A Mathematical Odyssey. Journey from the Real to the Complex
It may happen that mathematicians are confronted with the question what the heck they are doing all day except for computing pi since the Greeks had everything written down and there is nothing to be invented. One and one is two and that's all there is to it. With this book Krantz and Parks try to give a nontrivial answer to counter this opinion. Just like Homer's Odyssey describes the adventurous return of the hero Odysseus, presumed dead after the Trojan war, yet eventually slaying Penelope's suitors, this Krantz and Parks book reports on some astonishing successes of mathematicians in the past century, showing that mathematics is far from being dead. With these fourteen examples, the authors illustrate that mathematics is still very much alive and that it has great influence on our lives. Here is the list of the topics.
The four-color problem. Since its original formulation in 1875 many well known mathematicians had tried to find a proof that 4 colors were sufficient to color any map so that no two neighboring countries would have the same color. It was with a computer-assisted proof by Appel and Haken in 1976 and after many corrections and an acrimonious debate about this being a mathematical proof or not that in 1981 the Math Department of the University of Illinois acclaimed that Four Colors Suffice. By now this kind of proof has been accepted and it was applied in other situations, but long after 1981 alternative proofs were obstinately looked for.
Mathematics of finance is approached with a long history starting with the origin of interest, compound interest, before arriving at stock markets with the Dutch East India Company. Next, the reader is introduced to financial terms like derivative (a future contract), forward, option, arbitrage, call option, and most importantly how to determine the price of, for example, a call option. Several stochastic models existed already but the grand entrance was for the Black-Scholes equation for which Scholes and Merton got the Nobel prize in 1997 (Black died in 1995). However as the 1987 crash illustrated, models can and need to be improved, which is an ongoing topic of investigation for many mathematicians currently employed in the financial sector.
Ramsey theory named after F. Ramsey (1903-1930) may be a bit less known. Ramsey was convinced that absolute chaos did not exist. A typical question to answer is "How many elements of some structure must there be to guarantee that a particular property will hold?" For example P. Erdős formulated the following variant "How large should N be so that in a room with N people, at least k people are mutually acquainted or at least k are mutually unacquainted?" Of course the idea is to find the smallest number N. The inconceivably large Graham's number (powers of powers of powers...) is an upper bound for an equivalent problem in graph theory: If N points in the plane are connected with either a red or a blue line (nodes are acquainted or not), how large should N be so that either m of them are red or n of them are blue? The Ramsey number R(m,n) is the smallest value of N, and is, except for some simple cases, only known to be in some interval.
Dynamical systems are better known in a wider audience, via the popular Mandelbrot and Cantor sets, fractals, the Lorentz attractor, and the butterfly effect. Visual effects and chaotic effects that can astonish and fascinate the observer. A three dimensional Mandelbrot set, the Mandelbulb, is used an an illustration on the cover of the book.
The Plateau problem of minimal surfaces spanning a closed curve was originally posed by Lagrange in 1760, but it became popular and was named after the Belgian J. Plateau (1801-1883) who observed that soap films were good approximations to such surfaces. Plateau made several observations on curvature and the angle between soap bubbles where they meet. A formula was derived by Enneper and Weierstrass and Costa, Hoffman and Meeks designed a minimal surface that did not intersect itself. A beautiful shape that has inspired several artists. One of the first Fields medals were awarded to J. Douglas for his new approach for the solution of the problem, although an obscure but ill understood paper by Garnier preceded his. Proofs of Plateau's observations were given by J.E. Taylor in 1976 and generalizations of the problem are still under investigation.
Non Euclidean geometry arises when we leave the 5 Euclidean postulates. Exploring this possibility was a consequence of the efforts to show that the last postulate (only one parallel line is possible through a point outside a given line) was independent of the others. It stimulated J. Bolyai and N. Lobachevsky to come up with hyperbolic geometry where it is possible to draw at least two distinct parallel lines through a point. The consistency was only proved by Beltrami in 1868. It was B. Riemann who revolutionized the way we now think of geometry and Riemannian geometry is what was used by Einstein. Calabi's conjecture of the existence of some nice Riemannian metric on a complex manifold was proved by Yau (1977) which earned him a Fields Medal, and nowadays Calabi-Yau manifolds are an essential tool for theoretical physicists working on string theory.
Special relativity is another topic that is not too difficult to explain. Einstein got a Nobel Prize in 1921 especially for his discovery of the photoelectric effect. A major campaign was set up to award this Prize also to H. Poincaré (who, among many other things, developed relativity theory parallel to Einstein) but because, as Mittag-Leffer states, the Nobel committee "fears mathematics because they don't have the slightest possibility of understanding anything about it", Poincaré never got it.
Wavelets revolutionized Fourier analysis and they have a remarkable track in mathematics. Morlet, a geophysicist, came up with an alternative for the windowed Fourier transform, and with the help of Grossmann, a theoretical physicist, developed a theory of frames. Y. Meyer, a mathematician specialized in harmonic analysis, got hold of their papers and this was the start of a new approach to the domain. I. Daubechies later developed an orthogonal set of wavelets, which can only be described by an algorithm. The time between the mathematical formulation and the extensive applications in engineering (e.g. jpeg compression code) is remarkably short as compared to many other mathematical ideas. This is a relatively short chapter, discussing more the history of Fourier analysis while it is rather short on the wavelet stuff itself.
RSA encryption, named after Rivest, Ahamir and Adleman, is another widely used application. Most probably your browser uses RSA whenever you open a https website, where personal or private information is exchanged. The RSA code is freely available and has never been compromised in the 30 years that it has been around. It is remarkable that it completely rests on an old and simple idea of prime numbers. In particular the difficulty to find large prime factors is the fact that makes it work.
The P/NP problem follows immediately from the previous topic. P indicates the class of problems that can be solved in polynomial time, i.e., the execution time is a polynomial in the size n of the problem. NP however stands for nondeterministic polynomial, which means that it can be checked in polynomial time that a given solution is indeed solving the problem. It is generally believed that the class NP is strictly larger than P, but that has not been proved as yet. The chapter elaborates extensively on automata, Turing machines, and formal languages.
Primality testing is again related to the two previous problems. It was not until the AKS algorithm of 2002 by Agarwal, Kayal, and Saxena who generalized some ideas of Fermat (his little theorem) to get a polynomial time algorithm for primality testing, which places this problem in the class P. In the foundations of mathematics it is explained that a mathematical proof traditionally consists of a sequence of statements, in principle derived from some axiomatic system following some rules of (formal) logic. As seen in the first chapter, nowadays the computer can play an essential and active role in the concept of a proof, by algebraic computation or verifying an extensive set of possibilities. The whole chapter is building up via an introduction to formal logic to Gödel's incompleteness theorem.
Wiles' proof of Fermat's last theorem of 1995 is one of the last triumphs of mathematics in the 20th century. An introduction is given to polynomials over finite fields, elliptic curves, the Taniyama-Shimura-Weil conjecture, and Frey curves, to explain how Wiles, by proving Serre's form of the Frey conjecture actually proved Fermat's last theorem.
Ricci flow and Poincaré's conjecture are connected in the proof by G. Perelman that he published in three papers in the years 2002 and 2003 on arXiv. The conjecture dates from 1904 and says that every 3-dimensional surface on which a closed curve can be continuously deformed to a point is homeomorphic to a 3-sphere. It was one of the Clay Millennium Problems. It is of great importance because it relates to relativity theory and the shape of our universe. Hamilton introduces differential equations generating Ricci flows that can be considered as geometric evolution equations that result eventually in the geometric objects predicted by Thurston in his geometrization program to describe a possible classification of all n-dimensional manifolds. The mathematical community did not accept Perelman's proof immediately. Certainly too long to his taste and, deceived in the mathematical establishment, Perelman retired from the Steklov Institute and from mathematics. He was awarded the Clay Millennium Prize and the Fields Medal which he both declined.
Although there is some relation between some of the 14 subjects, each chapter can be read independently. The text is intended for the layman, but some knowledge and affinity with mathematical concepts is advised to help assimilate the material. Some average undergraduate mathematics from secondary school should suffice, and to really enjoy the texts, the reader should not have an aversion of mathematics. The chapters differ a bit in style and extent. Sometimes it starts with an extensive account of the very early history to introduce the topic, some are more mathematical than others, but they obviously serve the same purpose. The treatment is usually not very deep. It's just enough to enlighten the readers and give them an idea of what the topic is about and by whom and how the problem was eventually solved, and what impact it may have on society. It is also made clear that as mathematics advances, the problems can only be solved by applying interdisciplinary techniques. That obviously requires much more collaboration and hence ease of communication between mathematicians to finally crack the problem. Every chapter ends with a short list of references, but it is noteworthy that these are every time preceded by a section called "A Look Back". Why this is done is not very clear, but it seems to collect the crumbles of what has not been mentioned before. Most often that section contains some historical remarks or side stories about some of the main players in the theory or some future perspectives or generalizations. All in all, a book that is in the tradition a Krantz's previous books, which gives the non-mathematician an idea of what mathematicians get so passionately involved with and how that has resulted recently in successes.