Thursday, December 22, 2016

Genius from India: Mathematicians honor Srinivasa Ramanujan

Aiyangar Srinivasa Ramanujan was a mathematician born in Erode in Tamil Nadu. A child prodigy, he was largely self-taught and enrolled at the town high school in Kumbakonam in 1898, when he was 10, and there he came into contact for the first time with the mathematical formalism.

At fourteen, he not only gained merit certificates and academic awards throughout the school years, but helped his school in the logistics necessary to assign 1,200 students each with their own needs with thirty-five teachers. He completed his exams in half the time, showing familiarity even with the infinite series. However Ramanujan did not focus on other subjects and did not pass the exams in high school.

Once married, he had to look for a job. With the collection of his mathematical calculations, he moved to the city of Chennai looking for a job as a clerk. He finally found a job and an Englishman advised him to contact the Cambridge researchers. While being employed at the State Accounts Office, Ramanujan tried to get the recognition he hoped would allow him to focus on the study of mathematics.

He tenaciously solicited the help of local patrons, and published many papers in Indian mathematical journals, but failed to get a sponsorship. During this time Ashutosh Mukherjee tried to support his cause. In 1913 he sent a letter to three Cambridge professors HF Baker, EW Hobson and GH Hardy, to include a long list of theorems. Only Hardy, a member of the Trinity College of Cambridge in England, noticed the genius of Ramanujan's theorems. But the two others did, not even give an answer.

Hardy, along with colleague Littlewood, analyzed the letter. Hardy responded by requesting the demonstrations of some of the results cited in the letter and organized the arrival of Ramanujan in England. What followed was a fruitful collaboration, which Hardy described it as the only romantic episode in his life. Hardy said of Ramanujan's formulas, some of which he was not able to understand, that a single glance was enough to show that they can only have been written by a high-class mathematician. They must be true, because if it had been anyone he would have the imagination to invent them.

Paul Erdős said in an interview that the greatest contribution to mathematics was the discovery of Ramanujan and compared him to the mathematical giants such as Euler and Jacobi. Ramanujan was later appointed member of the Trinity, and received, the highest honor in science, as a member of the Royal Society.

Plagued by health problems all his life, away from home, and obsessively involved with his studies, Ramanujan's health worsened, perhaps exacerbated by stress. DAB Young concluded that he most likely suffered from liver amebiasis, a parasitic infection, because Ramanujan had spent much time in Madras, a coastal city where amoebiasis was widespread.

Srinivasa Ramanujan images wallpapers

He returned to India in 1919 and died soon after in Kumbakonam, leaving as his last gift the theta function.
Ramanujan's talent suggested a plethora of formulas that were then thoroughly investigated later. Consequently, they opened new directions for research. Ramanujan conjecture is an assertion on the size of the tau function coefficients, a typical cusp form in the theory of modular forms. It was eventually proven as a consequence of the proof of the Weil conjecture few decades later.

When he was still in India, Ramanujan had written many in three notebooks. The results were presented without calculation, probably the origin of the rumor that Ramanujan was unable to prove his conjectures and simply directly thought the final result. Berndt, realized in his review of Ramanujan's notebooks and work that he was almost certainly able to show many results, but chose not to.

This way of working can have many reasons. Since the paper was expensive, Ramanujan must have done most of his work and perhaps his proofs were on a blackboard which were then transferred on paper. The use of the blackboard was common in India between the time for maths students. It is very likely that Ramanujan was influenced by the style of one of the books from which he learned a lot of advanced mathematics, Compendium of Pure and Applied Mathematics of GS Carr. Also he collected many thousands of results without proof.

The first book contained 351 pages with 16 chapters and disorganized material. The second book had 256 pages in 21 chapters and 100 unorganized pages, and the third had 33 unorganized pages. The results of his books have inspired many mathematical articles. Hardy produced articles exploring material from Ramanujan's work, as well as GN Watson, BM Wilson and Bruce Berndt.

He mainly worked on analytic number theory and is known for many formulas of summations involving constants such as π, prime numbers and partition function. Frequently his formulas were set out without demonstration, and only later proved correct. His findings have inspired a large number of subsequent mathematical research.

In 1997, it was launched the Ramanujan Journal for the publication of work in areas of mathematics influenced by Ramanujan.
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