Open Problems in Mathematics

Description

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 Front Matter....Pages i-xiii \(P\mathop{ =}\limits^{?}NP\) ....Pages 1-122 From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond....Pages 123-171 The Generalized Fermat Equation....Pages 173-205 The Conjecture of Birch and Swinnerton-Dyer....Pages 207-223 An Essay on the Riemann Hypothesis....Pages 225-257 Navier Stokes Equations: A Quick Reminder and a Few Remarks....Pages 259-271 Plateau’s Problem....Pages 273-302 The Unknotting Problem....Pages 303-345 How Can Cooperative Game Theory Be Made More Relevant to Economics? : An Open Problem....Pages 347-350 The Erdős-Szekeres Problem....Pages 351-375 Novikov’s Conjecture....Pages 377-402 The Discrete Logarithm Problem....Pages 403-416 Hadwiger’s Conjecture....Pages 417-437 The Hadwiger–Nelson Problem....Pages 439-457 Erdős’s Unit Distance Problem....Pages 459-477 Goldbach’s Conjectures: A Historical Perspective....Pages 479-520 The Hodge Conjecture....Pages 521-543