Rediscover Mathematics: a guide to beginning graduate students in theoretical research

Pursuing theoretical research in fluid/solid mechanics, modelling, systems biology, and various other fields requires a strong foundation in mathematical methods. By this I do not mean just a knowledge of techniques and algorithms, but rather an appreciation for the ideas underlying the methods, an understanding of why the methods work, and an appreciation for their elegance. Unfortunately, mathematical courses in schools and colleges often reward calculation speed over understanding, and overemphasize the algorithmic aspects of the subject: we are taught how to calculate eigenvalues of a matrix without an understanding of what these values mean and why they are special. If this situation is not remedied, then doing original theoretical work will be difficult and taxing, rather than the joy it should be. So one of the first steps at the start of a PhD should be to strengthen one's mathematical foundations. This may require you to return to very basic topics, such as integration and differentiation---do not hesitate to do this as it is essential to get the basic ideas clear. For example, most of us know how to calculate an integral either analytically (as is drilled into us in school) or numerically (using trapezoidal rule for example), but do we understand what an integral means? How do we know that an answer even exists to be computed? How do we make sense of an infinite sum of infinitely small numbers? How did the idea of an integral arise? To understand these aspects it is important to read and understand the proofs that accompany theorems in mathematics books---the sections we all too often skipped in school and college in order to get to the exercise problems at the end of the chapter. (The explanatory value of proofs is highlighted in the wonderful book, The Mathematical Experience, mentioned below, and well as in the article Proving is convincing and explaining by Hersh.)

The good news is that the PhD program gives you time to think deeply and revisit old concepts. So go back, as much as you need to, and rebuild your understanding. Do not be discouraged if you take a day or more to understand one page or less. Grasping subtle mathematical ideas requires working through several examples along with deep reflection. The effort you put in compounds and will pay off both in terms of the quality of your work and the joy you derive from it. A possible outline of how to go about this process and recommendations for reading material follow. An important point to keep in mind while choosing books for learning mathematical topics is to look for books that are dedicated to the topic and develop the subject in a pedagogical way. Avoid large compendiums with titles like Mathematical Methods for Engineers---these are more suitable for use as a reference text once the material has been understood.

The sections that follow point to study resources (books, articles, lectures) for (i) mathematical concepts and methods, (ii) mathematical modelling, (iii) philosophy of mathematics.

Methods

1. Calculus: A clear understanding of functions, limits, integration and differentiation is essential for dealing with, for e.g., differential equations. A good book that helps build intuition without sacrificing rigour is the textbook by John and Courant, Introduction to Calculus and Analysis, Vol 1 (Vol 2 is on multivariable calculus).

2. Linear algebra or matrix algebra: Along with calculus, this subject forms the basis for most mathematical methods used in engineering, and is essential for understanding a range of advanced topics from differential equation models to modern techniques of data analysis. An excellent free online resource is the collection of MIT OCW lectures by Gilbert Strang, which is accompanied by his textbook, Introduction to Linear Algebra. 

3. Linear operator theory and functional analysis: The extension of ideas encountered in matrix algebra to infinite-dimensional spaces is essential for working with partial differential equations, which are ubiquitous in fluid and solid mechanics, transport phenomena, electro-magnetism, etc. A good book, especially for engineers, is Linear Mathematical Models in Chemical Engineering by Hjortso and Wolenski. They start with finite-dimensional linear algebra, but develop the subject in a more abstract way than Strang's book. This allows for the ideas to be more easily extended to infinite dimensions. A more advanced treatment that better reveals the beauty of the subject is the elegant, but now unfortunately out-of-print text, Linear Operator Methods in Chemical Engineering, by Ramkrishna and Amundson. It should be available in the IIT Bombay library. Another classic advanced-level book aimed at scientists and engineers is Linear Operator Theory in Engineering and Science by Naylor and Sell.

4. Vectors and Tensors: These objects and their associated concepts underlie continuum mechanics and the development of the equations governing fluid flows and solid deformations. An excellent text book is Vectors, Tensors and the Basic Equations of Fluid Mechanics, by Rutherford Aris. His detailed development of the governing equations of fluid flow is especially valuable when dealing with continuum models for complex non-Newtonian fluids. 

Modelling

Theoretical work rests on mathematical models, i.e. a consistent closed system of equations along with boundary and initial conditions. Having a sense of comfort and familiarity with the associated mathematics, a basic level of mathematical maturity, greatly helps not only in solving the models but also in formulating them, the latter probably being more important in an age with user-friendly symbolic and numerical computing software. In addition, however, one must also learn the art of modelling, which among other things involves identifying those aspects of the problem that are essential and must be retained and those that are merely incidental and may be neglected. By idealizing and simplifying, a model exaggerates certain "key" features of a system which we consider to be important. By analysing the models predictions we can then ascertain how central the "key" features really are. This is an iterative process that involves comparison with experience or experiment, and revision of the model where required. Here, it is important to point out the distinction between a model and a detailed simulation. In the latter, the goal is to maximise the quantitative accuracy of predictions, which naturally drives the simulation towards every increasing complexity, "towards the particular rather than the general". Such simulations, while important for applications, become too large to offer insight into the underlying principles. Moreover, they are tied to a specific situation and cannot help us distinguish fundamental aspects from accidental occurrences. A good model, in contrast, should capture the behaviour of the entire class, of which the system being simulated is just one realization. A vivid illustration of this distinction between a simulation and a model (and of their respective merits) is provided by large weather/climate simulators which generate huge amounts of data that can only be meaningfully interpreted and understood in the light of simpler models (Vallis, Proc. R. Soc. A. 472, 20160140). For example, a detailed simulation will accurately predict the meanders of the Gulf Stream current, but it requires a simpler theory from geophysical fluid dynamics to understand why such strong ocean currents, in the northern hemisphere, all appear on the eastern shore of continents. This feature is a fundamental aspect of ocean circulation to which the detailed shapes of the coastlines (carefully encoded into Earth simulators) are irrelevant. 

An effective model must therefore strike a balance between predictive and explanatory power: where exactly the line is drawn depends on the needs and aims of the modeller (several reasons for building a model other than prediction are discussed by Epstein in his engaging lecture Why Model?). It is this aspect of modelling which has caused it to be described as an art. Like other art forms, modelling is best learned through practice, ideally by apprenticeship (during a PhD perhaps). However, it also helps to think about the principles underlying the art of modelling and to revisit these from time to time as one works on concrete problems. An excellent discourse on modelling is given by Rutherford Aris (one of the early pioneers, along with Neal Amundson, of the use of mathematics in chemical engineering) in his book Mathematical Modelling Techniques. The dover edition of this book comes along with several of Aris's elegant papers on modelling. One of these that is both instructive and entertaining is Re, k and /pi : a conversation on some aspects of mathematical modelling (App. Math Model., 1(7), 386, 1977). His Danckwert's memorial lecture article, Manners makyth modellers (Chem. Eng. Sci., 46(7), 1535, 1991), is another nice exposition on the subject, wherein he concludes by discussing similarities between the practice of mathematical modelling and poetry.

On a similar note, I would like to draw a parallel between fantasy fiction and modelling, in terms of the way they help clarify and deepen our understanding of the world. J.R.R. Tolkein, whose Lord of the Rings (LOTR) has captured the imagination of generations of readers, rejected the popular notion of his time that fantasy fiction should be restricted to the children's nursery. In his essay On Fairy-stories, he argues that encountering aspects of of the world and of human experience in a fantastical setting, helps us to understand them from a fresh perspective. We can then translate this understanding back to our world. For example, in LOTR, the corruptibility of Power is a theme that is given vivid expression through the device of the One Ring, while Frodo's remarkable resistance to the Ring highlights the virtues of a Hobbit's simple life and his devotion to his friends. Similarly, Tolkein's Ents---ancient and majestic sentient trees---renew our appreciation for that old oak or banyan tree that we unthinkingly walk past everyday. Mathematical models similarly emphasize selected aspects of a system, while suppressing others, in the idealized, "fictional" world of the model. Another striking parallel is the requirement for intellectual self-consistency. Fantasy worlds, Tolkein says, should be believable; the peoples in it should have consistent histories and philosophical outlooks that tie into the story and explain their relationships to each other and the world (as exemplified in LOTR); magic if introduced should be logical, within its own defined system, and not used as an ad-hoc literary device. Only if the world is believable will the reader be able to invest in the characters and be moved by their story. Similarly, a mathematical model must also be self-consistent; approximations should be uniformly applied; the model's conclusions should not violate the underlying assumptions; the behaviour of the model in limiting cases should accord with the conceptual picture it was designed to represent. Only then can an exploration of the model's behaviour lead to meaningful insights. So, if you enjoy good fantasy fiction, you are likely to appreciate the beauty of simple mathematical models; and if you haven't read LOTR yet, it may make a good complement to your study of modelling.

Now, on to some practical aspects of mathematical modelling. Once a model is formulated, the next challenge lies in exploring its predictions and deducing underlying principles. Most phenomena of interest are governed by nonlinear equations which means, typically, that the model will not be amenable to standard analytical methods of solution. Of course the equations can be solved numerically, and this is becoming easier with continuing advances in computer hardware and computational software. However, numerical solutions come with a significant caveat: they offer limited insight and it is difficult to verify their correctness. Thankfully there are several techniques for educing meaningful qualitative information from an equation without actually solving it. These methods, which can help verify numerical solutions, are a standard part of a modellers toolbox and include regular and singular perturbation methods, symmetry transformations, graphical constructions such as nullclines, and coarse numerical calculations. These and other useful approaches are discussed by Aris in some detail along with examples in Mathematical Modelling Techniques. A nice summary is also presented by him, as 13 maxims for modellers, in How to get the most out of an equation without really trying (Chem. Engg. Edu., 10, 114, 1976). Another excellent book by Aris is Mathematical Modelling: A Chemical Engineers Perspective, in which he discusses some nice pedagogical problems and walks the reader through several of his seminal papers. The book ends with an autobiography of Aris which makes for fascinating reading, given his deep scholarship, wide range of interests, and non-traditional yet extremely successful career.

An aspect of modelling that deserves special mention is scaling and nondimensionalization. The purpose of scaling equations is to represent the magnitude of quantities in terms that are intrinsic to the system, which have physical significance, rather than in terms of arbitrary (from the problem's perspective) units of measurement (Aris points out that poetic imagery plays the same role in poetry, cf. Manners makyth modellers). This allows one to estimate the relative importance of terms in the equation and guide the process of approximation. Moreover, the nondimensionalization process reduces the number of parameters to a minimum, clearly revealing the dimensionless numbers which govern the system's behaviour. For example, scaling the incompressible Navier-Stokes equations for flow in a pipe leads to a system of equations with just one parameter: the Reynolds number. Thereby, one immediately sees that the predictions of the equations will be the same for fluids of different viscosities and densities, provided the channel dimensions are chosen so as to maintain the same value of the Reynolds number. A discussion of scaling is available in Aris's Mathematical Modelling Techniques. The first paper to treat this topic independently is Simplification and Scaling by Segel (SIAM Rev., 14(4), 547, 1972). The classic text Mathematics Applied to Deterministic Problems in the Natural Sciences by Lin and Segel also has a nice chapter on scaling. A more recent text which motivates the discussion with many examples from transport problems is Scaling Analysis in Modeling Transport and Reaction Processes, by William Krantz.

Another powerful technique is perturbation or asymptotic analysis, which enables us to find approximate solutions of nonlinear problems by taking advantage of small non-dimensional parameters in a model (these typically arise from the existence of processes with very different length or time scales). Two classic texts on the subject are Perturbation Methods by Hinch and Advanced Mathematical Methods for Scientists and Engineers by Bender and Orszag. A textbook that unifies several different asymptotic methods for multi-scale dynamical systems, within the framework of the center-manifold theorem, is Model Emergent Dynamics in Complex Systems by Tony Roberts. The book also shows how to use modern symbolic computing software (like Mathematica) to ease the, otherwise tedious, analytical calculations involved in deriving asymptotic solutions.

Apart from texts that focus on modelling in general, one can also learn a lot about modelling and its methods from books targeted at specific subjects. One such text is Advanced Transport Phenomena by Gary Leal, which takes the reader through several example problems in fluid dynamics and scalar transport, working out each step of the modelling and analysis process in detail, while taking care to explain the thought processes involved. A few more such text books, as well as articles on modelling, from practitioners in different fields are listed below:

- AC Fowler, The philosopher in the kitchen: the role of mathematical modelling in explaining drumlin formation, GFF, 140:2, 93-105, 2018.
- GK Vallis, Geophysical fluid dynamics: whence, whither, why? Proc. R. Soc. A., 47220160140, 2016.

A century ago the application of mathematics to understand and solve problems was predominantly restricted to physics and some aspects of chemistry. About half a century ago, modelling had started to make inroads into various engineering disciplines. Today, the impact of mathematical modelling has spread into almost every discipline, including the social sciences (Generative Social Science, Epstein). Biology has emerged as the exciting new frontier for the application of mathematics, with biological problems even inspiring the creation of new mathematics (The Mathematics of Life by Ian Stewart). There are several popular science books available which document the exciting breakthroughs achieved, in a wide range of fields, by the application of mathematics. Some of these are listed below:

- Fearful Symmetry, by Ian Stewart and Martin Golubitsky.
- Chaos, by James Gleick (a classic in this genre).

Philosophy

Mathematics has a rich and fascinating history and many deep ideas underlie its philosophy. Reading about both helps one to better appreciate the subject and its place in science and in human thought in general. A broad and engaging book is The Mathematical Experience by Davis, Hersh and Marchisotto. An indirect though possibly more concrete and certainly more entertaining route to understanding the evolution of mathematical thought is via biographies of famous mathematicians. In the Presence of the Creator, by Gail Christianson is an excellent biography of Newton. It describes how even Newton struggled in his first encounters with mathematics, but kept working on it over and over until he mastered the material. The intensity with which Newton worked is extraordinary. Another must-read biography is that of David Hilbert (by Constance Reid), after whom is named the function space that all fluid dynamicists love. Fermat's Last Theorem by Simon Singh gives a fascinating glimpse into the mind of a mathematician (Andrew Wiles) struggling with a deep problem and also illustrates how mathematical ideas take shape and evolve over time.  

While it is true that some branches of mathematics developed in tandem with physical science, the major portion of mathematics used in science today was actually developed without any thought to application. Rather, as indicated by the books mentioned above, most mathematics are motivated by the intrinsic beauty of their subject and choose areas of work based on purely aesthetic and intellectual considerations. However, time and again, such purely mathematical constructs have found application in unexpected areas of the natural sciences, leaving us to wonder at the "unreasonable effectiveness of mathematics" (Eugene Wigner, 1960). Furthermore, as Poincare points out in Science and Method, the most effective and useful methods (for science) often turn out to be based on particularly beautiful mathematics.

Indeed, a significant part of the joy of theoretical research derives from the sheer beauty and elegance of the mathematical equations that describe nature. In the context of fluid mechanics, one cannot help but be struck by the rich variety of phenomena described by the Navier-Stokes equation---from simple laminar flow all the way to chaotic turbulence. The equation is full of subtleties, like the singular nature of the viscous diffusion term that gives rise to boundary layers and the turbulent-dissipation anomaly. Considering free-surfaces introduces additional intriguing aspects, like surface tension forces that stabilize a planar interface but destabilize a cylindrical one. The concept of mathematical beauty and its role in the development of scientific theories is elegantly discussed in the article, Beauty and the Quest for Beauty in Science, by the great astrophysicist S. Chandrasekhar. His book Truth and Beauty expands further on these ideas, and discusses aesthetics and creativity in science in relation to that in music and literature. 

Some other good books that shed light on the nature and practice of pure and applied mathematics are listed below, in no particular order:

- A Mathematician's Apology, by G.H. Hardy.
- Letters to a Young Mathematician, by Ian Stewart.
- The Equation that Could'nt be Solved, by Mario Livio.