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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Geometric, algebraic, and analytic descendants of Nash isometric embedding theorems
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by Misha Gromov PDF
Bull. Amer. Math. Soc. 54 (2017), 173-245 Request permission


Is there anything interesting left in isometric embeddings after the problem had been solved by John Nash? We do not venture a definite answer, but we outline the boundary of our knowledge and indicate conjectural directions one may pursue further.

Our presentation is by no means comprehensive. The terrain of isometric embeddings and the fields surrounding this terrain are vast and craggy with valleys separated by ridges of unreachable mountains; people cultivating their personal gardens in these “valleys” only vaguely aware of what happens away from their domains and the authors of general accounts on isometric embeddings have a limited acquaintance with the original papers. Even the highly cited articles by Nash have been carefully read only by a handful of mathematicians.

In order not to mislead the reader, we try be open about what we do and what we do not know firsthand and to provide references to what is missing from the present paper.

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Additional Information
  • Misha Gromov
  • Affiliation: Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France; and Courant Institute for Mathematical Sciences, New York University, New York
  • MR Author ID: 77335
  • Received by editor(s): December 1, 2015
  • Published electronically: November 3, 2016
  • © Copyright 2016 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 54 (2017), 173-245
  • MSC (2010): Primary 58Dxx; Secondary 58Jxx
  • DOI:
  • MathSciNet review: 3619725