Open Problems in Mathematics

  The idea to compile this book started in 2014 from an originally casual conversation between the editors about the status of some open problems in mathematics. John Nash was at that time already 86, but he insisted to be actively involved in the selection of the subjects and the preparation of the volume. When the volume was almost ready in 2015, they wrote the preface, and shortly afterwards Nash left for Oslo where he was to receive the Abel Prize. Unfortunately he did not see this book published because he and his wife died in a taxicab accident in New Jersey on the way back home from Norway. Besides a short introduction by Misha Gromov about Nash's scientific ideas, several of the other papers refer to interests and work of Nash and the whole volume is dedicated to him.

The objective of the book was clear already in 2014 when the idea took off. Since Hilbert formulated his 24 problems in 1900 some of them were solved, some were reformulated in 2000 as one of the Clay Mathematical Institute's Millennium Prize Problems (MPP). Among the most notorious problems that feature on both lists we recognize the Riemann hypothesis. Another holy grail is the P vs NP problem that is more recent and is one of the toughest MPP. All other MPPs are discussed here too except the Poincaré Conjecture (solved by Perelman in 2010) and the Yang-Mills Problem about the mass gap in elementary particle physics. Of course these famous lists do not cover all the open problems since many different lists have been distributed and open problem need not be in any list to be an open problem of course. The editors of this book made a selection of problems they consider important and they have invited experts to write survey papers that give the state of the art of the problem or of the conglomerate of related problems, the historical attempts made to solve them, the methods currently applied, etc.

The book immediately opens with a strong and extensive survey of 123 pages on the notorious P/NP Problem. The problem was explicitly formulated by Cook and Levin in the early 1970's although its roots are usually assigned to Gödel in 1956. Here we learn that Nash already in 1950 gave a formulation. It has a meta-character since if one could prove that P = NP then it is in principle possible to write a program that formally solves all of the other Millennium Prize Problems too. The paper gives arguments in favor of P = NP and others in favor of P ≠ NP but the general belief is currently that P ≠ NP. Although the Turing machine is an essential element in the precise definition of the classes P and NP, it is not explained (think of any existing programmable digital computer), but the long list of all the different complexity classes is introduced (a glossary is given in an appendix) and theorems are formulated (no proofs) stating what inclusions hold for the respective classes. The larger part of the paper introduces many approaches and concepts of complexity theory like lower bounds, different barriers, oracles, etc. It only shows that so far nobody seems to have a clue on how to tackle the general problem. Some think the problem is just too difficult, but the author, Misha Gromov, is not so pessimistic. Even if the solution is still far away, high up there at the top of the mountain, so much insight is found already in the low vegetation that is being explored just now, that it justifies all the effort invested.

The other survey papers are somewhat shorter (on average some 20-30 pages) which doesn't mean they are less interesting. Some problems are well known too. The Riemann Hypothesis is one of them and several popularizing books on the subject are available. Alain Connes chose to be somewhat restrictive in his discussion and he describes three approaches to explicit formulas based on geometry (Riemannian, algebraic, and tropical geometry).
The Riemann Hypothesis is related to the distribution of prime numbers and involves zeros of the zeta function. The Generalized Riemann Hypothesis is about the spacing between prime numbers and requires the generalization of the zeta function to Dirichlet L-functions. In the corresponding survey paper this is linked with energy levels of quantum systems. This exposes a very surprising underlying universality that is still unexplained. The paper shows how random matrices is a valuable approach to tackle this problem. With its quantum physics component this paper comes in the neighborhood of the Yang-Mills Theory.
The Birch-Swinnerton-Dyer Conjecture (BSD) is another famous MPP. It asks about rational points on an elliptic curve and the relation with yet another kind of L-functions related to the zeta and the Dirichlet L-functions. The formulation of the problem and some recent results are given.
The Generalized Fermat Equation is again a number theory problem. It is a generalization of the famous Fermat's Last Theorem, solved by Wiles in 1993. Mathematicians have the urgent need to formulate and work on a generalization for every problem they solved. This problem asks for relative prime values $x$, $y$, $z$ and integers $p$, $q$, $r$ satisfying $x^p+y^q=z^r$. This survey concentrates on approaches for the case $1/p+1/q+1/r<1$.
The Goldbach Conjecture is another old number theory problem (every even integer larger than 2 is the sum of two primes). Different approaches used in the long history are briefly reviewed as well as closely related problems and generalizations.

More algebraic is the number theory problem on discrete logarithms. Note that $\mathbb{Z}/p\mathbb{Z}$, with $p$ prime and excluding zero fprms a multiplicative group. Hence for every $x$, there is some integer $a$ such that $x=g^a$ with $g$ a primitive root modulo $p$. This $a$ is then called the discrete $g$-logarithm of $x$. The problem is important for cryptography since it is difficult to compute this logarithm when $p$ is very large. Some algorithms to compute a discrete logarithm are discussed that link to elliptic curves and finite fields. However, so far no polynomial algorithm is known to compute a general discrete logarithm.

The remaining MPPs that are discussed include the Navier-Stokes Problem and the Hodge Conjecture. Navier-Stokes is about the existence and smoothness of the solutions of this equation of fluid dynamics. The problem is briefly recalled and linked to related problems. The Hodge Conjecture is a problem of algebraic topology. Although the paper is introductory, it requires some knowledge of topology for better understanding.
The Novikov Conjecture is another topology related problem. It says that higher signatures for smooth manifolds are homotopy invariant. The conjecture has intimate links with geometry, operator algebras and representation theory.

Other problems discussed include the Plateau Problem about minimal surfaces (the spontaneous shape of a soap film attached to a wire frame). Many generalizations of the simple basic problem are described and progress of the last 100 years is surveyed.
Two problems are attributed to Erdős. The Erdős-Szekeres Problem where it is conjectured that for $n>3$ one needs at least $N(n)=2^{n-2}+1$ points in general position in the plane to make sure that you will always find among them $n$ points that form a convex $n$-gon. Higher dimensional analogs all still unsolved. The Erdős unit distance Problem is asking how many pairs one my find among $n$ points in the plane that are at a distance 1 from each other. When such points are vertices in a graph connected by an edge, graph theory can be a trail to solve it and its generalizations.
More graph theory is used in the unknotting problem (how to compute the knottedness of a knot?) and in two generalizations of the 4-color problem proposed by Hadwiger. The Hadwiger's Conjecture saying that every graph can be t-colored or has a subgraph that can be contracted to a complete graph with t+1 vertices. And the Hadwiger-Nelson Conjecture which also relates to one of the Erdős problems mentioned before. It asks for the chromatic number of the plane, that is the smallest number of colors needed to color the points in the plane such that no two points with the same color are at a distance 1 apart.

It was the intention of Nash to write himself a survey on cooperative game theory. Unfortunately this paper never got written, but Eric Maskin wrote three pages on the topic to formulate an open problem in this context.

A general observation with almost all problems discussed in this book is that even though the problem is posed in one subdomain, its solution often requires application of a completely different domain. For example the Riemann Hypothesis started from a prime number problem, but ended up to be the formulation about zeros of a complex function, which can be tackled for example via algebraic geometry. Sometimes one has to invent and explore a completely new branch of mathematics to make some progress. The book contains an enormous load of information about just 17 of the open problems in mathematics, their history, relationships, generalizations, and approaches taken. Although the papers are written independently, some of the problems are related so that a global name or subject index might have been an appreciated feature. The aim of the surveys is usually to reach a general mathematical public, but the papers are not always easy reading if one is not familiar with some basics of the problem. They all have a very extensive list of references that can be very useful if you want to start digging somewhat deeper. Needless to say that you will not easily find a solution. The problems are hard and have resisted many different attacks for a longer time. The good point is that there are still open problems (and there are many more 'out there') and if they are hard, then they usually generate new mathematics as long as they are not solved. This is a wonderful book whose papers give us a guided tour along the heroic battle fields of mathematics. 

Adhemar Bultheel
Book details

The history, generalizations, and approaches taken in attempts to solve 17 open problems in mathematics are described in invited survey papers that should bring the reader to the current state-of-the-art. Seventeen such open problems were selected among which five of the Millennium Prize Problems. The selection was made by the editors. John Nash passed away shortly before the volume was ready, but he actively contributed to it and it reflects his interests. It is also a partial tribute to him. 

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978-3-319-32160-8 (hbk)
137.79 € (hbk)