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There are already several questions on MathOverflow that inquire about the many diverse relationships between PDE and several other 'areas' of mathematics (e.g., algebraic and differential geometry and topology, number theory, harmonic analysis, probability theory, dynamical systems, etc.); however, most of the answers give only a few particular examples.

The aim of this question is to collect a [big-list] of references (i.e., broad surveys or monographs) that specifically focus on the role played by PDE in various other areas of mathematics, or on methods "stemming from other topics" that are used in the analysis of PDE.

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    $\begingroup$ This question is impossibly broad. In some scientific subjects (much of physics, for example) the fact that the basic models are formulated in terms of ordinary or partial differential equations completely permeates the literature, to the point where this fact is not even mentioned explicitly. $\endgroup$
    – Igor Khavkine
    Commented Jul 25, 2015 at 12:04
  • $\begingroup$ @IgorKhavkine, you're clearly right. I'll edit the question to narrow its scope a bit. $\endgroup$
    – user75426
    Commented Jul 25, 2015 at 12:55
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    $\begingroup$ "...the fact that this PDE result also gives the Poincaré conjecture and the more general geometrisation conjecture makes it (again in my opinion) the best piece of mathematics we have seen in the last ten years. It is truly a landmark achievement for the entire discipline." - from arxiv.org/abs/math/0610903 $\endgroup$
    – Sam Hopkins
    Commented Jul 25, 2015 at 13:15
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    $\begingroup$ much of this is covered here: mathoverflow.net/questions/74183/… $\endgroup$
    – Carlo Beenakker
    Commented Jul 25, 2015 at 13:32

2 Answers 2

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• Geometry in Partial Differential Equations (A. Pràstaro, Th.M. Rassias)

This book emphasizes the interdisciplinary interaction in problems involving geometry and partial differential equations. It provides an attempt to follow certain threads that interconnect various approaches in the geometric applications and influence of partial differential equations. A few such approaches include: Morse-Palais-Smale theory in global variational calculus, general methods to obtain conservation laws for PDEs, structural investigation for the understanding of the meaning of quantum geometry in PDEs, extensions to super PDEs (formulated in the category of supermanifolds) of the geometrical methods just introduced for PDEs and the harmonic theory which proved to be very important especially after the appearance of the Atiyah-Singer index theorem, which provides a link between geometry and topology.

• For the question "What connections are there between number theory and partial differential equations?" see this MSE thread. (In brevity the answer given there as a comment can't be beaten :)

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  • $\begingroup$ The book by Pràstaro and Rassias is indeed a very interesting read. Thanks for your answer. $\endgroup$
    – user75426
    Commented Aug 13, 2015 at 16:46
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  • Klainerman, Partial Differential Equations, in Princeton Companion to Mathematics; a longer version is freely available here;
  • Klainerman, PDE as a Unified Subject;
  • Brezis and Browder, Partial Differential Equations in the 20th Century;
  • Yau, The Role of Partial Differential Equations in Differential Geoemtry;
  • Evans, Weak KAM Theory and Partial Differential Equations;
  • Stroock, Partial Differential Equations for Probabilists;
  • etc.
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