Page 34 - Ramanujan Yatra

Ellipses and elliptic curves are distinct. Circles, ellipses and families of curves are known as conical curves. They are curves that can be obtained by slicing a cone in different ways. Elliptic curves are graphs in a plane that has no cusps or self-intersections.
Recall your high school mathematics. The equation x2+y2=r2 represents a circle with radius r. Likewise, an equation x3+y3=z3+w3 in our modern perspective gives a rational elliptic surface. In his deathbed, Ramanujan had worked out infinite families of solutions to these curves— all trying to figure out Fermat’s last theorem.
DEEP INTUITION
Ono knew he had hit the jackpot. Once he returned to Emory University, he worked on the leads with his PhD student Sarah Trebat-Leder. “Together with my PhD student Sarah Trebat-Leder, we discovered that these identities can be reformulated as statements about two important areas of mathematics that did not even exist in Ramanujan’s day,” says Ono.
Ono and Trebat-Leder worked backwards to figure out his secret. Ramanujan had arrived at the formulae on that page by generating a more general identity. Ono recognised that the more general identity was an exceptional K3 surface. In wonderment, Ono says “mathematicians did not discover K3 surfaces until the 1960s, yet Ramanujan had already worked on it forty years before!” Using Ramanujan’s insights, Ono and Trebat-Leder were able to provide numerous examples of a particular category of an elliptic curve, far beyond anyone had done. “It turns out that Ramanujan’s work anticipated deep structures and phenomena which have become fundamental objects in arithmetic geometry and number theory,” says Ono.
THE LEGACY
From exploring the quantum world to secure internet transactions, Ramanujan’s legacy plays a role. K3 surfaces and rank of elliptic curves, two prominent subjects are hot topics today. They are indispensable mathematical tools for string theory and cryptography, respectively.
We are familiar with the three space dimensions (up-down; left-right and forward-backwards) and the fourth time. Curiously the string theory, if true, says that the world we live in consists of more than the three spatial dimensions we can see. Like the ant crawling on the surface of the ball never feels the third dimension, the extra dimensions are folded up tightly into too small space for us to perceive. For that, the tiny wrapped spaces must have particular geometric structure, called Calabi-Yau manifolds. K3 surfaces, which Ramanujan was the first to discover, are the simplest classes of Calabi-Yau manifolds. The enigmatic K3 surfaces are named in honour of Kummer, Kähler and Kodaira, three
Ramanujan
YATRA
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