This is a guide to writing up a math paper in the field of Geometric Analysis. Although it focuses on the final writeup, I do breeze through the earlier stages of conducting math research. - Christina Sormani
Stage 1 Start the Project:
1A: Find a problem (often recommended by a doctoral or postdoctoral advisor).
1B: Read up key existing papers related to the problem (often recommended by a doctoral or postdoctoral advisor). It is important to ensure that you know the latest work that has been completed in the area of the project by searching on mathscinet and the arxiv. If your advisor permits you to speak to other mathematicians about questions related to the project they might also be a resource as to what recent results have been completed in the area. Workshops and Conferences also keep you up to date as to what is known.
1C: Proposal: In some settings one must write up a research proposal for the project. It is essential that this proposal include a summary of related literature both as a description of what is done and as an explanation why the project is important. Even if a proposal is not required it is worth writing one.
!D: Advisors: Be sure to stay in close contact with the advisor to report your progress and ensure you are not too far off from the planned project. It can be exciting to explore new directions tangential to the original project but you must be sure not to veer into something that has already been done or that has been attempted for fifty years using the same methods you are now trying. To work on an old unsolved conjecture you need a new technique. Most advisors have a research program in mind with a variety of related projects to assign their students and postdocs who then benefit from learning the advisor's techniques.
Stage 2: Completing the Project
2A: Break up the Project:
As you complete the research, break up the project into lemmas. Often in geometry we have a vision of how a proof will proceed. That vision can be translated into a series of numbered and labelled lemmas which must be proven to reach the goal. Write the statements precisely into a tex file with warnings that some are NOT PROVEN. This is then a tex file that you can share with your advisor regularly so that they can see your progress towards the goal. Some projects are more open ended, but even so, write each proven fact as a lemma, and then write in the proof in a more or less detailed way. Ultimately the proofs must be correct for each lemma, but you may not keep all the lemmas in the final version.
2B: Caution:
Only speak with those you and your advisor trust about the project. If someone gives you a lot of ideas then they become a coauthor. That may not be permitted for a dissertation.
2C: Rigor:
Be sure to write in all the proofs of your lemmas rigorously checking all possible cases. Some people have a second tex file which only contains rigorously proven material. I prefer to continue with the same tex file carefully marking what is proven or not.
Stage 3: Finalizing the Project;
3A: Is it Worthy? You have completed the proof of a theorem or two and you may now decide to write it up as a paper. Check that your advisor feels it is good enough to publish or if more should be done first. Sometimes you may not feel it is worth anything but your advisor feels it is strong enough. Usually in that case it is strong enough to publish but you can discuss with the advisor whether it is strong enough to publish in a good journal, if it is worth improving on, or if maybe it is best to finish quickly and continue with a new paper. Delays in publication can lead to others completing the same project first, so do not delay. Sometimes, especially before a job hunt, a paper may be posted on the arxiv as long as it is rigorous even if it is not as good as you would like. It might be replaced by a better version before submitting to a journal. This is part of the worthiness assessment. Note that some parts of your project that are incomplete may be postponed either to be included later before submission to a journal or put in a different paper altogether. Once you choose what will be in the paper: create a new file for that paper.
3B: Examples: It is good now to add a section of examples to your paper clarifying how each hypothesis of each theorem is essential. That is, if your theorem says hypothesis A and hypothesis B together imply conclusion C, you should add examples where hypothesis A or B fails but the other hypothesis holds and conclusion C fails. These can be very well known examples that you fins easily, or through a search of the literature, or it might take some time to find an example. If there is no time to find an example then you add a numbered remark stating whether you think such an example exists and might be found and how. Examples stated as labelled and numbered examples must have proofs rigorously demonstrating they are indeed examples as claimed. Sometimes additional lemmas may need to be proven in order to complete the proof of an example.
3C: Context and References: It is time to go back to the literature and redo the proposal adding papers that you missed before and new ones that may have appeared since you began the project. This will be repeated again after posting on the arxiv and presenting the work.
3D: Applications: Does your result have natural corollaries? If so include them even if they are just new proofs of existing theorems. You may wish to add a section at the end of the proofs you wrote up in Stage 2 that is just for the applications.
3E: Future Work: Keep a list of ideas as to how this work may be expanded upon or applied in the future. Some of these you might include in your paper (especially if you are not interested in implementing them yourself) but many you might only reveal in future research proposals.
3F: Writing the Abstract: The abstract contains a rough description of your key theorems and a sentence or two of context. It is aimed at a complete expert.
3G: Writing the Introduction: The first 1-3 paragraphs should give a grand context: the big picture of the field and essential definitions often the same as in your research proposal. You should then state your theorems as numbered theorems. You should choose your most exciting and easily read theorem to state first as a numbered theorem. Sometimes this is one of your corollaries. Below that statement, you can have a paragraph mentioning the examples "example 2.3 has hypothesis A but fails hyp B and the conclusion". You may also point to a remark. You should also add a paragraph mentioning related results by those who have similar theorems explaining how yours is different. You will then state your next theorem and repeat this process. Some theorems are highly technical and might just be stated inside the paper with a description in the introduction explicitly mentioning the number of the theorem. At the end of the introduction, give an outline of how the paper proceeds. At this point you may include a brief description of the proof mentioning the works you have applied.
3H: The citations: it is important to provide the last names of the authors the first time you mention a citation (Schoen-Yau proved in [3] that....). You should repeat the names again even if you also have another citation by the same set of authors. But if you repeatedly mention the same paper you do not need to mention their names again if its on the same page. Try to cite the original sources of the theorem not just a textbook although you might add (see also \cite{textbook}) when the original source is not clear. If you are applying a specific theorem or lemma from a textbook, provide the number of that theorem or lemma. If it is a key result you might choose to include it written out in full in a background section. Note that the act of looking up the citations is also verifying that you have correctly applied them.
3I: The background: If your paper is using advanced techniques that are not well known and might require a reader to run to the library and study for hours before following your paper, then you need a background section. Otherwise it is not necessary. The background section includes the key definitions and theorems and lemmas that you expect the reader to know. Proper citations are given for everything.
3J: The rest of the paper is already written if you followed the guidelines in Stage 2 above. It is also already sorted into sections. Any theorems that have been stated in the introduction, should no longer be stated inside. Just write above the proof "We now prove Theorem 1 stated in the introduction...". It is a good idea to start each section with the statements of theorems in that section that were not mentioned in the introduction. Always state any lemma, theorem, or proposition before proving it. Below the statement of any theorem there should be a discussion of examples and related work as in the introduction. Propositions and lemmas don't need this.
3K: Acknowledgements: Here is where you thank your advisor, any others who discussed the project with you, and any funding sources or workshops that enabled your work on the project. You might also thank people who invited you to speak on the paper, but that is usually added later after the first arxiv posting before submitting it. Mentioning many names is a good thing here as it helps editors find referees.
4: Submission: Before posting on the arxiv or submitting yoir paper, show it to your advisors. Even after posting on the arxiv wait for feedback. People will often contact you recommending additional citations. Then submit to the journal that publishes papers similar to your paper of a similar length and topic, with an editor that is likely to be interested. You need to check the editorial boards of a few journals. High quality specialist journals may be needed to publish a highly technical paper. Be sure to check with your advisor if there is any politics you need to be aware of when submitting to a journal. I find it is best not to reveal your first name on a submission. Be sure to include your email address on the file.
5. Delays Delays: Every three months check with the editor the status of your paper.
6: Rejections and Revisions: Most papers are rejected 1-2 times before acceptance. The rejections can sometimes be harsh. If they claim it is uninteresting: ignore them. If they claim it has been done before: find out where it has been supposedly be done before and be sure to change your introduction to clarify the distinction between the papers. If one of your theorems has been done before you may wish to move it inside the paper, being sure to properly credit those who have done it before, and showcase a different original theorem instead. Otherwise you need to showcase that you have a new proof of the old result. Generally it is a waste of time to object to a rejection, so take the feedback, apply it, and submit elsewhere. If you are asked to revise a paper, the easiest thing is to just revise exactly as suggested and send it back. I know you may personally object to some of the revisions especially if they ask you to drop a theorem or application, but again the publications take long enough not to cause a delay. Generally one reposts the paper on the arxiv with a new version when the revision is complete.
7. More delays: after acceptance there is often a long delay before the article appears in a journal, so again check in every few months. By the time you are asked to check the final proofs, often the bibliography needs to be updated and possibly even your affiliation. Be sure to update both.
Good Luck!
Any further suggestions I may be happy to include. Let me know.