## The power of small things

Submitted by Marianne on November 5, 2021Life is the art of approximation. If we took into account all the details of every aspect of the things we experience, we'd never get anywhere. We need to be careful about which things to ignore, though, because if those details contain a proverbial devil, they can come back to bite us.

Mathematicians have known this for a long time. Indeed, this year the Isaac Newton Institute in Cambridge ran a virtual research programme, called *Applicable resurgent asymptotics* (ARA), on a related topic. It concerned very small — exponentially small — quantities, which over time and space can grow to be exponentially large. Understanding these potentially explosive quantities is essential in all sorts of areas beyond maths: from building jet engines to theoretical physics. The programme comes in two parts, with the second part taking place in 2022.

Here are some articles exploring the motivation and aims of the research programme. Enjoy!

Stokes's phenomenon: An asymptotic adventure — Nearly 200 years ago the astronomer George Biddell Airy had a question about rainbows. Little did he know that it would set the ball rolling towards an entire field of maths.

Stokes's phenomenon: An asymptotic adventure continued — It was George Gabriel Stokes who answered Airy's question using an ingenious method which showed that exponentially small terms are not to be trifled with.

The power of small things: Resurgent asymptotics — Following on from Stokes, many people have worked on the maths behind exponentially small terms and their power to explode. Today this maths is used to model water waves, build jet engines, and to glimpse the fundamental secrets of the Universe.

### Background reading

Maths in a minute: Asymptotes — Curves can be crazy, but asymptotes give a nice and straight approximation.

Maths in a minute: Asymptotic expansions — You may know that you can approximate a function with a convergent series. But did you know you can do this with divergent series too? What you need is an asymptotic expansions.

Maths in a minute: Complex numbers — To unravel Stokes's phenomenon, and understand the mathematics explored by the ARA programme, you need a little help from complex numbers.

Euler's formula — When you're dealing with complex numbers a change of viewpoint can be useful. Euler's formula comes to the rescue.

Complex square roots — Taking square roots isn't hard, but defining a square root function of complex numbers is tricky. Understanding why helps you understand Stokes's phenomenon.

Maths in a minute: The prime number theorem — Finally, it's got nothing to do with Stokes's phenomenon, but it's one of the most important results in number theory — and it's asymptotic in nature.

### About this package

*This collection of content forms part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI), an international research centre and our neighbour here on the University of Cambridge's maths campus. INI attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more.*