Active Stocks
Thu Mar 28 2024 15:59:33
  1. Tata Steel share price
  2. 155.90 2.00%
  1. ICICI Bank share price
  2. 1,095.75 1.08%
  1. HDFC Bank share price
  2. 1,448.20 0.52%
  1. ITC share price
  2. 428.55 0.13%
  1. Power Grid Corporation Of India share price
  2. 277.05 2.21%
Business News/ Opinion / Chaos of growth and vice versa
BackBack

Chaos of growth and vice versa

In the chaos that itself grows from simplicity, there must necessarily also be the occasional oases of calm, of order

Photo: ThinkStock Premium
Photo: ThinkStock

You’ve heard it said, I have no doubt, or words to the effect—if a butterfly flaps its wings in Bandra today, it will cause a hurricane in Brazil next week. That idea has been repeated so often, it’s almost a cultural icon. The point, though, is a serious one mathematically—that small changes can produce huge disturbances in a system, in this case weather. Of course, nobody has seriously tried to track down which frenetic individual butterfly set off a given hurricane. But the idea is valid: some complex systems, such as weather, are remarkably sensitive to initial conditions.

Another way to look at it: wild complexity can arise from utter simplicity. This is called chaos, and it’s become an entire field of mathematical inquiry.

Yet what’s fascinating here is that inside the complexity of chaos, you will find oases of simplicity too. Distilled to essentials, this is one of several profound mathematical results the Brazilian mathematician Artur Avila has proved through his career. His investigations have brought him the Fields Medal this year.

This idea of oases is not so unfamiliar either. The classic case is the phrase that’s part of the language now—the eye of a storm. Meaning, somewhere inside the vast fury of a great storm, you will find a spot that’s calm and quiet. But it holds elsewhere too. Somewhere in the middle of a roiling crowd, you’ll find a woman standing still, watching everyone roil around her. Somewhere in the seemingly random collection of digits that is the decimal expansion of pi, you’ll come across 9999999 (in fact, for the first time at position 1,722,776 after the decimal point). Or your phone number, or any pattern of digits you choose, etc.

Here’s a necessarily simplistic—and I use that word deliberately—way to look at Avila’s work. What he proved is that in the chaos that grows from simplicity, there must necessarily also be the occasional oases of calm, of order. In a real sense, chaos implies the existence of some order. If you didn’t have it, you wouldn’t truly have chaos.

It’s one of those thoughts that tends to grab and then grow on you. The more you contemplate it, the more you see it as a deep, complex idea.

A metaphor, you might even say, for itself.

Martin Hairer, the fourth winner of the Fields Medal this year, is also interested in systems that are inherently random. Think, for example, of using an absorbent piece of tissue paper to blot up spilled water. You hold the paper at one end and dip it into the water. You see the paper getting progressively wetter; eventually the soaked up water will reach your fingers.

Now you will agree that this process is not the same as water flowing through a pipe. The next piece of tissue you use won’t absorb water in precisely the same way. A whole branch of mathematics seeks to understand random, even haphazard, processes such as these. It’s called stochastic analysis, and it is Hairer’s speciality.

One way to consider this is that Hairer is studying particular kinds of growth. In the case of the tissue paper, how does the patch of wetness grow, or spread, so that eventually the whole sheet is moist? Such growth is not smooth and linear, because it proceeds through and past obstacles—imperfections or folds in the paper, uneven gaps between its fibres, and so on. Mathematicians have long formulated certain equations that describe such growth. But before Hairer, they could not easily solve the equations.

His insight was that since the imperfections are on a smaller scale than the growth itself, you can model the growth at each imperfection by squashing the overall equation into smaller fractions and adding those up. But importantly, you don’t need to go to ridiculously small fractions to find an adequate solution.

Think of travelling from Mumbai to Delhi. The journey is made up of your taxi ride from Parel to the railway station, lazing in the Rajdhani for 16 hours, and then a taxi ride to your friend’s place in Jangpura. That’s probably enough to describe it to anyone interested.

No need to mention the steps to your front door, the pause to grab an apple, the car making a U-turn that you waited 30 seconds for, etc. Hairer called this regularizing the noise. Imagine listening to your favourite music and blocking out sundry car horns and your neighbour yelling: that’s regularizing the noise for you.

Is there any connection between the work of these two gifted mathematicians? A naive dilettante such as me would suggest this: the little wisps of air set off by a butterfly’s flapping wings eventually grow into a cyclone. How can we describe and then understand that growth? And if we can, will we learn anything about spots of calm in the cyclone?

Or, is there sense that we can find in the randomness of the world?

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences.

Comments are welcome at dilip@livemint.com. To read Dilip D’Souza’s previous columns, go to www.livemint.com/dilipdsouza

Follow Mint Opinion on Twitter at https://twitter.com/Mint_Opinion

Unlock a world of Benefits! From insightful newsletters to real-time stock tracking, breaking news and a personalized newsfeed – it's all here, just a click away! Login Now!

Catch all the Business News, Market News, Breaking News Events and Latest News Updates on Live Mint. Download The Mint News App to get Daily Market Updates.
More Less
Published: 28 Aug 2014, 07:13 PM IST
Next Story footLogo
Recommended For You
Switch to the Mint app for fast and personalized news - Get App