The best books on Applied Mathematics

Your book, The Princeton Companion to Applied Mathematics (2015), seems to have got some very high praise: a ‘tour de force,’ ‘The treasures [in this book] go on and on.’ Is it a unique book, in terms of getting together lots of experts in the field?

Yes, I don’t think it’s been done before in applied mathematics. The idea was to produce a book that gives a flavour of what applied mathematics is about, what it’s used for, and what the future might be. In no sense can it be comprehensive, even in about 1000 pages, but we tried to have a selection of topics that we thought were the most interesting and to get the very best authors we could from around the world: people who are both leaders in their field but also good writers. It was a five-year project. We had a brilliant copy editor and production editor who worked on it as well. I’m very happy with the result in terms of the coverage but also the quality of the articles: from the writing and even down to the figures. A lot of effort went into producing a good selection of figures to enliven the text. So, it is not just prose and equations but illustrations as well.

Do you want to explain, for the layperson, what applied maths actually is?

This is something we had to think about. The fact is that there is no generally agreed definition of what applied mathematics—or pure mathematics—is. In the opening section of the book, titled “What Is Applied Mathematics?”, I quote some luminaries from the past to give a flavour of their views. My favourite quote is from Lord Rayleigh. He said that applied maths is about using mathematics to solve real world problems ‘neither seeking nor avoiding mathematical difficulties.’ That means you’ve got some real world problem in mind and you don’t oversimplify it so that you can solve it. You’ll do whatever you have to do to come up with either a solution or an approximation. That is a nice way of giving the flavour of applied mathematics. We’ll search out whatever tools we can, and we may have to make approximations, but in the end we’ll come up with something useful. That doesn’t mean to say that we don’t sometimes solve problems just for the sake of it, like, traditionally, a pure mathematician would. Also, there are pure mathematicians who have solved problems that have gone on to be incredibly important in the real world. In banking now, for example, with public key cryptography, the underlying security algorithms are very much based on number theory. So, it goes both ways.

Are there topics or research fields that sit on the boundary between applied and pure mathematics?

Yes, certainly in the sense that things that at one time were thought to be very pure now have lots of applications. In the world of information, which is becoming increasingly important, a lot of the underlying theory is very pure but can be applied to the real world – to cracking codes, for example. Then there is the whole issue of computation. In the past, before we had computers, people did computations by hand. Now, with digital computers, we can do much more and this often gives insight into problems that were very hard. For example, the famous four colour map theorem was eventually proved with the help of a computer, which was used to check hundreds of thousands of cases.

Are there still debates about the validity of computer proof?

In terms of the ones that generated a lot of publicity, they seem to be accepted now. There is always the question of ‘What do you really mean by a proof?’ To what extent can you check a computer program to be sure that it has produced what you thought it was going to? In computer science itself, that is still an area of research: proving programs correct. With things like safety issues in airliners that is a real problem. So, some programming languages have been designed with the aim of making programs easier to verify.

Tell us more about safety in aircraft…

Modern airliners have fly-by-wire systems instead of entirely manual controls. There are miles of wires on the airplane and the pilot might say ‘do this’ to move a flap and a computer algorithm checks that the movement won’t do something to cause the plane to crash. But that algorithm had better be really reliable or it might disallow a plane-saving manoeuvre. There are lots of examples where, due to special conditions in the air, the computer has given the wrong information to the pilot or made the wrong decision. There was the Air France flight that crashed on the way from Brazil to Paris when frozen pitot tubes led to wrong airspeed readings. You can imagine how an algorithm might fail because nobody thought of that particular circumstance.

Let’s go through the books you’ve chosen. The first one on your list is X and the City: Modeling Aspects of Urban Life (2012) by John Adam. Tell us what you like about this book.

This book has a nice theme, which is using mathematics to understand the city. It’s amazing how many different aspects of city life can be investigated using mathematics. There are aspects of driving, congestion, populations, air pollution, light, gardening — there’s even some description of rainbows.

Taking just one of those as an example, how can maths help solve the problem of congestion?

This is quite tricky. As with many cases in this book, mathematical models are constructed to try to understand the problem. A mathematical model is, in essence, a set of equations that you can use to try to understand a process and to predict what’s going to happen in the real world. If you take a problem like congestion, you want to understand how things like the speed or the density of the traffic affects it. So you come up with equations, that you play around with, that will model that situation. With your equations, you can vary the speed of the traffic or the density and use that to predict what will happen. Of course, there’s always the question of whether your model faithfully matches the real world behaviour. How do you check that? How do you choose your parameters? Ultimately, the aim is to gain more understanding. You might, for example, choose speed limits for certain roads in order to try to reduce the congestion. What should those speed limits be? You don’t necessarily want to try different speed limits over a period of months and then see what happens. You want to do some computations and say ‘Ah! 25 miles-per-hour looks like the right speed limit for this particular road’ — thereby saving a lot of time and effort.

What’s the difference between a model and a simulation?

In the most general usage, modelling and simulation are sometimes used as synonyms, but a distinction is usually made between developing and analysing the model (modelling) and carrying out computational experiments with the model (simulation).

I like this book because it is inventive — with a lot of great examples of applied mathematical modelling. The ‘X’ in the title refers to an unknown. I did know of this book before I actually got my hands on it. I had been rather put off by the apparent connection to the TV series, Sex and the City, which I have never watched. But actually there’s not even the slightest connection. There’s no allusion or mention of it even in the chapter entitled “Sex and the City.”

What’s that chapter about, then?

He is making use of the logistic differential equation to model the populations of bed bugs and rats. This is a very old topic in mathematical modelling – modelling the growth of populations. One of the examples that we often use in undergraduate courses is the ‘fox and rabbit’ simulation. This is where you’ve got a population of foxes and a population of rabbits. The foxes eat the rabbits and the rabbits reproduce, both of which can happen at different rates. What’s going to happen? Will all the foxes eat the rabbits and there be no rabbits left? Will there be a steady state? Or will the populations oscillate up and down forever? These so-called predator-prey equations go back about a hundred years. There’s a long tradition of mathematical modelling through differential equations to understand how populations vary over time. This is talking about those sorts of ideas.

The book also gives an estimate of the answer to the question, ‘How many people have lived in London?’ Estimation is well used — for example, to answer ‘How many squirrels live in Central Park?’ And some practically important questions are also addressed, such as should you walk or run in the rain?

Are the equations in a book like this hard? How much maths do you need to know for the book to be enjoyable?

This is one of the two most mathematical books out of the five I have chosen. There are quite a lot of equations and mathematical derivations. I think having a first course in linear algebra and in differential equations or calculus is probably what you need to make much headway. But quite a lot of the book is readable anyway. It introduces the problems in a nice gentle way, setting the scene before introducing the technicalities.

Let’s go on to book number two on your list, which is William J. Cook’s In Pursuit of the Traveling Salesman (2012).

The title refers to the travelling salesman problem (TSP), which is the problem of working out the shortest route that visits a specified group of cities and then returns to the starting point. The problem gets its name from the travelling salesman who wants to visit all the target cities while minimising the total distance he or she travels. It sounds like a very narrow problem but actually, as the author shows very clearly, there are many instances where it arises and it’s very important to solve it. For example, suppose you’re looking at the night sky with a telescope and you want to visit a certain number of stars. In what order do you move between the stars to minimise the movement of the telescope and its wear and tear? Similarly, when you’re laying out computer chips: how do you move the device—that’s coming down on that chip and plonking things on the circuit—to make the process as fast as possible?

It hasn’t been solved yet?

No. First of all, the problem is very hard. Even with the fastest computers and the best algorithms that we have today, we aren’t able to solve it yet. We can’t find the best route in any reasonable time, except for a small number of cities. More interestingly, from the mathematical point of view, despite the intense efforts of mathematicians and computer scientists, we don’t even know how to quantify how hard the problem is, in a technical sense. In maths, we’re always trying to say: here’s a problem, here’s some measure of the difficulty of it. We don’t really know how difficult this problem is. We don’t know where to put it in the hierarchy from easy problems to almost insoluble problems. Is it just that we haven’t thought of the best algorithm or is it that there is no efficient algorithm for solving it?

But if you did ask a travelling salesman wouldn’t he know—from bitter experience—what the best route is?

Well, in a sense, this is how the problem is solved at the moment, with heuristics. A salesman might think he or she has got a good route but it may still be far from optimal. How would they know? Of course, the more cities there are—we’re talking here about a few hundred cities—the harder it becomes to know whether you’ve found anything like the best route. There’s a long history of development of algorithms, each building on the previous one, from the 1960s onwards.

Is that when this problem was first talked about?

The original statement of the problem goes back to the 18th century, with Euler and Hamilton, but it was really only with the advent of computers that people started to make any headway. The 1950s onwards, I would say, is when it started to become a real area of research.

This book is by one of the leading researchers on the TSP and gives a very readable and wide-ranging treatment of the problem. Before reading it, I hadn’t realized just how widespread the TSP problem is. Cook also has a fascinating discussion of various heuristics that have been proposed for TSP over the years. These are illustrated with lots of helpful diagrams. The subtitle is ‘Mathematics at the Limits of Computation’ and it does an excellent job of explaining how large TSPs are tackled, harnessing the best available algorithms with the fastest computers.

Do you get the sense that this problem will be solved through a gradual development of better and better algorithms? Or do you think we need a qualitatively different approach to make significant headway?

First of all, I should say what it means to solve the problem. We’ve been talking about the very best solution. That is what’s hard. What the algorithms available at the moment will do is offer approximations. They will give upper and lower bounds. So, they’ll say, if you want to do a 1000 city travelling salesman tour of this group of cities, then here’s a tour that I can guarantee is within 5% of the best. In practical terms, that’s probably okay, it’s enough. The mathematical difficulty is in saying how hard the problem is, and giving the very best route. That’s what people in this research area try to do. Large amounts of computer time are being spent trying to tackle this problem. There’s also the possibility that a completely new idea will come along and revolutionise the field. This has happened in other areas of mathematics: a famous conjecture gets solved with a completely new approach. This may be one area where that might happen.

When you were saying that it is not even known how difficult the problem is, is that the ‘P versus NP’ problem?

Yes, that’s the framework that I was referring to, the ‘P versus NP’ problem, which is one of the seven Clay Institute Millennium Problems, for each of which a $1 million prize is available for a solution.

The third book I’ve got on my list is the one by Amy Langville and Carl Meyer — Who’s #1? The Science of Rating and Ranking (2012). This is absolutely key at the moment isn’t it? Everything, these days, seems to rely heavily on ratings and rankings.

Yes. Google searches, all the shopping websites that say ‘you liked that, so you may like this,’ the ranking of teams (particularly in the US, where there is a lot of interest in ranking sports teams), or movie sites where people rate movies and they use that to predict what movies you might like. These are the sorts of applications the authors have in mind. The same authors wrote an earlier book purely about internet search engines, and Google in particular. Then they were led to think more generally about ranking and hence this book came along. It’s quite mathematical. There are a number of equations in there. But the mathematics is all elementary undergraduate mathematics. If you are familiar with matrices, vectors and basic probability, you should be able to follow much of the book. Indeed it’s an excellent read for anyone doing a linear algebra course, as it beautifully illustrates the power of linear algebra for solving real-life problems.

Is ranking a fully-fledged area of mathematics now? I was surprised to see it as an undergraduate course book.

I wouldn’t call it a field of mathematics, as such. There have been papers on this topic for many years though this might actually be the first book dedicated to ranking. But the sort of questions they’re asking are pretty fundamental. For instance, if you’ve got two different rankings how far apart are they? If you and I both come up with a ranking of the best football teams in the UK and then someone else comes up with a ranking, which pair of us are the closest? How do you compare different rankings? That’s one of the questions they address here. Also, if there are ties, how can you break them? If your ranking has two football teams on the same level, what extra information can you put into the equation to decide how to order those? How sensitive are those rankings to the underlying data that you’re working from? That’s particularly relevant to internet search engines when people try to spam. They try to create lots of pages purely to generate extra links that can influence a ranking. How much do rankings change when you introduce those sorts of additional nodes or make perturbations to the data? That’s another important question that is addressed here.

Is there a method of ranking or rating that you could explain to us, as an example?

The one that I’m most familiar with is the internet search engine rankings. They are based on representing the whole web as either a graph or matrix. So there are several billion pages on the web now. Any one page may link to any other page, so you have to form a massive array where each row is a webpage. If I’m the n-th row, the question is which of the other several billion pages I link to. You put a 0 there if I don’t link to the page and a 1 if do. So, you end up with a huge matrix of noughts and ones.

So you’re talking about a billion rows and a billion columns?

Yes, with just noughts and ones in there. What Google does—or what people think they do, and they certainly did in 1998 when they published an algorithm—is use the properties of that matrix to produce an ordering of pages based on importance. It’s a clever argument based on if I link to an important page – say the New York Times – I’m linking to a page that clearly has some authority. If they link to me then that gives me authority because they’re important and they’re linking to me. So, by using those sorts of ideas, and a little bit of mathematics, they came up with a way of converting this massive matrix into an ordering of essentially all pages on the web. That is an example of a ranking. The maths behind that is relatively elementary. The maths had been known for a long while but it had not been used in quite this way before.

It sounds to me like a much more complex computation than the travelling salesman.

It’s actually quite a simple computation because all you have to do with that matrix is multiply it into vectors a few times and that can be done quite efficiently. It sounds like it’s harder because of the sheer size of the matrix, but it’s actually much easier. It has not got the combinatorial explosion property that the travelling salesman problem has where there are so many routes you can take that you couldn’t possibly check them all. Whereas here, the hidden properties of the matrix are not too hard to pull out with a little bit of relatively well-known mathematics.

So when people talk about how search engines work and what Google is up to, the mystery is more about what Google is choosing to do rather than the complexity of the maths they’re using to do it?

The thing with Google is that they published an algorithm in 1998. A lot of what they’ve done since then is catering to advertisers and avoiding spamming. They have all sorts of techniques that will juggle the list to achieve their many aims. It will not be a simple algorithm any more, I’m sure.

I hadn’t realised they had actually published it.

Well this was in 1998 when they were just starting up. Brin and Page were PhD students in computer science at Stanford University. They actually never finished their theses because they decided to set up the company. At that point, they didn’t know what was going to become of the company so they had no qualms about publishing a paper. That was the only time they’ve ever, as far as I’m aware, published anything like that.

Because now if they make it public then everybody would be trying to game the system?

Exactly. They cannot reveal their precise methods.

I find Google search quite interesting in that it works well for lots of things but it doesn’t work for hotels. When you’re looking for a hotel, all the ads advertising the hotel take priority so it’s hard to find the hotel you’re trying to get to. For most other things, it seems to work.

Maybe hotels don’t fit in too well to this idea of linking to other important pages. They’re not going to be linking to a lot of things other than local maps and tourist sites.

Do you think anybody setting up a website, like ourselves, should read this book?

For a website, probably their earlier book on page ranking and specifically about search engines would be the first place to look. This certainly would be a good follow-up. It’s looking at ranking in general, not just webpages.

Your next choice is 50 Visions of Mathematics (2014) edited by Sam Parc, which was published for the fiftieth anniversary of the foundation of The Institute of Mathematics and its Applications (IMA).

This book, appropriately, has 50 articles in it. It was edited by a number of people in the IMA. Sam Parc is not a person. It is a pseudonym for the editors, constructed from their first names, which are listed on page six of the preface. What’s really nice about this book is the wide range of topics and the fact that the articles are all very short, typically three or four pages. It’s a fantastic book to just dip into. There’s a whole variety of articles—from biographical and autobiographical articles, which I particularly like— through to topics like “Motorway Mathematics,” which explains how you can use modelling to understand traffic jams. On the motorway, you slow to a crawl, and then a little later you think, ‘Why did I slow down? There’s no obvious accident or problem.’ Mathematical modelling helps you understand the wave effect of that slow-down through the traffic. That’s why we’re getting these smart motorways now. Ultimately, they will be able to adjust the speed limits with the aid of—hopefully—mathematical theory to minimise traffic jams.

There’s also a good article about how if you’re at a murder scene and look at the blood stains on the floor you can find out what happened by working out where the body was when the blood started spurting out. Just using elementary mathematics, sines and cosines—it’s almost ‘A’ Level maths in a way—to try and work backwards. This is by somebody who has a job doing this kind of thing: Graham Divall, independent consultant forensic scientist with 35 years of bloodstain examination experience. There’s also one by Simon Singh on Simpson’s rule. Of course, Simon Singh has published an entire book on the mathematics behind the TV series The Simpsons. This is just a short article relating to one aspect of that. I’ve never watched The Simpsons so I don’t particularly connect with that one. The editors did a really good job of finding interesting articles.

Is the IMA, and hence this book, all about making maths accessible?

The members of the IMA are teachers at schools all the way through to university researchers and people in industry. It caters for all those. It organises conferences, it has journals, and it has a nice magazine that comes out every month.

But is it trying to appeal to the general public? I notice there are no equations in the book.

Yes, the IMA is very strong on outreach, on showing the importance of mathematics to the general public. I think one aim of this book was to do that. So, yes, this is very accessible. It is published by Oxford University Press and they’ve clearly gone to a lot of effort to make it read well and be widely accessible.

When did you get interested in maths? Is it something you were always passionate about?

When I was doing my ‘O‘ Levels, I would have said my best subject was English. But I somehow knew that I’d be better off going into science so I did maths, physics, and chemistry at ‘A’ Level. I never really regretted it. In general, scientists are not very good at writing. It’s not hard to be a better-than-average writer as a scientist, especially as a mathematician. So, if you have an interest in writing and you’re reasonably good at maths that’s a great position to be in. There’s no point in doing great science if you can’t explain it to other people. You won’t even get published if you can’t explain your work to your peers and write it in a compelling way. So I’ve always been interested in maths but liked writing as well. I actually spend more of my time writing. Probably every scientist would say the same, because you have to write emails, write lecture notes, write papers, write referee reports, and somehow you spend more time doing that than doing the actual science.

Do you wish you had more time for maths?

Well, always, yes. But the act of writing is not something you can hand over to somebody else, because writing is part of understanding. Quite often I’ll write something up and realise that I don’t fully understand it. I’ll have to go back and look at the maths again. It’s an iterative process. It’s not what many students think is the process when you’re doing a PhD — spending three or four years doing the maths and then, at some point in the last year, saying ‘I’ll write it up and then I’ve got my thesis.’ That’s certainly not the way I work with my students. I get them writing immediately. If you leave it to the end then (a) you’ll run out of time and (b) you’ll have forgotten some of the work you did three years ago — you can’t understand it anymore and it probably contains flaws. Writing in maths has to be part of the process.

Let’s go on to your fifth and final book: Ian Stewart’s Seventeen Equations that Changed the World (2012).

This is not the first book about equations to be published but what I liked about it is that there’s lots of applied maths in it. And, of course, being by Ian Stewart, it’s very well written. He is a brilliant writer and one of the most famous people in the world for popularising mathematics. For every equation, he starts off with a page that shows the equation and explains what all the terms are in a diagram. He then talks about the equation, where it came from, applications of it, and gives explanations of it in a very readable way with minimal use of equations (other than the equation itself).

One thing I should say that is a bit of an unusual comment is that I love the index of this book. It has a 12-page index which is the best index of all the books I’ve selected — and yet this is the most popular book, the sort that usually doesn’t even have an index. I’ve got a bit of a thing for indexes because they really are a great way into topics. I was flicking through this book’s index and I saw an entry that said ‘digital photography, 157-160.’ That’s a particular interest of mine. Now, you’d never know from the chapter title, ‘Ripples and blips. Fourier Transform,’ that this chapter includes digital photography. But it does – it’s about how images are stored on the computer. The so-called JPEG format.

How are they stored?

It’s a complicated process but it involves breaking the image into little squares and then doing some maths on each square in such a way that you can throw information away without the human eye seeing that you’ve thrown information away. It exploits the fact that the human eye is more sensitive to changes in brightness than changes in colour. You can fiddle about with the colour in a photo more than you can fiddle with the brightness. JPEG exploits that. As Stewart points out, though, although JPEG is used for virtually everything—all the images on your computer will be stored as JPEGs—there’s one example, fingerprints, where JPEG is not very good. Fingerprints have got lots of edges and JPEG is very bad at handling edges. So for fingerprints, there’s a different method called wavelet compression that he explains and, again, it is relevant to the Fourier equation. Back in the 1990s, the FBI realised that this wavelet idea was the right one for storing fingerprints.

And he’s also looking at the history and who Fourier was, presumably?

Yes. Fourier was a French mathematician. In 1807 he submitted his now famous paper to the French Academy of Sciences, based on a new partial differential equation. As usual, it wasn’t appreciated at the time. They declined to publish the work. Now, it’s absolutely crucial to many areas of science. Digital photography, as we said, but also mobile phones – there will be Fourier transforms in them. It has all kinds of applications. Sound can be understood by using Fourier analysis. It really is a very important topic.

He says, at some point, that human history has been redirected by an equation “time and time again.” Is that a credible claim, do you think?

You don’t necessarily need the equation to be benefitting from what the equation is telling you. But I think for things like relativity, Einstein couldn’t have done it without writing down his equations, analysing them, and making predictions from them. So I guess it’s true, to a large extent.

Stewart has also got the wave equation in there and the Navier–Stokes equation, which governs fluid flow. These are important because, say, if you’re designing an airplane you want the wings to minimise drag. Or if you’re designing a boat for the America’s Cup – you want a boat that goes through the water in the best way. Waves are incredibly important.

Do you think you can read this as a non-mathematician?

Yes. Of all the books I’ve presented here, this is the one that would be the most readable for someone who has not got a strong mathematics background. It’s also worth noting that in a recent interview Stewart said this is his favourite out of all the books he’s written.

Can you tell me about the logistic equation, which he also discusses?

This equation is relatively simple. There was a famous paper by the ecologist Robert May in the 1970s that showed that this equation had the property that it could have very unpredictable solutions – this idea of chaos. One aspect of chaos is that you just make a small change in where you start and what happens further on can change dramatically. The weather, for instance, is chaotic. There’s the idea of a butterfly flapping its wings in another part of the world and changing the weather here in the UK.

Is that true?

Yes. Some systems are very, very sensitive to changes in the underlying data. The point is that this is a simple equation where this has been shown to be the case. Normally it’s quite complicated equations that are required to produce chaos. So, this is chaotic but it also has applications. In modelling and understanding population growth, this is the sort of equation that has been used. As Stewart says, it models how a population of living creatures changes from one generation to the next.

And would that count as chaotic?

Well, the fact that the equation can be chaotic means that the population might display all sorts of strange oscillations so that you might think, ‘What’s going on here?’ — in the population of fish or insects of whatever it might be. That might be what happens because of the underlying mathematics.

There is a famous phrase, ‘the unreasonable effectiveness of mathematics.’ Does your own book answer the question of why it is so?

When I was working on the book, I dug out some of the papers that discuss this theme. It was Eugene Wigner who wrote the original paper with that title, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences. “ I hope that we give a bit of a feel for that in our book, but I wouldn’t want to make any great claims that, even in 1000 pages, we could fully answer that question.