# What is applied mathematics?

Earlier this year, I gave a talk on Invisible Mathematics at the British Science Festival. The majority of the talk was devoted to my recent work on Invisibility Cloaks and Seismic Metamaterials. However, the other type of *Invisible Mathematics *running through my talk was the idea that Mathematics and, more specifically, Applied Mathematics underpins much of the work in all Scientific fields.

## So, what is Applied Mathematics?

Trying to define an entire field of research is always a dangerous proposition; it is almost certain that at least one person will think that you’re talking nonsense. However, to me, Applied Mathematics is the study of integral and differential equations and their methods of solution.

## So, why do we study equations?

As soon as I mention *equations *everyone who isn’t a mathematician (and more than a few mathematicians for that matter) immediately groan and roll their eyes. Many non-mathematicians can’t see the point of studying equations rather than the actual physical/biological/chemical processes themselves.

A nice way of addressing this point is to look at a specific example. Most of my research revolves around the study of wave propagation in complex materials. And, in some sense, the Helmholtz equation can be thought of as ** the** equation for modelling wave propagation.

## The Helmholtz equation

The Helmholtz equation (and its solutions) has several nice Mathematical properties and interesting features, such as only being solvable in certain coordinate systems. These attributes make the formal study of the Helmholtz equation interesting in its own right.

However, from my point of view, understanding the Helmholtz equation gives us a powerful tool to model and understand a huge range of physical phenomena, from light, to earthquakes, microwaves, radio waves, watches, gravitational waves, and sound. All these physical phenomena are governed, in some way, by the Helmholtz equation. So, by understanding the properties and solutions of the Helmholtz equation, I can work with a broad range of physical problems across several fields.

Now, I am not nearly arrogant or naïve enough to suggest that understanding one equation makes me an expert in all these fields - far from it. Despite being an *Applied *Mathematician, I am still quite partial to studying spherical cows in vacuum. When I collaborate with colleagues from other disciplines, I am awed by their intimate level of subject knowledge and, quite frankly, embarrassed by my lack of it!

Nevertheless, the above example does serve as a rather nice illustration of the power of Mathematics and the broad range of applications that it has. It also underscores that Mathematics is the invisible framework within which all the quantitative sciences can be understood and the toolbox that my many wonderful colleagues in physics, engineering, acoustics, and seismology can use to do their thing!

I have also recently adapted the talk I gave at the British Science Festival for Open Days at the University of Liverpool. Hopefully I can convince a few prospective students, and their parents, that Mathematics isn’t just big sums.

*The photo at the top of this post was taken by William Bout at the San Francisco Museum of Modern Art.*