what’s the hardest math problem

what’s the hardest math problem

Unsolved Mathematical Problems: Exploring the World’s Toughest Challenges

1. Introduction to Unsolved Mathematical Problems

The status of mathematical work done to solve basic problems is generally far more clear-cut. Control of basic ideas as well as interesting technicalities are what have defined or built much of mathematics. Control of difficult problems was, in general, very much oriented toward this end. On the other hand, unsolved problems have something of the mystique often associated with unsolved problems in science, and they are generally more ancient, as well. Despite this variety, the problems in this book have the following types of motivation: 1) that they are, and have been seen as, basic problems for a substantial length of time; 2) that their interest has not waned – they are either still frequently discussed, or lots of people with apparently no contact with the above class of old problem refers to old themes; and 3) that many different types of mathematics are brought together in the problems.

Why would anyone want to study unsolved problems? That is a hard question. Indeed, upon reflection, it is surprising that mathematical problems are ever solved at all. Yet, somehow, the great majority of the basic problems of mathematics have been solved, and it is the frontier problems which seem to be hard. What will be the role of these problems in mathematics? The reader will find many diverse opinions about the role of unsolved problems in other fields. For example, it is quite commonly supposed that the great unanswered questions of science – in physics and chemistry – are of some mystical importance, and biologists do not seem troubled by the fact that many aspects of evolution and embryology are poorly understood. This does not, however, suggest that study of frontier problems in other fields will not be fruitful. Indeed, some of our most solid scientific theories were developed in order to answer almost purely academic questions. Quantum theory is the classic example here.

2. Famous Unsolved Problems in Mathematics

Mathematics has an excellent way of moving, primarily away from any cutting-edge issues. The fundamental reason why true questions in mathematics are never answered is that they are irrational. What we do in mathematics is to state specific problems. Since the mid-1980s, there has been no work on ‘most raised’ famous problems, and it would seem that the list is ‘consolidated’ in some sense. Hilbert’s list is different from the rest, and there are several proofs. There are many books and articles about famous problems – what they are and how they remained unsolved. There are also books about general methods for tackling such problems and about the history of their development.

Mathematics is full of unsolved problems – challenges for mathematicians in the past and, in many cases, still unyielding to their attack in the present time. A number of these unsolved problems have been collected in the compendium of famous unsolved problems in mathematics. The first collection of this kind appeared in the 19th century. Today the most famous of these collections is the one created and maintained by David Hilbert, perhaps the greatest mathematician of his time. He presented his list at the beginning of the 20th century. Although David Hilbert was primarily a very practical geometer, his list includes all sorts of topics that today are viewed more as pure than as applied mathematics. According to Hilbert, all these problems have very deep, unification value.

3. Theoretical Frameworks and Approaches to Problem Solving

Philosophically, the goal of mathematics is to solve all mathematical problems. Thinking about such problems is actually a large part of mathematical research. At a certain point, all mathematics was unsolved. In the preface to Karnaugh (1958), the author describes Socrates’ premise that human greed has produced many artifacts yet there exists no end to material consumption. Less greedy are mathematicians – they never have enough solutions to problems. Some problems have infinitely many solutions; others have no solution. Some have a finite number of solutions but no known solutions. Meanwhile, new problems continuously emerge. Of course, not all unsolved problems are worth solving. However, mathematics would be so interesting if all its problems were already solved.

The unsolved mathematical problems provided in the introduction are fairly comprehensive. These are problems for which there exists no solution satisfying stringent conditions; opinions on which problems should be included in this class vary. This book consists of an introduction describing such unsolved problems from many perspectives and an annotated history and chronology of important mathematical concepts, highlighting unsolved problems. The history is only a summary, however, a pointer to various sources. Additional biographies were included in the hope of exposing more French contribution to the mathematical heritage.

4. Implications and Applications of Solving Hard Mathematical Problems

A related benefit from solving difficult problems comes from the collaborations that evolve from an effort to advance the state-of-the-art in a particular part of mathematics (or science). Obviously, the best talent is attracted to a field where one can prove theorems whose solutions escape popular imagination and application. The research identified in this volume was completed by 92 world-class mathematicians, actively engaged in the quest to expand research borders. Research students, academic staff, and other mathematicians worldwide also contributed comments and research on 10 World’s Toughest Problems identified in my 2002 book.

Apart from the personal satisfaction and acclaim that accrues to those who solve difficult problems, the mathematics community benefits in several ways. First, attention drawn to current unsolved problems stimulates research in a field and can provide overall cohesiveness as problems are solved. Results from fields where techniques were developed to address challenging mathematical problems have frequently been transferred to solve practical problems. More subtly, the methods and tools developed while attempting the solution of a hard problem can inspire a new way of approaching another problem that is similar in some ways. Indeed, sometimes the key insight supporting the solution of an outstanding problem comes from a field completely unrelated to the original topic.

5. Current Research and Future Directions

The problem I am most recently interested in is related to the Gibbs-Thomson effect and spinodal decomposition. Let me briefly relate the open question which is stated in formal definitions and propositions in the 20-page preprint by DeLellis, Szekelyhidi, and myself. The Gibbs-Thomson effect describes that in regions of high curvature, the interface of a liquid mixture will have lower free energy than in regions of lower curvature. This will, in general, have the effect that one large droplet melts before many small droplets with the same total volume. Spinodal decomposition refers to a completely different mechanism of dissipation of a binary liquid mixture described by the Cahn-Hilliard model. Our question is: Can the Cahn-Hilliard model describe the curvature-driven interface dynamics of the Ginsburg-Landau model (which, in turn, approximates the sharp interface limit of the Allen-Cahn equation)?