The great mathematical problems

The popular science writer Ian Stewart tackles the Great Problems of Mathematics, i.e., the conjectures that gained fame because they remained unsolved for many years or that are still unsolved. They are important since they often generate a whole new body of mathematics so that prestigious prizes are rewarded for their solution. Stewart describes the origin, the history, and the development of some of these. It is not easy, but Stewart at least tries, with some success I must say, Stewart largely succeeds in transferring the ideas without the plethora of formulas and difficult mathematical concepts that one would expect. He also illustrates that often a breakthrough is triggered by the development in a seemingly unrelated piece of mathematics.

The first topic is the Goldbach conjecture (1742): every integer larger than 2 is the sum of 2 primes. Stewart takes his time to introduce prime numbers, their history and their computation. This pays because several of the subsequent problems are related to prime numbers as well. He explains what has been shown and what is not, how it has been linked to the Riemann conjecture and the open problems left for the present and future generations.

Squaring the circle is a problem that dates back to the Greeks and is clearly linked to $\pi$. This is a nice illustration how an obviously geometric problem, is reformulated as an algebraic one, solving polynomial equations, leading to real and later complex analysis (to prove that $\pi$ is irrational and transcendental). Much more recent is the four colour problem (1852) Again, the solution is not important for cartography because there are many other reasons to choose a colour for the map, but it started research is networks and graphs, and it has been generalized to colour problems on much more complex topological surfaces. It was finally proved in 1976 by Appel and Haken and it revolutionized the concept of a proof, since it was the first time that a proof relied on the verification by a computer of many cases to which the problem was reduced. Too many to be checked by humans.

Although the original 1611 version of Kepler's conjecture appeared in his booklet on 6-pointed snowflakes, it is asking how dense spheres can be packed. Every grocer knows how to mount oranges in a pyramid, but it took 387 years for a proof to be found. Again, a computer was needed to solve the global optimization problem. A formal proof avoiding the computer is still an ongoing project. The Mordell conjecture (1922) was proved by Faltings in 1983. It is about Diophantine equations but has a geometric formulation stating that a curve of genus $g>1$ over $\mathbb{Q}$ has only finitely many rational points. Stewart uses this on his path towards Fermat's last theorem, since one may start from Pythagoras equation $x^2+y^2=z^2$ with integers to the equation of a circle $(x/z)^2+(y/z)^2=1$ with rational numbers and subsequently to a generalization where the circle is replaced by an elliptic curve.

Although Poincaré got the award of the Swedish King Oscar II in 1887, he did tot really solve the originally posed three body problem that was suggested by Mittag-Leffler. Nowadays there are numerical techniques to solve equations with a chaotic solution approximately, rigorous proofs and many questions remain unanswered. The answers are directly related to fundamental questions about the stability of our solar system.

Back to prime number distribution with the Riemann hypothesis (1859). Again number theory is lifted to complex analysis in the study of the ζ-function (Stewart needs quite some pages to come to this point). It is explained how this leads to the conjecture that the zeros of the ζ-function are on the critical line $x=1/2$ in the complex plane. This is one of the most famous open problem in mathematics today. It survived Hilbert's 1900 unsolved problems and it is reformulated as one of the Clay millennium problems. Most mathematicians believe it to be true as numerics seem to indicate but a proof is still missing.

The other six millennium problems are the subject of the following 6 chapters, in which the mathematics that Stewart needs become tougher. The Poincaré conjecture (1904) was solved by Perelman in 2002, but because it took 8 years for the math community to verify his proof, the introvert and eccentric Perelman, totally disappointed with that situation, has withdrawn from mathematics and refuses all contact with the media. He declined the EMS prize (1996), Fields Medal (2006), and the Clay millennium prize (2010).

The P/NP problem is still open and the outcome is uncertain: are hard problems such as the traveling salesman problem solvable with polynomial time algorithms? The answer to this question seems to be NP-hard itself.

Solving the Navier-Stokes equation is a problem from applied mathematics. Can one verify that the small changes made by numerical procedures don't miss some turbulent solution because the approximation is not fine enough. In January 2014, Otelbaev claims to have solved this problem. The proof is still under review. The mass gap hypothesis relates to quantum field theory of elementary particles. These quantum particles have a nonzero lower bound for their mass even though the waves travel at the speed of light. In relativity theory, the mass would be zero. The Birch-Swinnerton-Dyer conjecture is another millennium problem about rational solutions of certain elliptic curve equations. Finally the Hodge conjecture connects topology, algebra, geometry and analysis to be able to say something about algebraic cycles on projective algebraic varieties.

 

Although Stewart tries very hard to introduce the unprepared reader to the problems and the techniques for the latter four problems, the much more advanced mathematical needs make these chapters definitely harder to read than the earlier ones. As a conclusion, he gives his own opinion of what will and what will not be proved in the (near) future. Just in case the reader gave up on the Riemann hypothesis and is looking for inspiration to find another really challenging problem, Stewart provides a list of 12 somewhat less known open questions that are as yet unsolved.

Stewart's entertaining style, his meticulous sketching of the historical context, his sharp analysis of the importance and consequences, his broad insight in the wide spectrum of mathematics and, being a mathematician himself, his understanding of the human behind the mathematician, struggling for solutions and recognition, makes this book a very interesting and highly recommendable read.

Reviewer: 
A. Bultheel
Affiliation: 
KU Leuven
Book details

Ian Stewart, in his well known style, guides us through the history and the developments of the great conjectures of mathematics, the approaches undertaken and the kinds of mathematics that were developed in the attempts (and successes) to solve them. Not only the Goldbach and Riemann conjectures and Fermat's last theorem, but also several others, including the Clay Mathematical Institute millennium prize problems.

Author:  Publisher: 
Published: 
2013
ISBN: 
978-1-84668-1998 (pbk)
Price: 
£9.99 (pbk)
Pages: 
352
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