## 17 Oct Finding equations to explain the world

While the world at large is always talking about the cosmic phenomena, particle accelerator or the space exploration, chances are no one has ever talked about what goes on behind the scene. The central piece which goes to the heart of any scientific expedition is mathematics, which however, remains invisible to most of us. It’s a paradox: On the one hand, Mathematics is woven in every fabric of our daily lives. As you are reading this article on internet or sending a text message to your friend, the mathematics formulas and algorithms are in action passing the data accurately and reliably from one end to other. Yet, this treasure of mathematics has been kept hidden from most of us, not many are aware of its key contribution.

Consider Mangalyaan, India’s Mars Orbiter mission, which was a spectacular feat. While millions of Indian celebrated as the Spacecraft entered the Martian orbit, most of us hardly knew the backside of the operation which had been as dramatic as anything that stood on frontier of progress. The mission can simplistically be equated to a 300 day game of chess in which all the moves were planned before the first move was made. It’s a complex game of mathematics; each of the moves almost is an outcome of a mathematical equation. Right from November 5, 2013, when PSLV-C 25 lifted off from Sriharikota, to September 24, 2014, when the spacecraft entered the Martian orbit some 400 million km away, every bit of calculation was made and fed into the onboard computers. The most significant challenge was determining the position of Mars at a given time. Both our Earth and the Mars are revolving around the Sun. So a rocket, when launched, must not point to the position of Mars as it was during the launch, but the aircraft should point to the position where Mars would be around the time the spacecraft reaches near the planet. This problem statement itself is mind-boggling; yet the interplanetary measurement, relying directly on the mathematical models for orbital motion by Johannes Kepler and on the integral and differential calculus developed by Sir Isaac Newton, was accurate enough in the nail-biting launch phases.

There are a few standout mathematicians whose brilliant work and intuitions propelled our world in leaps and bounds. Greek Mathematician Pythagoras is considered as founding father of modern mathematics. Because so much of our world is based on rectangles and right triangles, the Pythagorean Theorem is a very special and important relationship in geometry. There are many relevant applications that require the use of the Pythagorean Theorem. In Mangalyaan mission, engineers and astronomers use the equation of Pythagorean to calculate the paths of spacecraft. Architects use the Pythagorean Theorem to calculate the heights of buildings and the lengths of walls. Athletes even use the theorem when they are calculating distances, which are important in determining how fast they can run or where a ball needs to be thrown.

One of the greatest mathematicians of 20^{th} century and the focus of this article is John Nash. While Dr. Nash solved intractable problems in differential geometry and pure mathematics, his more remarkable contribution was to the Game theory for which he won the Nobel Memorial Prize in Economic Sciences in the year 1994. The generation prior to John Nash, accepted a narrower definition of economics concerned mostly with production and allocation of material goods. Those economists primarily addressed the economics issues as so called zero-sum games in which one player’s gain is another’s loss. Is the real-world interaction so simple? In his deceptively simple extension of Game Theory, the equations of John Nash called “Nash equilibrium” addressed this question providing a way for possible outcomes in which there are opportunities for opposite players for mutual gains. This work of John Nash is a major turning point in the history of economic thought; today economics define their field as being about the analysis of incentives of all social institutions.

Mathematician develops ingenious formalism which often, decades after its invention, suddenly finds a use in science – and therein lies the mystery and joy of mathematical creation. Nash didn’t receive the Nobel Prize until forty years after his theorem was first published in the late fifties. The importance of Nash theorem was not initially appreciated by many even in the mathematical community. It was the era of cold war, the concept of cooperative game was poorly understood. Then there were personal setbacks. In late sixties, his brilliance turned malignant, for more than three decades he was stricken with paranoia and delusion. The epic tale of Dr. Nash’s life miraculous transformation – his brilliant rise, the years lost to schizophrenia, his return to rationality and his receiving the Nobel Memorial Prize in Economic Sciences was vividly captured in the Oscar-winning film, “A Beautiful Mind”. The theory of Nash came into prominence post-cold war as we had commenced living in a world of globally interconnected, democratic, technically advanced societies. Today, the Nash equilibrium is providing a powerful mathematical tool for analyzing a wide range of competitive situations, from corporate megamergers to massive foreign direct investment, from auctioning of airwave licensing to oil leases and radio spectrum. His greatest contribution was nowhere more powerfully felt than in the Nobel Academy who went on to recommend changing the very nature of economics prize. Dr. Nash’s equations essentially redefined the Nobel economics prize as an award to social sciences, open to great contributions in fields like political science, psychology and sociology.

Life, just like mathematics, is a continuous Learning; my quests for John Nash lead me to an unexpected finding and intense joy. Dr. Surajit Borkotokey is a young Math enthusiast who had the distinction of winning the national award for the Best Research Paper in Mathematical Science at the Indian Science Congress as well as the Indo-US Research Fellowship at Louisiana State University, USA. He is currently a Professor of Mathematics, Dibrugarh University, India. In the summer of 2014, Surajit was invited to present paper at the International workshop on game theory and economic applications of the game theory society, University of São Paulo, Brazil. The conferences provided Surajit an excellent opportunity to hear from world-renowned mathematicians – the most memorable was the plenary lecturer by John Nash. “*He was by far the most charismatic speaker in the conference*”, Surajit was sharing the incredible meeting with the Nobel laureate. “*Dr. Nash was warm, easy to approach, and easy to talk to. Our brief conversation covered everything from my work in the Cooperative Game Theory to Assam tea. I am very excited about meeting John Nash in person, moments which I will cherish all my life.*”

Mathematics is the source of timeless profound knowledge, which goes to the heart of all matter and unites us across cultures, continents, and centuries. Let’s the students of Assam see, appreciate, and marvel at the magic of axioms, formulas and equations, for who are fluent in the language of mathematics will be in the cutting edge of progress.