Ramanujam an everlasting wonder
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Searching for: modular functions and string theory
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Here is an essay on Ramanujan and his connection to modular functions and string theory:
Ramanujan was a self-taught Indian mathematician who made remarkable contributions to various fields of mathematics, such as number theory, analysis, infinite series, and continued fractions. He was born in 1887 in Erode, a small town in Tamil Nadu, and showed an extraordinary talent for mathematics from a young age. He obtained a copy of Carr’s Synopsis of Elementary Results in Pure and Applied Mathematics, a book that contained thousands of theorems without proofs, and used it as a source of inspiration and challenge. He verified many of the results in the book and discovered new ones on his own. He also developed a deep interest in modular functions, which are special functions that have symmetries under certain transformations of the complex plane.
Ramanujan faced many difficulties in his life, such as poverty, ill health, lack of formal education, and social isolation. He tried to communicate his findings to other mathematicians, but most of them ignored or rejected his work as incomprehensible or trivial. He finally found a sympathetic correspondent in G.H. Hardy, a prominent British mathematician who recognized Ramanujan’s genius and invited him to Cambridge in 1914. There, Ramanujan collaborated with Hardy and other mathematicians on various topics, such as partitions, prime numbers, modular equations, hypergeometric series, and mock theta functions. He also became the first Indian to be elected as a Fellow of the Royal Society and a Fellow of Trinity College.
Ramanujan’s work was highly original and unconventional, and often lacked rigorous proofs. He claimed that many of his ideas came to him in dreams or visions from the goddess Namagiri of Namakkal, whom he worshipped devoutly. He said that the goddess would write equations on his tongue or on a slate that he would see in his dreams. He also said that he had access to a “lost notebook” that contained secrets of mathematics that were revealed to him by the goddess. Some of his statements were so mysterious and profound that they baffled even Hardy and other experts.
One of the most fascinating aspects of Ramanujan’s work is his connection to modular forms and string theory. Modular forms are functions that have symmetries under certain transformations of the complex plane, such as scaling, rotating, or flipping. They are important objects in number theory, as they encode information about various arithmetic properties of numbers, such as congruences, divisibility, and quadratic reciprocity. They also have applications in physics, especially in string theory, which is a framework that attempts to unify all the fundamental forces and particles of nature by describing them as vibrations of tiny strings.
String theory requires that the strings move in a higher-dimensional space-time that has 10 or 11 dimensions. However, we only perceive four dimensions: three spatial dimensions and one temporal dimension. The extra dimensions are assumed to be curled up into very small shapes called Calabi-Yau manifolds. These manifolds have special geometric properties that determine the physical characteristics of the strings, such as their mass, charge, spin, and interactions. One way to study these manifolds is to use modular forms, which can describe their shapes and symmetries.
Ramanujan was one of the first mathematicians to discover and study some of the modular forms that are relevant for string theory. For example, he found an infinite series expression for the inverse of the Dedekind eta function1, which is a modular form that generates many other modular forms by using simple operations. The Dedekind eta function also appears in the partition function2 of string theory3, which counts the number of possible states or configurations of a string.
Another example is Ramanujan’s tau function4, which is defined by the coefficients of another modular form called the Ramanujan delta function5. The tau function has many remarkable properties, such as being multiplicative6, satisfying certain congruences7, and obeying certain differential equations8. The tau function also appears in the counting of black hole microstates9 in string theory10, which is related to the entropy or disorder of a black hole.
A third example is Ramanujan’s mock theta functions, which are functions that resemble modular forms but do not have all their symmetries. They were discovered by Ramanujan shortly before his death in 1920, and he left behind only a few examples and vague hints about their properties. They remained mysterious for decades until they were understood by using the theory of harmonic Maass forms, which are generalizations of modular forms that allow non-holomorphic terms. The mock theta functions also appear in the counting of black hole microstates in string theory, where they capture some subtle quantum corrections.
Ramanujan’s legacy is immense and enduring. His work has inspired and influenced many mathematicians and physicists, who have continued to explore and extend his ideas. His notebooks and letters contain many formulas and conjectures that are still being verified or proved. His life and achievements have also been celebrated in books, films, and documentaries. He is widely regarded as one of the greatest mathematicians of all time, and a symbol of the power and beauty of mathematics.
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