## Open Problems in MathematicsThe goal in putting together this unique compilation was to present the current status of the solutions to some of the most essential open problems in pure and applied mathematics. Emphasis is also given to problems in interdisciplinary research for which mathematics plays a key role. This volume comprises highly selected contributions by some of the most eminent mathematicians in the international mathematical community on longstanding problems in very active domains of mathematical research. A joint preface by the two volume editors is followed by a personal farewell to John F. Nash, Jr. written by Michael Th. Rassias. An introduction by Mikhail Gromov highlights some of Nash’s legendary mathematical achievements. The treatment in this book includes open problems in the following fields: algebraic geometry, number theory, analysis, discrete mathematics, PDEs, differential geometry, topology, K-theory, game theory, fluid mechanics, dynamical systems and ergodic theory, cryptography, theoretical computer science, and more. Extensive discussions surrounding the progress made for each problem are designed to reach a wide community of readers, from graduate students and established research mathematicians to physicists, computer scientists, economists, and research scientists who are looking to develop essential and modern new methods and theories to solve a variety of open problems. |

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### Contents

1 | |

Pair Correlation Statistics and Beyond | 123 |

The Generalized Fermat Equation | 172 |

The Conjecture of Birch and SwinnertonDyer | 207 |

An Essay on the Riemann Hypothesis | 224 |

A Quick Reminder and a Few Remarks | 259 |

Plateaus Problem | 272 |

The Unknotting Problem | 303 |

The ErdősSzekeres Problem | 351 |

Novikovs Conjecture | 376 |

The Discrete Logarithm Problem | 403 |

Hadwigers Conjecture | 417 |

The HadwigerNelson Problem | 439 |

Erdőss Unit Distance Problem | 458 |

A Historical Perspective | 479 |

The Hodge Conjecture | 521 |

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3SAT abelian algebraic arithmetic circuits Boolean boundary chromatic number circuit lower bounds class number cohomology combinatorial complexity theory convex convexly independent defined denote diagram discrete logarithm elliptic curve embedded equation equivalent Erd˝os Euler example exists explicit exponential Fermat finite field formula geometry given graph G Hodge class Hodge conjecture Hodge structures homology homotopy implies input isomorphism Jones polynomial K-theory knot L-functions Lemma linear manifolds Math Mathematics minimal surface modular Mulmuley multiplication natural proofs neutron NEXP Novikov conjecture NP-complete number theory Open Problems oracle P D NP P=poly plane points polynomial-time algorithm prime number proved PSPACE quadratic quantum random matrix theory Reidemeister moves result Riemann Hypothesis Riemann zeta function Sect smooth solutions space subset there’s topological Turing machine unit distance unknot upper bound Vassiliev invariants version in Proc vertex vertices zeros zeta function