hardest math problem

Unraveling the Mysteries of the Hardest Math Problems

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This article provides a survey of several problems that have attracted the interest of a number of elite mathematicians in the last century while still remaining open. We discuss some of their history, their vital statistics, why they are important, and what is known about each problem. The survey focuses on the seven problems known as the Millennium Prize Problems. These are problems that were posed by members of the Clay Mathematics Institute in the year 2000 and for which the institute maintains a $1 million reward for each solution. The seven problems pose a broad spectrum of mathematical challenges and are of interest across the entire mathematical community of problem solvers. The problems are (1) the Riemann Hypothesis, (2) the Birch and Swinnerton-Dyer Conjecture, (3) the Hodge Conjecture, (4) P versus NP, (5) the existence of Yang-Mills Connections, (6) the existence and smoothness of the Navier-Stokes Equations, and (7) the Riemann Hypothesis for Function Fields. Each of these problems will be the subject of an essay in this volume.

Mathematical problems have, for centuries, been a staple of human intellectual activity. Not only have they been mankind’s first encounter with much of modern mathematics, but mathematical problems are still being formulated today. In part, problems are interesting because of their etymology, that is, their history. We often study problems that were meaningful, advanced problems in Euler’s time and were naturally posed by the original, sometimes 18th-century, discoverer. Beyond their historical context, problems remain of interest to human beings because of their intrinsic mathematical structure. Often, the setting up of a problem requires the process of abstraction – the ability to see past the inessential details and focus on the essential structure. Almost as an epilogue, the actual problem itself becomes of interest.

Our theme of hardest open math questions, however, transcends the influences of mankind’s historical development. Many combinatorial problems can be geometrically oriented, so that, to some extent, the mathematics of the past may—or may not—influence the future. Likewise, understanding the discrete landscape will inevitably have implications for understanding our continuous world. Theoretical questions arise without direct practical roots and may cause practical ripples far beyond that which was originally envisioned. However, the way in which these questions are formulated often depends on cultural issues. However, time and again, it is the mathematical creation for the sheer beauty of formulating the process, the insights, and the solutions, rather than the practical consequences, that captures the creative talents of liberal-arts-educated people.

The historical evolution of the most difficult math problems has been shaped, in part, by their practical applications. Questions about the natural world have, unsurprisingly, always captured our curiosity. We wanted to predict events. When is the next growing season? Birth of the next lunar cycle? These questions evolved into practical knowledge about the natural world that, in turn, required systems of abstract laws to describe the universal phenomena around us—from classical mechanics to quantum physics. As humankind matured and came into more regular contact with one another, practical enforcement of social mores became important, requiring legal foundations to maintain social order. Interesting questions emerged regarding geometry and simple arithmetic. Rules of inheritance and trade required the development of algebra and number theory to model. To a smaller extent than, for example, the development of early civilizations, the push and pull between the mathematics of the abstract and the mathematics of the practical continues to mold current mathematical practice.

If a problem is hard or has been unsolved for a long time, then new approaches to it might also be needed: this might involve new ways to use the tools one already has or the surgery of the problem to reduce its difficulties in such a way that one is then able to see a solution. There are many approaches in this direction that can be beneficial, and mathematicians have both natural instincts and leaden rules to help them do so. These include the method of turning a continuous into a discrete problem (whatever that might mean), the creative use of both the symmetries of the original problem and the violation of such symmetries that arise in attempts to resolve it, and attempts to produce a reduced version of the problem, constructing the solution then stepping back up towards the original problem. The road may be bumpy, and a disproportional amount of effort inserted to obtain some decrease of dimensionality in the problem’s complexity, but reductionism and separation of time-scales are a powerful tool in the hands of humans and may make everything accessible.

The six major strategies in this paper are general and require nontrivial experience and expertise to correctly apply. They have, however, the advantage that they manifestly apply to a very wide range of problems. But they may be unsuitable when the overall guide is quite clear but problems which fit the search are difficult to find. In such situations, an alternative approach to devising new ideas may aid problem-solving and converge on other approaches: in what follows, we consider other methods, symmetries, and generalizations.

Theorems are specific, demonstrably true mathematical statements. Proofs describe techniques for demonstrating the truth of statements. Mathematicians view the identification, design, and development of proofs as a creative, consummately human enterprise; the truth of a proof, once demonstrated, enables mathematicians to probe ever-deeper questions in the discipline. In a handful of instances, a proof has fully penetrated into the statement, demonstrating, deepening, and magnifying the truth of connections between concepts posited. In the six problems chosen, deep connections between diverse areas of mathematics, both within mathematics and at its boundaries, are both surprising and part of the core inheritance of currently understood mathematical results. Unsolved, these problems pose challenges heretofore not encountered and hint at remaining untapped richness in the interconnectedness of mathematical ideas.

All scientific disciplines share the vexation of outstanding major unsolved problems, the solution of which will advance beyond expectations in stunning ways the state of the discipline. Among the small but deep collection of outstanding unsolved problems in mathematics, only a few are easily compellingly stated to the uninitiated. The six unsolved problems chosen here are of particular historic significance. Solved, each of the six problems will deepen the collective understanding of core mathematical notions, the interconnectedness of mathematical ideas, and the extent to which powerful tools can be brought to bear to penetrate deep problems. Also, each unsolved problem is of major scientific import, going to the very heart of the modern view of mathematics as a living, vibrant field, whose ideas and methods continue to develop and intersect, broadening and deepening our collective understanding of algebraic and analytic structure.

The mathematical culture of problem-solving aspires to capturing students’ hearts and enthusiasm, building on a person’s ability to persevere, rely on one’s subjectivity, believe in one’s own perception, engage emotionally with the search for answers and an understanding of the phenomena. By working out intricate mathematical questions, people aim to locate a place in the world of mathematical creation, tend to find contacts between interior processes and creative looking for the unknown, rely on life principles not devoid of glamorous feeling, like permanence, eternity, and harmony, and also communicate with the unknown by formulating questions and awaiting their resolution. These are also reasons for the creative scientist’s dilettantism: to share the pleasure of solving intricate issues.

The qualitative and quantitative study of the unique problem-solving experiences that befall students is still in its infancy, and the serious study of the generation of advanced mathematical content is also a relatively little-treaded area. Theoretical consideration of how learners might participate in mathematical inquiry and in the advancement of mathematics has been seriously lacking. Only if we understand the intricacies of the problem-solving process in mathematics will we be able to provide appropriate curricular support for the creative endeavors of intelligent students interrelating with rich mathematical domains.

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