Open problems in mathematics [electronic resource] /
Nash, Jr., John Forbes.editor.
Rassias, Michael Th.editor.
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.
Includes bibliographical references at the end of each chapters.
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. Mihăilescu, 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 Erdős-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. Erdős's Unit Distance Problem (E. Szemerédi) -- 16. Goldbach's Conjectures (R.C. Vaughan) -- 17. The Hodge Conjecture (C. Voisin). .
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.
Number theory.
Algebraic geometry.
Differential geometry.
Partial differential equations.
Game theory.
Operations research.
Management science.
Number Theory.
Algebraic Geometry.
Differential Geometry.
Partial Differential Equations.
Game Theory, Economics, Social and Behav. Sciences.
Operations Research, Management Science.