## Real Algebraic Manifolds

FIRST EDITION IN ORIGINAL WRAPPERS of one of Nash's most important papers.

Nash “was bent on proving himself a pure mathematician... Even before completing his thesis on game theory, he turned his attention to the trendy topic of geometric objects called manifolds. Manifolds play a role in many physical problems, including cosmology. Right off the bat, he made what he called ‘a nice discovery relating to manifolds and real algebraic varieties.’ Hoping for an appointment at Princeton or another prestigious math department, he returned to Princeton for a post-doctoral year and devoted himself to working out the details of the difficult proof.

“Many breakthroughs in mathematics come from seeing unsuspected connections between objects that appear intractable and ones that mathematicians have already got their arms around. Dismissing conventional wisdom, Nash argued that manifolds were closely related to a simpler class of objects called algebraic varieties. Loosely speaking, Nash asserted that for any manifold it was possible to find an algebraic variety one of whose parts corresponded in some essential way to the original object. To do this, he showed, one has to go to higher dimensions.

“Nash’s theorem was initially greeted with skepticism. Experts found the notion that every manifold could be described by a system of polynomial equations implausible. ‘I didn’t think he would get anywhere,’ said his Princeton adviser.

“Nash completed ‘Real Algebraic Manifolds,’ his favorite paper and the only one he concedes is nearly perfect, in the fall of 1951. Its significance was instantly recognized. ‘Just to conceive the theorem was remarkable,’ said Michael Artin, a mathematician at MIT. Artin and Barry Mazur, who was a student of Nash’s at MIT, later used Nash’s result to resolve a basic problem in dynamics, the estimation of periodic points. Artin and Mazur proved that any smooth map from a compact manifold to itself could be approximated by a smooth map such that the number of periodic points of period p grows at most exponentially with p. The proof relied on Nash’s work by translating the dynamic problem into an algebraic one of counting solutions to polynomial equations...” (Kuhn and Nasar, The Essential John Nash).

IN: Annals of Mathematics, Vol. 56., No. 3, pp. 405-421. Baltimore, MD: Princeton University Press, 1952. Octavo, original printed wrappers. Toning to spine and some light toning to rear wrappers; a hint of edgewear. An extremely well-preserved copy of one of Nash’s greatest work. RARE in original wrappers and without any institutional stamps.

Note:: A custom box is available for this item for an additional \$225.

Price: \$1,900 .

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