Survey papers on the role played by PDE in mathematics 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.
 A: &bullet; 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.

&bullet; 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 :)
A: *

*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.
