono says.

S: Maths News and Blog

 

[Mad Indian]

Awesome thread Bro :thumb:. Truly mind boggling @Razor

 

[alphacentury]

ENCOUNTER WITH THE INFINITE

HOW DID THE MINIMALLY TRAINED, ISOLATED SRINIVASA RAMANUJAN, WITH LITTLE MORE THAN AN OUT-OF-DATE ELEMENTARY TEXTBOOK, ANTICIPATE SOME OF THE DEEPEST THEORETICAL PROBLEMS OF MATHEMATICS—INCLUDING CONCEPTS DISCOVERED ONLY AFTER HIS DEATH?

http://www.believermag.com/issues/201501/?read=article_schneider_phelan

 

[Peter]

Please check this video out.