Disclosure: I've never been on a hiring committee or a grant review committee. I can't guarantee that following this advice will get you hired. But I think it will help you get some words on the page, which is an important and sometimes terrifying step.
THE FIRST PAGE
The first sentence should tell a hiring committee who should read your research statement.
My research is focused on . The aim of (RESEARCH AREA) is to .
I research , which is the study of .
My research is in , using techniques from and with applications to .
Some words you might use to describe your area:
investigate the behavior of...
study connections between...
examine (object) through the lens of (other object)
apply (technique a) to (area b)
model (object) through (other object)
Activity One: Choose a sentence structure above and fill it out for you. Congratulations, you have a first sentence!
THE FIRST PAGE
Your first page should be understandable to nonspecialist (i.e, mathematicians not in your field).
Activity Two: If you wanted a mathematician not in your field to get one concept out of your research statement, what concept would you want it to be? Write down a sentence. For example, "X has many equivalent definitions, which means there are many paths to approach the same problem," or "objects Y have a lot of structure, allowing us to prove much stronger results than in the case of object Z."
Activity Three: State one of your own personal results or main questions. Underline every word that a non-expert would need defined.
Brainstorm informal ways you can define some as many of the underlined words as possible. For example, ``in certain cases, this notion coincides with vector space dimension''.
HISTORY AND CONTEXT
(RESEARCH AREA) emerged in (CENTURY) to answer questions about...
(OBJECT) was invented to describe...
(RESEARCH AREA) had an explosion of popularity when (PERSON) proved...
Answer the following questions:
What questions were being asked when your field first emerged?
Are there any famous theorems or results from your field, or that inspired the beginning of your field? (for example., ideals were initially invented as a tool to prove Fermat's Last Theorem).
What other fields does your field support and inspire?
What fields is your field supported and inspired by?
Who can you ask about this if you have no idea?
Do not be afraid of using
Depending on your research, it may make sense to lump all the background into one section, or it may make more sense to have different sections that motivate different problems.
Each concept will need a little bit of its own motivation, but whether this is a paragraph or a page will be particular to your work.
One potential structure:
All Background and Motivation
All Future Directions
Another potential structure:
Background and Motivation to Question/Result 1
Future Directions About Question/Result 1
Background and Motivation to Question/Result 2
Question and Result 2
Future Directions About Question/Result 2
Break the research you have done up into chunks. Brainstorm section titles for each of the questions you've investigated.
What previous theorems are necessary to include in order for your work to make sense?
WORDS AND PHRASES THAT SOUND NICE
fertile ground for further research
extend existing results
extend techniques from...
exploit techniques from...
can be understood through the lens of...
In order to understand X it is crucial to Y
Inspired by the results of...
AUTHOR X has had success applying technique Y
A natural follow-up question is...
there’s a growing trend towards...
I am particularly well-prepared to answer these questions because...
construct a theory on...
the (algebraic/combinatorial/analytic/etc) nature of this question allows for...
combination of (technique x) and (technique y)
at the heart of...