Small Margins, Big Ideas: How Innocent Equations Grow To Run The World 

by Dr Kaneenika Sinha - Dec 31, 2016 01:39 PM +05:30 IST
Small Margins, Big Ideas: How Innocent Equations Grow To Run The World Image credits (<a href="http://www.freepik.com/free-vector/digital-lightbulb-background_717318.htm">Designed by Freepik</a>)
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  • Starting January 2017, Swarajya will bring you a monthly column which tells the story of some of the most important ideas and discoveries of mathematics. Here is the author, Dr Kaneenika Sinha, introducing her series.

In 1637, a French lawyer and mathematician by the name of Pierre de Fermat immersed himself in the study of the Greek textbook "Arithmetica" .  Written in the third century AD by the philosopher Diophantus, this book contained several problems which could be encoded in polynomial equations. To solve these problems, one needed to find solutions to such equations in whole numbers.  While reading this book,  Fermat made a remark in the margin of a page:

“It is impossible for a cube to be written as a sum of two cubes, or a fourth power to be written as the sum of two fourth powers, or, in general, for any number which is a power greater than the second to be written as the sum of two like powers ... I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.”

It would be more than 350 years before Fermat's assertion could be proved (and it is now believed with near certainty that the complete proof is not what Fermat had in mind).  In this quest, deep and beautiful facets of numbers were discovered.  Simple properties of numbers like divisibility and factorization were viewed from a tantalisingly new perspective.  This led to wider generalizations and development of new structures in mathematics.  There were big ideas lurking in the small margins of Fermat's copy of "Arithmetica".  As an independent byproduct, while struggling with the beautiful and intricate questions that these structures gave rise to, mathematicians ended up making some observations which ensure safe digital transfer of information in such a way that only the intended receiver of the information can intercept it.

A century later, in 1736, Leonhard Euler (an immensely talented mathematician, physicist and engineer) looked at a network of seven bridges connecting the islands and mainlands of the city of Königsberg (now known as the Russian city of Kaliningrad).  He wondered if one could walk through the city by crossing each bridge exactly once. That is, irrespective of where you lived in the city (islands or mainland), could you leave your home, travel through each bridge exactly once and return home? It did not take Euler very long to deduce that this is not possible.  To answer this question, he interpreted the bridges as edges between the islands and mainlands, which he denoted as points.

This framework, now known as graph theory, helps us to view various phenomena in terms of a graph, that is, a mathematical model consisting of various nodes and edges (or bridges) that connect these nodes.  This viewpoint has inspired fundamental research that links graphs to other areas in mathematics, theoretical computer sciences and biology.  It also enables us to build well-connected communication networks at optimal cost, surf the internet smoothly and navigate efficiently through road networks to reach our destination.

There are several other instances of beautiful ideas which originated in "innocent" circumstances and were developed over centuries of laborious efforts.  These now yield rich dividends for us, either through a better understanding of the universe or through technological breakthroughs without which many of us cannot imagine our lives.

What attracted Euler, Fermat and many others to these ideas and problems? What made the mathematical community persist with these problems and develop these ideas for multiple centuries? Did they categorise their ideas into "pure" or "applied" or "translational"  or "marketable" as we tend to do today? Did they perceive depth in these ideas which would help them (and others) to improve their understanding of nature? Or were they simply motivated by the sheer inherent beauty of these ideas and observations?

This column seeks to answer these questions.  The goal of this column is to delve into some ideas and discoveries that have laid the foundations of mathematics as we practise it today.  We will attempt to explore these ideas, trace their evolution and analyze what these ideas mean to us today.

As we welcome a new year, our nation prepares to embrace digitization at a scale much larger than any other nation has attempted before.  So, to start with, over the next few months, let us try and understand the mathematics that goes behind this grand venture.

Dr Kaneenika Sinha (@kaneenikasinha) is an Assistant Professor of Mathematics at IISER-Pune.
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