3. Do you believe in models? Simplicity and complexity

Models are central to the world of applied mathematics. In its simplest sense, a model is an abstract representation of a system developed in order to answer specific questions or gain insight into a phenomenon. In general, we expect a model to be based on sound principles, to be mathematically consistent, and to have some predictive or insight value. Models are the ultimate form of quantification since all variables and parameters that appear must be properly defined and quantified for the equations to make sense. ‘Do you believe in models? Simplicity and complexity’ discusses the complexity of models; the steps involved in developing mathematical models—the physics paradigm; and collaborative mathematical modelling.

1. What’s so funny ‘bout applied mathematics? Modelling, theory, and methods

What is applied mathematics and how is it different from mathematics or any other scientific discipline? ‘What’s so funny ‘bout applied mathematics? Modelling, theory, and methods’ argues that applied mathematics includes the modelling of natural phenomena and human endeavours, the study of mathematical ideas originating from these models, and the systematic development of theoretical and computational tools to probe models, handle data, and gain insight into any problem that has been properly quantified. Applied mathematics is best characterized by three intertwined areas: modelling, theory, and methods. Any work in applied mathematics fits in one of these three categories or combines them judiciously.

6. Can you picture that? X‐rays, DNA, and photos

Applied mathematics is also concerned with the manipulation and analysis of signals and data. In our digital world, where huge amounts of data are routinely collected, transmitted, processed, analysed, compressed, encrypted, decrypted, and stored, efficient mathematical methods and numerical algorithms are needed. ‘Can you picture that? X-rays, DNA, and photos’ outlines how applied mathematics is used to extract information from this data using the examples of computed tomography, X-ray crystallography, and the compression of digital image files. Modern developments have seen the discovery of new methods to extract information and process it efficiently. Examples are the theory of wavelets and compressed sensing, which has been implemented in Magnetic Resonance Imaging scanners.

4. Do you know the way to solve equations? Spinning tops and chaotic rabbits

In applied mathematics it is of the greatest importance to solve equations. These solutions provide information on key quantities and allow us to give specific answers to scientific problems. ‘Do you know the way to solve equations? Spinning tops and chaotic rabbits’ describes the ways to solve equations and differential equations, outlining the key work of mathematicians Sofia Kovalevskaya, Pierre-Simon Laplace, Paul Painlevé, and Henri Poincaré, whose discovery led to the birth of the theory of chaos and dynamical systems. The difference between an exact and a numerical solution is also explained. Numerical analysis has become the principal tool for querying and solving scientific models.

8. Where are we going? Networks and the brain

Applied mathematics sits at the convergence of collaborative modelling and data research. Its role is fundamental to how we understand our world and develop new technologies. Many applied mathematicians see new challenges as an opportunity to expand their mathematical horizon, and in our rapidly changing society, such new challenges abound. ‘Where are we going? Networks and the brain’ considers recent developments in applied mathematics such as network theory, used in modelling the internet and for studying epidemiology, and its use in studying the brain, an organ of extreme complexity. Applied mathematics is being used to create a mathematical theory of the brain, which is crucial for understanding of a host of neurological conditions.

7. Mathematics, what is it good for? Quaternions, knots, and more DNA

Mathematics has an important place in society, but one that is not always obvious. ‘Mathematics, what is it good for? Quarternions, knots, and more DNA’ considers examples of mathematical theories—quarternions and knot theory—that have found unexpected applicability in sciences and engineering. Quaternions, introduced by William Rowan Hamilton in 1843 during his work on complex numbers, are now routinely used in computer graphics, robotics, missile and satellite guidance, as well as in orbital mechanics and control theory. Knot theory has found unexpected applications in the study of many physical and biological systems, including the study of DNA. Another example is the use of number theory in cryptography.

2. Do you want to know a secret? Turkeys, giants, and atomic bombs

When observing and trying to quantify the world around us, two simple facts can be appreciated and readily agreed upon. First, different physical quantities are characterized by different quantifiers, such as length and time. Second, objects come in different sizes. These two facts have far-reaching consequences that can be appreciated when we understand their mathematical implications and study the constraints that dimensions and size impose on physical processes. Obtaining information in this way is known as dimensional or scaling analysis. ‘Do you want to know a secret? Turkeys, giants, and atomic bombs’ considers several examples of scaling analysis and explains that dimensional analysis relies on a simple universal principle, the principle of dimensional homogeneity.

5. What’s the frequency, Kenneth? Waves, quakes, and solitons

Since we live in a world where interesting things happen both in time and space, the description of many natural phenomena requires the study of functions that depend on multiple variables. These laws can be expressed as partial differential equations. Just like ordinary differential equations, these equations are the natural mathematical tool for modelling because they express, locally, basic conservation laws such as the conservation of mass and energy. ‘What’s the frequency, Kenneth? Waves, quakes, and solitons’ introduces the wonder of partial differential equations by considering two generic spatiotemporal behaviours: wave propagation through linear waves on a string and earthquakes, and non-linear waves and the discovery of solitons, which behave like discrete particles.