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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.

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Nonsmooth calculus
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by Juha Heinonen PDF
Bull. Amer. Math. Soc. 44 (2007), 163-232 Request permission


We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singular spaces where calculus is possible parallels the classical development from smooth functions to functions with weak or generalized derivatives. Moreover, there is a new way of looking at the classical geometric theory of Sobolev functions that is useful in more general contexts.
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Additional Information
  • Juha Heinonen
  • Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
  • Email:
  • Received by editor(s): November 7, 2005
  • Received by editor(s) in revised form: June 16, 2006
  • Published electronically: January 24, 2007
  • Additional Notes: This paper constitutes an expanded version of the AMS invited address given by the author in Boulder, Colorado, in October 2003
    The author is grateful for the support and hospitality of MSRI and UC Berkeley, where the bulk of this paper was prepared during a visit in 2002-2003. Supported also by NSF grants DMS 0353549 and DMS 0244421.
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Bull. Amer. Math. Soc. 44 (2007), 163-232
  • MSC (2000): Primary 28A75, 49J52, 53C23, 51-02
  • DOI:
  • MathSciNet review: 2291675