Submissions to journals

Journal-specific information:

• Submissions to Journal of the American Mathematical Society: Please use the online submission form; this will be faster and more direct than submitting the paper to me.  I will assume that the first author listed is the corresponding author.  Some additional submission information can be found here.  Note that JAMS sets very high standards for its articles (comparable to Annals or Inventiones), and in fact rejects 80%-90% of its submissions, including many good and otherwise publishable research papers.  If you do not think it likely that your peer reviewers will rate your submission as within the top 10%-20% of papers in the field, then I would not recommend sending your submission to JAMS, in the interests of rapid publication of your paper (it generally takes at least six months for JAMS to reach a decision on a submission).   We are currently experiencing severe backlog problems, and thus are declining all papers longer than 50 pages (except in truly extraordinary circumstances) and will be applying particularly stringent standards to papers just below this threshold.  In particular, we cannot guarantee that any given paper, even one of the highest quality, will necessarily be accepted for publication for JAMS at this time.  [NB: please do not offer to artificially compress your paper below the 50 page limit, or to otherwise satisfy the letter of this rule rather than its spirit.  At this point in time, a lengthy paper is probably best processed by a journal other than JAMS.]  For submissions regarding major unsolved problems (Riemann hypothesis, Navier-Stokes, Goldbach, etc.) please read on to the bottom of this page. 
• Submissions to the American Journal of Mathematics: Submission by email (in DVI, PS, or PDF format) is greatly preferred and will be processed much faster than a print submission.  Please specify “AJM” in the subject header and cc: your submission to ajm@math.jhu.edu as they will handle all the technical details of the submission process.   AJM is a generalist journal; papers of an overly specialized and technical nature should be sent to a more focused journal (e.g. Dynamics of PDE). 
• Submissions to Dynamics of Partial Differential Equations: Once again, submission by email (in DVI, PS, or PDF format) is greatly preferred.  Please specify “Dynamics of PDE” in the subject header and cc: your submission to an editor-in-chief such as Charles Li (cli@math.missouri.edu).
• Submissions to International Mathematical Research Surveys: IMRS articles are by invitation only; we do not accept unsolicited submissions. 

General advice on submissions:

• Proofread and double-check your paper before submission.  Nobody, least of all referees, enjoys a paper which is badly written and full of errors.  Even if the result itself is first-class, a cavalier attitude to expository or accuracy issues (or the presumption that it is solely up to the editor and referee, rather than the author, to address these issues) will negatively impact both the referee reports on the paper, and the editor’s final decision on the paper.  In particular, having an error discovered after the submission process, which then necessitates sending a revised version onward to the referee, tends to tie up a lot of time and thus slow down an already sluggish process.  Investing some time before submission to try to eradicate these sorts of problems (and to give the paper one last polish, in the meantime) is usually well worth it; it earns the goodwill of the editor and referee, thus speeding up the published process, and also improves the final product. 
• Submit a final draft, not a first draft.  This is a corollary to the previous point.  If you are still finding typographical or other errors in the paper while reading it, or if you are still adding results and commentary to the paper, then it is not yet time to submit the paper to a journal.  Even if you are for some reason in a hurry to publish your paper, a poorly prepared paper will have a greater chance of rejection, may incur demands for major revisions, and will in general slow down the processing of your paper to the extent that it would have been faster to wait until the paper was in a more final form before submission.
• Use the introduction to “sell” the key points of your paper.  Every now and then, I see an author upset at a rejection of a paper because the referee “clearly did not grasp the key point of the paper”.  In many cases this is because the key point is not stated prominently enough in the introduction, instead being buried in a footnote, an obscure remark, a lemma, or even not explicitly mentioned at all.  This can be as much the fault of the author as it is of the referee; it is incumbent on the author to state as clearly as possible what the merits, novelties, and ramifications of the paper are, and the fact that an expert in the field could read the introduction and not see these is a sign that the introduction is not yet of publication quality.  In particular, the introduction should spend some time comparing and contrasting the paper to other literature, and demonstrate why the paper’s results and techniques are new, interesting, and/or surprising given this context.  For instance, if new difficulties had to be resolved here which were not present in previous work, or if counterexamples indicate that the result or proof cannot be improved in various obvious directions (e.g. by dropping a hypothesis, strengthening a conclusion, or by using a simpler method in the literature), then these points need to be made prominently.  The introduction should also clearly state (or at least paraphrase) the main results of the paper, and ideally should also outline how and where these results are to be proved. 
• Submit to an appropriate journal.  It is always tempting to submit a paper to a prestigious journal, but if the paper makes only a borderline case for publication in this journal, then the net result may be a lengthy process, critical reviews holding the paper to a very high standard, and ultimate rejection of the paper.  For instance, with JAMS, a paper really has to do something that makes referees excited and enthusiastic; a paper which is merely a good, solid application of mostly standard techniques to solve a moderately interesting problem will unfortunately have a rather low probability of being accepted into JAMS, even if it would have been readily published elsewhere.  Similarly, a journal devoted to research mathematics is unlikely to accept any paper whose primary focus lies in recreational mathematics, physics, philosophy, biology, computer science, or anything else outside the scope of research mathematics.
• Write professionally.  (This advice is mainly for first-time authors.)  A journal article is going to be a permanent record of your work; thus it is important that these articles are as professionally written as possible, so as not to cause embarrassment several years from now.  Thus the majority of the paper should be objective and factually based; informal remarks and opinions are permissible, but should be clearly labeled as such to distinguish them from formal, rigorous assertions.  (For instance, they can be placed in footnotes.)  Overly philosophical, witty, or otherwise “clever” comments should generally be avoided (they may not seem so clever to you ten years from now).  Papers need to have a properly formatted title, abstract, introduction, and bibliography.  The references should be current, showing all recent related work; even if these works are not directly used in your paper, a comparison between your approach and others in the literature is expected.  These references should be cited within the text whenever appropriate, giving an accurate assignment of credit, provenance, and precedence.  The standard format for mathematical papers is TeX, AMSTeX, or LaTeX; other formats such as Word or Mathematica can cause technical difficulties (and will ultimately need to be converted to a TeX format), and so should be avoided.  Spelling and grammar should be checked, especially if the language used (most probably English) is not one’s native language; in such cases, one might consider using spell-checking software.  
• Organize the paper.  Some thought should be given as to the logical layout of the paper; a stream-of-consciousness format, in which results are presented in the order in which they occurred to the author, are generally a very bad idea, as they project an image of carelessness and are hard to follow (or enjoy) by readers or referees.  For instance, peripheral results which are not strictly necessary to the main argument should be moved to remarks, footnotes or discussion sections.  Results which are necessary to the main argument, but which are very different in nature from the rest of the paper (e.g. they use material from a different field of mathematics, or consist entirely of dull computations), may be placed in an appendix.  Each time there is a major turning point in the argument, or a shift to a different component of the argument, one should start a new section; conversely, a collection of closely related facts should probably be placed within a single section. Whenever possible, each major milestone in the argument should be formalized in a self-contained and prominently located proposition or theorem, in order to facilitate a “high-level” understanding of the argument, and to allow the reader to mentally divide the argument into simpler non-interacting components.  Try to group related sections together; thus for instance, the statement and proof of lemma X should ideally be placed close to where lemma X is actually used (especially if the lemma is only used once in the entire paper).  Of course, executing all of these suggestions may require some initial planning and thought, and possibly some significant reshuffling of the paper from its first draft, but with modern text editors (and especially with LaTeX, which has automatic theorem numbering and similar tools to facilitate this reshuffling) this is not as difficult as it used to be, and it does significantly increase the readability (and hence influence) of your paper.
• Motivate the paper.  A paper should not just be a sequence of formulae or logical steps.  It should also be organized and motivated in such a way that the reader is always aware what the near-term and long-term objectives of any portion of the paper are, how the current arguments are advancing towards these goals, how crucial they are to those goals, and why the claimed results at each step are at least plausible (or if they are surprising, to indicate exactly why and how they are surprising).  Informal, heuristic, or motivational reasoning is therefore very welcome, but should be clearly indicated as such to distinguish it from formal, rigorous reasoning (for instance, these portions of the paper can be placed in remarks or footnotes).   At the start of each section, it is often a good idea to give a brief paragraph describing the purpose of that section.  For instance, if a section is devoted to proving a key milestone in the paper, the milestone can be stated near the start of the section, next to a discussion as to why this milestone is important and perhaps a brief sketch as to how one is going to prove it in this section. 
• Use good notation.  Good notation can make the difference between a readable paper and an unreadable one.  Such notation should emphasize the most important parameters and features of a mathematical expression or statement, while downplaying the routine or uninteresting parameters and features.  For instance, if one does not care much about the exact values of constants in estimates, then notation which conceals these constants (such as \lesssim, \ll, or O()) are useful; conversely, these notations should be avoided if the precise values of these constants are of importance to the paper.  Notation which is used globally should be defined in a notation section near the front of the paper, or in the introduction; notation which is only used locally (e.g. within a single section, or within a proof of a single lemma) should be defined close to where it is used (possibly with a reminder that this notation is not used elsewhere in the paper; this is helpful when there are many sections, each with their own extensive notation.  Note that notation or statements which are introduced within a proof of a lemma are already understood to be localized to that lemma; it is bad form to then recall that notation or statement outside of that lemma, except perhaps as a remark or as motivation).  In some cases it is worthwhile to define the notation once near the start of the paper, and then recall it whenever necessary.  One should strive to make one’s choices of notation compatible and consistent with notation already in the literature, so that the readers who are already familiar with prior notation will adapt easily to your paper and will not be confused.  There is an issue of where to strike the balance between too little notation and too much notation.  A good rule of thumb is that any expression or concept which is used three or more times will probably benefit from introducing some notation to capture that expression or concept; conversely, an expression which is only used once probably does not need its own special notation.  (An exception would be for particularly crucial theorems or propositions in the paper; here it might be worthwhile to invest in some notation in order to make the statement of those theorems clean and readable.)
• Describe the results accurately.  A paper should neither understate nor overstate its main results.  If the main result is very surprising or a substantial breakthrough compared with the previous literature, these facts should be noted (and justified in detail, for instance by explicit comparison with prior results, examples, and conjectures).  Conversely, if there are unsatisfactory aspects to the result (e.g. hypotheses too strong, or conclusions a little weaker than expected) these should also be stated honestly and openly, e.g. “We do not know if hypothesis H is actually necessary”.  Similarly, it is worth noting down any interesting open questions remaining after your result. If you are using a famous unsolved conjecture (or the prior work of famous mathematicians) to motivate your own work, one should give a candid evaluation of the extent to which your work truly represents progress towards that conjecture (or is truly related to the prior work), so as to avoid the impression of “false advertising” or “name-dropping”.  If for some reason you need to assert a non-trivial statement without proof or citation, it should be made clear that you are doing so (e.g. “It can be shown that…” or “Although we will not need or prove this fact here…”), so that the reader does not then hunt through the rest of your paper for the non-existent justification of that statement.  Titles of sections should be descriptive (e.g. “Proof of the decomposition lemma” or “An orthogonality argument”), as opposed to uninformative (e.g. “Step 2” or “Some technicalities”). 
• Give appropriate amounts of detail.  A paper should dwell at length on the most important, innovative, and crucial components of the paper, and be brief on the routine, expected, and standard components of the paper.  In particular, a paper should identify which of its components are the most interesting.  Note that this means interesting to experts in the field, and not just interesting to yourself; for instance, if you have just learnt how to prove a standard lemma which is well known to the experts and already in the literature, this does not mean that you should provide the standard proof of this standard lemma, unless this serves some greater purpose in the paper (e.g. by motivating a less standard lemma).  Conversely, some computations which you are very familiar with, but are not widely known in the field, should be expounded on detail, even if these details are “obvious” to you due to your extensive work in this area.

• Michèle Audin's, "Conseils aux auteurs de textes mathématiques"
  
• Oded Goldreich's "How to write a paper".
  
• David Goss' "Some hints on mathematical style"
• Paul Halmos' "How to write mathematics" (the book also contains similar pieces by Dieudonne, Schiffer, and Steenrod)
  
• Ashley Reiter's "Writing a research paper in mathematics"
  
• Jean-Pierre Serre's "How to write mathematics badly"

As JAMS editor, I receive a large number of submissions regarding either famous open problems (e.g. Riemann hypothesis, Goldbach conjecture, Navier-Stokes regularity, etc.), or famous theorems (Fermat’s last theorem, Four-color theorem, Cantor’s theorem, Goedel’s theorem, etc.).  Such papers are held to an exceptionally high standard, and doubly so for a premier journal such as JAMS; extraordinary claims require extraordinary evidence, especially in view of the very many failed attempts to prove these types of problems.  In order to conserve limited refereeing resources, and to avoid possible embarrassment and damage to reputation for the submitter, I am thus imposing extremely strict quality standards on any such submission.  In order to even be sent to a referee, any such submission must

• Be fully proofread and checked to be free of errors of any sort (mathematical or otherwise);
• Be completely finalized in form;
• Adhere to all professional mathematical writing standards (see the previous section of this web page);
• Demonstrate full awareness of relevant recent literature.

Any submission which does not attempt to satisfy these requirements in good faith will be rejected without refereeing.  All such decisions will be final. I will not consider any further revisions or resubmissions beyond the first when it comes to these sorts of submissions; it has to be perfect the first time, or it will not be considered at all.

Due to many existing time constraints, I will be unable to assist any prospective submitters with help in improving their mathematical exposition.  If you are not a practicing research mathematician, my advice would be to first build up experience (and credibility) by working on less famous problems in the same area, in order to practice exposition skills, to learn basic techniques and literature, and to avoid common errors in the field.  You may also wish to seek out a professional mathematician in your local area to collaborate or discuss mathematics with.  See also my career advice page.