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The 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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Preface (J.F. Nash, Jr., M.Th. Rassias).- Introduction (M. Gromov).- 1. P Versus NP (S. Aaronson).- 2. From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond (O. Barrett, F.W.K. Firk, S.J. Miller, C. Turnage-Butterbaugh).- 3. The Generalized Fermat Equation (M. Bennett, P. Mihailescu, S. Siksek).- 4. The Conjecture of Birch and Swinnerton-Dyer (J. Coates).- 5. An Essay on the Riemann Hypothesis (A. Connes).- 6. Navier Stokes Equations (P. Constantin).- 7. Plateau´s Problem (J. Harrison, H. Pugh).- 8. The Unknotting Problem (L.H. Kauffman).- 9. How Can Cooperative Game Theory Be Made More Relevant to Econimics? (E. Maskin).- 10. The Erdös-Szekeres Problem (W. Morris, V. Soltan).- 11. Novikov´s Conjecture (J. Rosenberg).- The Discrete Logarithm Problem (R. Schoof).- 13. Hadwiger´s Conjecture (P. Seymour).- 14. The Hadwiger-Nelson Problem (A. Soifer).- 15. Erdös´s Unit Distance Problem (E. Szemerédi).- 16. Goldbach´s Conjectures (R.C. Vaughan).- 17. The Hodge Conjecture (C. Voisin).

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"It presents a collection of open research problems in pure and applied mathematics, but each article is written by a different specialist. ... certainly you may find some appealing and engaging presentations in this collection of problems that somehow got the attention of John Nash." (Felipe Zaldivar, MAA Reviews, maa.org, November, 2016)

"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. ... This is a wonderful book whose papers give us a guided tour along the heroic battle fields of mathematics." (Adhemar Bultheel, European Mathematical Society, euro-math-soc.eu, August, 2016)

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