ISSN 0002-9920 (print)
ISSN 1088-9477 (online)of the American Mathematical Society
May 2016

In Memoriam
John Forbes Nash Jr.
page 486

The Quantum Computer Puzzle
page 508

Volume 63, Number 5

Open Problems in Mathematics

Photo Princeton University, Office of Communications,
Danielle Alio (2015).

Just before he left to collect his Abel Prize in Oslo in far from ‘classical’—it is something that brings about aMay 2015, Nash was working with Princeton postdoc dramatic alteration of our understanding of the basic
Michael Th. Rassias to finish up the preface to an ex- logic of analysis and differential geometry. Judging fromtraordinary book they edited together called Open Prob- the classical perspective, what Nash has achieved in hislems in Mathematics. The book will be published later
papers is as impossible as the story of his life… [H]isthis year by Springer.
work on isometric immersions…opened a new world of
The book consists of seventeen expository articles,
written by outstanding researchers, on some of the cen- mathematics that stretches in front of our eyes in yet
tral open problems in the field of mathematics today. unknown directions and still waits to be explored.”Nash and Rassias first met in September 2014 in theEach article is devoted to one problem or a “constellacommon room of the Princeton mathematics building,tion of related problems,”
Fine Hall. Nash was eightythe preface says. Nash andRassias do not claim thesix years old and probably
book represents all of thethe most famous mathemamost important problems in
tician in the world, and Rasmathematics; rather, it is “asias a twenty-seven-year-oldcollection of beautiful mathPrinceton postdoc who hails
ematical questions whichfrom Greece and had justwere chosen for a variety offinished his PhD at the ETH
reasons. Some were chosen
in Zurich. A chemistry develfor their undoubtable imoped between the two mathportance and applicability,
ematicians and precipitatedothers because they constitheir collaboration on Opentute intriguing curiosities
Problems in Mathematics. Awhich remain unexplained
Rassias talks to 2014 Abel Laureate Yakov Sinai
mysteries on the basis of
Princeton News article thatas 2015 Abel Laureate Nash looks on.
current knowledge andappeared on the occasion oftechniques, and some for more emotional reasons. Nash receiving the 2015 Abel Prize discussed RasAdditionally, the attribute of a problem having a some- sias’s interactions with Nash (www.princeton.edu/
what vintage flavor was also influential in our decision main/news/archive/S42/72/29C63/index.xml?secprocess.”tion=topstories). Rassias is quoted as saying: “WorkHere is another taste of the book, this one from theing with him is an astonishing experience—he thinks
introduction, titled “John Nash: Theorems and Ideas”
and written by Mikhail Gromov: “Nash was solving differently than most other mathematicians I’ve everclassical mathematical problems, difficult problems, met. He’s extremely brilliant and has all this experience.
something that nobody else was able to do, not even to If you were a musician and had an opportunity to work
imagine how to do it… But what Nash discovered in the with Beethoven and compose music with him, it’d becourse of his constructions of isometric embeddings is astonishing. It's the same thing.”

Table of Contents of Open Problems in Mathematics edited by John F. Nash Jr. and Michael Th. Rassias
Preface, by John F. Nash Jr. and Michael Th. Rassias
Introduction: John Nash: Theorems and Ideas, by Misha Gromov
P versus NP, by Scott Aaronson
From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond, by Owen Barrett, Frank W. K. Firk,
Steven J. Miller, and Caroline Turnage-Butterbaugh
The Generalized Fermat Equation, by Michael Bennett, Preda Mih˘ailescu, and Samir Siksek
The Conjecture of Birch and Swinnerton-Dyer, by John Coates
An Essay on the Riemann Hypothesis, by Alain Connes
Navier Stokes Equations: A Quick Reminder and a Few Remarks, by Peter Constantin
Plateau's Problem, by Jenny Harrison and Harrison Pugh
The Unknotting Problem, by Louis H. Kauffman
How Can Cooperative Game Theory Be Made More Relevant to Economics?: An Open Problem, by Eric Maskin
The Erdős-Szekeres Problem, by Walter Morris and Valeriu Soltan
Novikov's Conjecture, by Jonathan Rosenberg
The Discrete Logarithm Problem, by René Schoof
Hadwiger's Conjecture, by Paul Seymour
The Hadwiger-Nelson Problem, by Alexander Soifer
Erdős’s Unit Distance Problem, by Endre Szemerédi
Goldbach's Conjectures: A Historical Perspective, by Robert C. Vaughan
The Hodge Conjecture, by Claire Voisin
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