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I am in the pure math camp but was invited to referee an applied/interdisciplinary paper because I'm a specialist in the underlying mathematical tool. I want to ask for general guidance about evaluating an applied math paper compared to a pure math paper. For reference, the authors seem to be serious and well-published people, although only one is affiliated with a mathematics department.

About the paper itself: The main mathematical work comprises writing out a bunch of (fairly lengthy) equations that are meant to govern the process being modeled, then walking through a computational procedure (implemented outside the paper) to find approximate solutions, along with some supporting examples and figures. There are no theorems, proofs, or formal definitions, or really any serious attempt to justify the model, which has an "ad hoc" feel to it. The paper is loose on technicalities (e.g., minimal attention to the regularity of functions), yet at the same time uses lots of fancy jargon.

For these reasons, I'm not particularly enthusiastic about the paper, although I imagine it is par for an applied paper and I want to avoid any improper bias on my part. What are reasonable expectations and standards that I should apply?

Also: I am only asked to review the portions of the paper within my expertise. That said, I would hope to give some sort of opinion regarding the paper's merit.

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    $\begingroup$ I don't think you should try to pretend you're an applied mathematician in your review (and in particular, knowing nothing about the paper in question I don't think that serious applied mathematicians like things that are super ad hoc or an over-reliance on fancy jargon). You should give your professional opinion on the paper, but explain two things: 1. the parts of it that you feel qualified to review and the parts that you don't; and 2. that you're a pure mathematician, so a cultural outsider to this area whose opinion should be interpreted as such. $\endgroup$
    – Andy Putman
    Commented Nov 10 at 2:25
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    $\begingroup$ Many journals have guidelines for reviewers. You could also ask the editorial board member who invited you to review the article for specifics on what they're looking for. (I still think this is a reasonable question though and would be interested in seeing public answers.) $\endgroup$
    – Sam Hopkins
    Commented Nov 10 at 2:26
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    $\begingroup$ By the way, you are not obliged to accept all referee requests, and when I get requests that are really far from things I am interested in I often decline them. In particular, I have to at least somewhat care about a paper before I'm willing to look closely at it. $\endgroup$
    – Andy Putman
    Commented Nov 10 at 2:27
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    $\begingroup$ “Applied mathematics” covers an extremely broad range of fields, encompassing (in my impression) a wider range of paper structures/expectations than pure maths does. A typical paper on quantum error-correction looks extremely different from a paper in compiler verification, which in turn looks different from a paper in mathematical biology or financial modelling. Could you give a little more guidance, e.g. en example of a somewhat comparable paper (close enough to illustrate the general form without breaking confidentiality)? Otherwise I don’t think this is really answerable. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Nov 10 at 10:55
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    $\begingroup$ @mdr My guess is that the editors do expect you to point out where the paper is "loose on technicalities." If they have asked you to be a referee, they probably want you to evaluate the rigor according to pure mathematics standards (even if they subsequently decide that they are willing to accept a lower standard of rigor). The other concerns you listed are also worth mentioning, but the editors can probably evaluate those aspects of the paper themselves. (I say all this as a pure mathematician who has spent most of his working career with applied mathematicians and engineers as colleagues.) $\endgroup$
    – Timothy Chow
    Commented Nov 11 at 14:09

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