Open Problems in Mathematics

1. Front Matter

   Pages i-xiii

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2. \(P\mathop{ =}\limits^{?}NP\)

   • Scott Aaronson

   Pages 1-122

3. From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond

   • Owen Barrett, Frank W. K. Firk, Steven J. Miller, Caroline Turnage-Butterbaugh

   Pages 123-171

4. The Generalized Fermat Equation

   • Michael Bennett, Preda Mihăilescu, Samir Siksek

   Pages 173-205

5. The Conjecture of Birch and Swinnerton-Dyer

   • John Coates

   Pages 207-223

6. An Essay on the Riemann Hypothesis

   • Alain Connes

   Pages 225-257

7. Navier Stokes Equations: A Quick Reminder and a Few Remarks

   • Peter Constantin

   Pages 259-271

8. Plateau’s Problem

   • Jenny Harrison, Harrison Pugh

   Pages 273-302

9. The Unknotting Problem

   • Louis H. Kauffman

   Pages 303-345

10. How Can Cooperative Game Theory Be Made More Relevant to Economics? : An Open Problem

    • Eric Maskin

    Pages 347-350

11. The Erdős-Szekeres Problem

    • Walter Morris, Valeriu Soltan

    Pages 351-375

12. Novikov’s Conjecture

    • Jonathan Rosenberg

    Pages 377-402

13. The Discrete Logarithm Problem

    • René Schoof

    Pages 403-416

14. Hadwiger’s Conjecture

    • Paul Seymour

    Pages 417-437

15. The Hadwiger–Nelson Problem

    • Alexander Soifer

    Pages 439-457

16. Erdős’s Unit Distance Problem

    • Endre Szemerédi

    Pages 459-477

17. Goldbach’s Conjectures: A Historical Perspective

    • Robert C. Vaughan

    Pages 479-520

18. The Hodge Conjecture

    • Claire Voisin

    Pages 521-543

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.

• John Nash Jr. publication
• open problems mathematics
• generalized Fermat equation
• quantum systems
• Riemann hypothesis
• Navier Stokes equations
• Novikov's conjecture
• Hadwiger's conjecture
• partial differential equations

“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)

• Department of Mathematics, Senior Research Mathematician, Princeton University, Princeton, USA

  John Forbes Nash, Jr.

• Department of Mathematics, Princeton University, Princeton, USA

  Michael Th. Rassias

John Forbes Nash, Jr. was Senior Research Mathematician at Princeton University. Professor Nash was the recipient of the Nobel Prize in Economics in 1994 and the Abel Prize in Mathematics in 2015 and is most widely known for the Nash equilibrium in game theory and the Nash embedding theorem in geometry and analysis. He was also the recipient of the John von Neumann Theory prize in 1978. Nash's groundbreaking works in game theory, algebraic & differential geometry, non-linear analysis and partial differential equations have provided insight into the factors that govern chance and events inside complex systems in daily life. Moreover, Nash's theories are widely used in economics, computing, evolutionary biology, artificial intelligence, accounting, politics and other disciplines.  

Michael Th. Rassias during his collaboration in 2014-2015 with John Forbes Nash, Jr. for the preparation of this book was a postdoctoral researcher at the Departments of Mathematics of Princeton University and ETH-Zurich, working at Princeton. He is currently a postdoctoral researcher at the Institute of Mathematics of the University of Zurich and a visiting researcher at the Program in Interdisciplinary Studies of the Institute for Advanced Study, Princeton. His research interests lie in mathematical analysis and analytic number theory. Rassias has received several awards in national and international mathematical Olympiads. He was also awarded the 2014 Notara Prize in Mathematics from the Academy of Athens. He has also authored, co-authored and co-edited five other books with Springer.

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