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Recently it has been brought to my attention that the subject "geometric analysis" is not even well-defined (unlike the subject partial differential equations, algebraic geometry, etc). Can someone give me a pointer what is a rigorous definition of geometric analysis? It seems to me geometric analysis is not even a subject, but a loosely connected network of analytic techniques used in differential geometry. Is this true? If not, what are the central distinguishing elements that made it an independent subject worth of study?

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    $\begingroup$ for the pragmatic minded, there is a good enough definition :-) $\endgroup$
    – Willie Wong
    Commented Apr 28, 2017 at 3:04
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    $\begingroup$ Your question is somewhat ill-defined. What do you mean by "subject" and "well-defined"? Clearly geometric analysis makes enough sense to the people that use the term. If you articulate more clearly what it is you are looking for, perhaps people can help you. $\endgroup$
    – Ryan Budney
    Commented Apr 28, 2017 at 4:36
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    $\begingroup$ I hardly believe that "the subject partial differential equations" or "the subject algebraic geometry" are well-defined. It always depends on your point of view. For example, is arithmetic geometry part of algebraic geometry or of number theory? Many areas of research are in the intersection of different subjects and if person A tells you that person B is doing algebraic geometry, this tells you almost as much about person A as about person B. Therefore I think that the best you can hope for is a list of descriptions of what people think geometric analysis is, not an objective definition. $\endgroup$
    – Helene Sigloch
    Commented Apr 28, 2017 at 9:05
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    $\begingroup$ @RyanBudney: I mean something written down as "Definition:....". You can define PDE to be the subject related to solving partial differential equations. I do not know how to define geometric analysis. $\endgroup$
    – Zhexiu Tu
    Commented May 1, 2017 at 16:07
  • $\begingroup$ I think it is a very good question. The "official" definition of geometric analysis pointed out by Willie Wong is not necessarily what many people think geometric analysis really is. I think my research is in geometric analysis, but I never dare to mention geometric analysis in my NSF proposals, because I do not want to be placed in the panel mentioned by Willie Wong. I always end up in the "Geometric Function Theory" panel, although I really think my research should be called "Geometric Analysis". I think a honest discussion about meaning of geometric analysis would be a good thing to do. $\endgroup$
    – Piotr Hajlasz
    Commented May 14, 2018 at 22:28

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