Applied Mathematics: A Very Short Introduction
If you are a mathematician, try to define what exactly is applied mathematics and in what sense is it different form (pure) mathematics, and you will realize that it is not that easy. Existing definitions are not univocal. Although in most cases you recognize it when you see it. To that conclusion comes also Alain Goriely, who is an applied mathematician himself, in the introduction of this booklet. Yet he isolates three key elements characterizing the topic. First there is the modelling: some phenomenon is (approximately) described by choosing variables and parameters brought together in equations. Then there is of course a whole mathematical machinery to support and analyse the model theoretically, and finally there are the theoretical as well as the algorithmic and computational methods that solve the equations. The digital computers that emerged after WW II have certainly contributed to the development of applied mathematics bifurcating from pure mathematics. These three elements (model, theory, methods) form the framework for the rest of this (very short) introduction to applied mathematics which is intended for a mathematically interested outsider. Like the other booklets in this series it is a compact (17 x 11 x 0.6 cm) pocket book that is entertaining to read, even on a commuting train or during some idle moments.
The data that an applied mathematician has to deal with are numbers, but these numbers have a certain dimension (length, weight, time,...) and they need to be expressed in proper units (like mks) and at a proper scale. Only when all this has been taken care of in a proper way, one can start building a model to, for example, predict the cooking time of poultry as a function of their weight or try to solve the inverse problem: how fast mammals can loose heat. With the answer to the latter problem, one may deduce something about their metabolism as a function of their volume. Keeping track of the proper dimensions throughout the modelling and the computations is called dimensional analysis.
Choosing a model is a matter of deciding which are the most influential variables. The finer the model, the more computing time it will require while its predictive power or insight will not increase correspondingly. A simple mechanism to arrive at a model is illustrated with the model describing our solar system. First there was the geocentric system, but anomalies in the observations made Copernicus propose his heliocentric alternative. The more precise observations provided when telescopes were being used (a lot of data were provided by Tycho Brahe), allowed Kepler to derive his laws which fit the data, but it was only Newton's gravitational theory that gave the eventual explanation, not only for Kepler's laws, but for gravitation in general. Nowadays, models are constructed in a similar although a more interactive and more complex way. Observations lead to simple models, that are checked by experiments, which require subsequently refinements of the simple model, which is then checked against new observations, etc.
Once the model is shaped in the form of equations, it requires theory to analyse under what conditions there exist solutions and what properties these solutions will have. For example one may analyse when they have an explicit solution (in terms of simple functions). If not, the equations can be considered as defining equations for new (less elementary) functions. The celestial gravitational problem of two mutually attracting bodies was generalized to the three-body problem, which was only solved partially by Henri Poincaré who, by doing so, created chaos theory. A deterministic world view had to be left behind and a qualitative analysis of (nonlinear) differential equations was born. The Lotka-Volterra equations describes a prey-predator model has periodic solutions, but with three species involved they will have chaotic solutions. Also the Lorenz equations, a set of three simple differential equations, originally describing an atmospheric convection problem is a famous model generating chaotic solutions.
When it comes to periodicity, then the wave equation is the example that pops to mind. However when non-linearities are involves, like with seismic P-waves that travel trough the earth mantle, or phenomena like rogue waves, then solitons are involved, which are bump-like shapes that travel along without changing shape. They have a particle-like behaviour, and thus they have potential as carriers of digital information in optical communication, which is an exciting recent research field.
The applications mentioned in the remaining chapters are computer tomography, the discovery of DNA, and the use of wavelets in JPEG2000 for image compression. Other examples are illustrating that what originally were theoretical developments, eventually turned out to be of the highest importance for applications like complex numbers, quaternions, and octonions (this line of complication was eventually replaced by the concept of a vector space), and knot theory (which found application in DNA modification). Finally large networks and big data are fairly recent topics that are used for describing global phenomena. Even with the complexity and magnitude of these networks, they are still inferior to what a human brain is capable of. Accurate modelling of our brain is momentarily still a (distant) target exceeding our current computational capacity but we are closing the gap.
The previous enumeration is just a selection of some of the topics discussed that should illustrate what applied mathematics is about. Of course this limited booklet cannot be exhaustive. The approach is partially historical and still manages to refer to topics of current research. While examples are rather elementary in the beginning, towards the end, the topics tend to be more advanced. But even when discussing these more advanced subjects, Goriely tries to convince the reader that even if math is not always simple, still it is fun to do. The many quotations from the Marx brothers (most of them from Groucho) sprinkled throughout the text are funny of course. Goriely even provides a play-list of pop music that you could play in the background while reading (at least some of these he used while writing). This makes it clear that he has enjoyed writing the book and some of this joy radiates from the text when you read it.