The Research Statement required of all applications for Research Postdoctoral positions requires careful consideration in writing, and especially the help of your faculty advisor(s). On the one hand, you need to state clearly what your research topic is, and what you have accomplished to date as part of your thesis research. Since you may not have begun writing your thesis, this demands that you step out of the process to describe it to another mathematician, not necessarily in your exact field. On the other hand, you must also express clearly some sort of program of future research which is compelling, yet not so overly ambitious as to raise doubts. We offer here a few opinions on what to strive for in your preparation of this document.

You should also read the discussion on writing a Research Statement in the AMS Academic Job Search page.

The Research Summary

One part of your Research Statement is a summary of your thesis work. Some job applications may require a "Thesis Abstract" which is essentially this. Whatever, keep in mind you are writing for an audience that consists of the the members of the "Faculty Search Committee" at the various universities to which you have applied. The goal is thus to start the Research Summary with as broad as possible explanation of your research area and work. Of course, it should not be so broad as to simply declare "I do Mathematics". That is OK for Milnor to write, but not for a new doctorate.

The latter part of the Statement should then state precisely what you have proved in your thesis work, as the audience now consists of specialists, who want to know exactly what you proved. If you have a preprint written from your work, and either posted to your web site or to the arXiv, then you may direct the reader to this paper for further details.

Expect to rewrite this document many times, the first time to simply get something written on paper. Show this first draft to your advisor and get her or his comments, and then revise. Give yourself some time, at least a week or possibly a month, to keep working on this document, revising it until it reads well and seems clear to someone besides yourself. At some point, you might even let selected faculty or graduate student friends also make comments on what you have written, to see how clear it is to a non-specialist.

The revision process is much easier if you prepare the document in LaTeX. Here is a sample LaTeX 2e file that you might use to start the report with. It illustrates several "TeX-niques" which make for a nice math document.

Proposed Research

The Statement of Proposed Research is a delicate mixture of fact and fiction. On the one hand, you know what you would like to keep working on, based on your current research. This gives a nice basis for starting to write your statement. However, keep in mind that when you arrive at your new position, and have attended a few seminars, it may dawn on you that there is no one in the department you are now a member of, who knows about what you do. So how do you keep working on a subject that you have no one to discuss it with? Typically, you don't.

When you start a position at a Research Department, you may (and probably should) start to collaborate on projects in research areas that are of common interest with someone on the faculty there. Although, you may find yourself in a department that specializes in exactly what you love to think about, and there you go. You will follow up on every idea you planned to work on, or at least on something of similar taste. The point is simply that it is impossible to know what the future will bring, so do not write with such conviction that a reader at another university thinks you are not interested in working with anyone there.

After describing what immediate projects you plan to work on, for example during the Summer & Fall semesters after graduation, then you need to write about some research projects you may have in mind for the following year. Your advisor can be a great help on this sort of question, as she or he has spent many years writing grant proposals, which ask the faculty member to do exactly this. Perhaps, your future may well include working on related projects to those of your thesis advisor, and you can describe some aspect of these as "Proposed Research". Predicting the future can be fun, especially if any of it happens to later come true. Good luck!