Introduction
Srinivasa Ramanujan (1887-1920), epitomizes one of the superlatively great mind in mathematics. In the world of math, he stand out as an enigma. His equation transcend human imagination. He was one of the greatest mind of all time ever lived. The life of Ramanujan was only of 33 years but in his life of 33 years he achieved what many of scientists together can't achieve in their life time. Let's explore the enigmatic life story of Srinivasa Ramanujan
Birth Of Srinivasa Ramanujan.
Srinivasa Ramanujan was born on 22 December 1887 in Erode a small village about 400 km southwest of Madras (now Chennai) In India. His father's name was Kuppuswami and mother's name was Komalatammal. His father worked in Kumbakonam as a clerk in a cloth merchant's shop. When he was nearly five years old, Ramanujan entered the primary school in Kumbakonam although he would attend several different primary schools before entering the Town High School in Kumbakonam in January 1898. At the Town High School, Ramanujan was to do well in all his school subjects and showed himself an able all round scholar. In 1900 he began to work on his own on mathematics summing geometric and arithmetic series.
It was in the Town High School that Ramanujan came across a mathematics book by G S Carr called Synopsis of elementary results in pure mathematics. This book, with its very concise style, allowed Ramanujan to teach himself mathematics, but the style of the book was to have a rather unfortunate effect on the way Ramanujan was later to write down mathematics since it provided the only model that he had of written mathematical arguments. The book contained theorems, formulae and short proofs. It also contained an index to papers on pure mathematics which had been published in the European Journals of Learned Societies during the first half of the 19th century. The book, published in 1886, was of course well out of date by the time Ramanujan used it.
Contributions Of Srinivasa Ramanujan In Mathmatics.
One of Ramanujan's remarkable capabilities was the rapid solution of problems, illustrated by the following anecdote about an incident in which P. C. Mahalanobis posed 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."
The Ramanujan Conjecture
Although there are numerous statements that could have borne the name Ramanujan conjecture, one was highly influential on later work. In particular, the connection of this conjecture with conjectures of André Weil in algebraic geometry opened up new areas of research. That Ramanujan conjecture is an assertion on the size of the tau-function, which has as generating function the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms. It was finally proven in 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The reduction step involved is complicated. Deligne won a Fields Medal in 1978 for that work.
In his paper "On certain arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called Ï„(n) (the Ramanujan tau function).[72] He proved many congruences for these numbers, such as Ï„(p) ≡ 1 + p11 mod 691 for primes p. This congruence (and others like it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to conjecture that there is a theory of Galois representations that "explains" these congruences and more generally all modular forms. Δ(z) is the first example of a modular form to be studied in this way. Deligne (in his Fields Medal-winning work) proved Serre's conjecture. The proof of Fermat's Last Theorem proceeds by first reinterpreting elliptic curves and modular forms in terms of these Galois representations. Without this theory there would be no proof of Fermat's Last Theorem
The Enigma Of Ramanujan
Srinivasa Ramanujan (1887-1920), epitomizes one of the superlatively great minds in mathematics. In the world of math, he stands out as an enigma. His equations transcend human imagination. The scribblings from his Lost Notebook, during the last few years of his short life, continue to contribute to research in the fields of mathematics and physics, including String Theory. Michio Kaku, the bestselling author of the book, Hyperspace, and one of the original contributors to the development of String Theory, says of Ramanujan's genius, "Working in total isolation from the main currents of his field, he was able to re-derive 100 years worth of Western mathematics on his own... Scattered throughout the obscure equations in his notebooks are these modular functions, which are among the strongest ever found in mathematics... This bizarre function contains a term raised to the twenty-fourth power... Miraculously, Ramanujan's function also appears in string theory, each of the 24 modes in the Ramanujan function corresponds to a physical vibration of the string..." Some of Ramanujan's equations continue to haunt today's world-class mathematicians. Let's quickly review String Theory to know the immense impact of Ramanujan's functions (also called modular functions). From the world of Quantum Mechanics, we know that the matter we see in the classical world, such as a tree or a mountain, is represented by its wave function. The electrons and protons making up these objects exist in their waviness state. String Theory tries to consolidate QM and Relativity (such as gravity), and goes beyond the idea of electrons and protons. As we also saw earlier, the latest standard model of an atom prescribes that protons and neutrons are not fundamental particles any more but made up of further smaller sub-atomic particles called quarks, leptons and mesons. Three quarks make up a proton or a neutron. Whereas an electron is a part of the lepton category comparable to quarks.
Now, according to String Theory, these quarks and leptons are nothing but one-dimensional oscillating or vibrating strings in the multi-dimensional space-time continuum. These one dimensional vibrations of the strings render mass to particles such as quarks and leptons. So the mass of an electron comes from the oscillations of a one-dimensional string in a particular pattern. String Theory further postulates that in order for these strings to exist, space-time must have 10 dimensions and not merely 4 dimensions of space and time.
Michio Kaku noted that in the works of Ramanujan, the number 24 (called 24 modes), appears quite frequently. He says this is a magical number that appears in places in the equations of String The I where it is least expected. When used in String Theory, Ramanujan's 24 modes correspond to the 24 physical vibrations of the one-dimensional string. String Theory arrives at a total of 24+2=26 dimensions. But when Ramanujan's functions are further generalized, these 24 modes miraculously reduce to number 8, giving 8+2-10 dimensions to space-time. So, the strings vibrate in 10 dimensions of space-time. Physicists don't have answers why strings need 10 dimensions to vibrate. All that Kaku concludes is that, "Perhaps the answer lies waiting to be discovered in Ramanujan's lost notebooks".
What do we conclude as laymen? To what can this deep mystery be attributed? How did what Ramanujan created decades ago, purely from his mental abilities with no prior idea of the theories of Relativity, Quantum Mechanics and String Theory, relate to the underlying structure of space-time? The 10 dimensions of String Theory have not yet been practically proven, unlike the 4 dimensions of space-time. But the mathematics employed to arrive at these results is fool-proof according to the scientific community. String Theory is the best bet in coming up with the Theory of Everything (TOE), a theory that promises to reconcile all the great ideas of physics and present one final picture of how, in reality, our universe is designed from both microscopic and macroscopic perspectives. The research goes on unabated to fulfill the dreams of many scientists and who knows what secrets lie hidden away in the Lost Notebooks of Ramanujan's eternal mathematical works, that can one day help to unravel one of the greatest mysteries of all time.
Plagued by health problems throughout his life and living in a foreign country away from India, Ramanujan was entirely dedicated to mathematics. He often said for me has no meaning, unless it represents a thought of God G.H. Hardy, himself a great mathematician, was Ramanujan's at, "An equation mentor and collaborator at Cambridge, UK, and when once asked what his greatest contribution to mathematics was, he replied "Discovery of Ramanujan".
How Did Ramanujan Died?
Many may wonder that any mystery attends Ramanujan’s illness, for until 1984 it was
generally believed that tuberculosis was the cause of death, and that his illness had been
explicitly treated as such in the various English sanatoria and nursing homes. However
that diagnosis originated, not in England, but in India with Dr P. Chandrasekhar of the
Madras Medical College, who attended Ramanujan from September 1919 until his death
on 26 April 1920. The ever-worsening emaciation and pulmonary symptoms that
followed his relapse on arriving in India have been powerfully persuasive. The verdict of
the English doctors was quite the opposite, but its publication had to wait until Rankin
brought the hitherto unpublished material together in his 1984 papers. It quite clearly
contradicts, for example, the statement in Ranganathan’s book that ‘by the end of 1918, it
was definitely know that tuberculosis had set in’. It was disappointing has set in. It was
disappointing, therefore, that Kanigel in his biography, although acknowledging the
doubts, nevertheless chose to perpetuate the undue emphasis on tuberculosis.
Although we have but little information about Ramanujan’s illness, there is much that can
be inferred from it. Then illness began with a acute episode that was diagnosed as gastric
ulcer. Later, the condition eased and the symptomatology must have changed
significantly, for this diagnosis was rejected and that of tuberculosis favoured. The sign
prompting this was probably the onset of intermittent pyrexia, which eventually became
the regular nighttime fevers described by Hardy. Intermittent pyrexia is found in a group
of otherwise diverse diseases, by far the most important of which at that time was
tuberculosis. Expert opinion was accordingly sought from Dr H. Battty Shaw (1867-
1936), a London specialist in consumption and chest diseases. His verdict, given
probably in August 1917, was that it was not tuberculosis but metastatic liver cancer,
derived, he believed, from a malignancy of the scrotum excised some years before. Time
proved him wrong, but it must be very significant that he did not favour tuberculosis. By
the summer the bouts of intermittent fever had become less frequent, and they ceased
altogether during Ramanujan’s stay at Fitzroy Square. Here, doubtless, advantage was
taken of the wealth of medical expertise and facilities available in London. The consensus
of medical opinion mentioned by Hardy in his November letter, namely that Ramanujan
had been suffering from some obscure source of blood poisoning, entails that, at a
minimum, blood counts were carried out. These are procedures that were, even then, a
matter of routine. Then following discussion is a diagnosis by exclusion, and is limited to
clinical detail and diagnostic procedures available in 1918, as presented in the 1917
edition of Munro’s textbook.
Blood poisoning here would refer to septicaemia or toxaemia due to either an abscess or a
source of inflammation. Both would give rise to intermittent pyrexia, but more
specifically to neutrophil leucocytosis, an absolute increase in the polymorphonuclear
fraction of the white blood cells. This fact would rule out several other diseases
associated with intermittent pyrexia: tuberculosis, brucellosis, kala-azar (an example of
Dr Kincaid’s ‘obscure Oriental germ’) and pernicious anaemia. The last, whether due to
absence of intrinsic factor or to a dietary deficiency of vitamin B12, causes symptoms
similar to Ramanujan’s and has been suggested as a diagnosis. Malaria must also have
been considered and blood smears examined accordingly, together with the sensitivity of
the fever to quinine.
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