Ramanujan Conjecture and some diophantine equations (TIFR, Mumbai)
One of Ramanujan's most influential conjectures concerns the magnitude of the Fourier Coefficients of a modular form. These were made on the basis of his calculations as well as a far-reaching insight as to their usefulness in the study of some diophantine problems. Today his original Conjecture is a Theorem and its generalizations constitute one of the central unsolved problems in number theory. We will review some of these basic developments and highlight some recent far reaching applications of his remarkable insights.