The abstract of your project is the most important part of your application. The mathematicians on our review panel will make up their mind on how to rank your research and proposed presentation mainly based on what they read in the abstract. Keep in mind also that the reviewers will have to look through a large amount of abstracts in a rather short time. Although they will be highly experienced mathematicians with a great amount of experience, they will thus greatly appreciate it if you can make your case as clearly and efficiently as possible.
If accepted you can still edit your abstract during the confirmation phase shortly after the invitation email. This allows you to update results or correct errors. The final version of your abstract will be published in the conference abstract booklet, which will be distributed to all participants and posted on the YMC web pages.
State the mathematical results of your research efforts as clearly as you can. Your description should clearly distinguish which parts are an exposition of previously known results and which parts are new research and/or new results. If you can phrase a main result as a formal theorem, by all means describe it or (if short enough) simply state it! In other cases try to give a summary of your efforts that focuses on noteworthy mathematical observations and conclusions.
Reviewers and participants are interested in what you did (rather than a longer than necessary exposition about others did) , and would prefer to have that explained in a concise fashion.
Primary criterion for reviewers will be the mathematical content of your abstract. Applicability of your result to other areas of science or "real life" may be a plus and appealing. You should, however, make sure the description of applications do not overshadow or obscure the underlying mathematical problem you have dealt with.
Keep in mind that, in order for reviewers and participants to understand your results, they need to have at least a vague familiarity with the mathematical objects and notions you are using. So you will need to make decisions which terminology requires additional explanations and definitions.
Generally, readers of your abstract are mathematically well educated so that the basic notions in most subject areas of mathematics can be assumed. However, if your abstract starts cold with specialized terminology, acronyms, and references as in "Let H be a quasi-triangular ribbon-algebra obtained from an FRT reconstruction ..". or "We improve the algorithm proposed by Korzybski ('97) for nucleotide linkage data, and relate it to the QTL analysis ..." only very few people will read to the end.
Most notions can be explained briefly in informal terms omitting formal details so that the reader has an understanding what the general nature of the terminology is. It is a good idea to ask your mentor what most mathematicians would expect to have explained.
Put your work into a context. It should be almost immediately clear from your abstract which general subject area in mathematics your research is in. How do the problems you work on fit into the general goals of your subject area? Why is it interesting? What are further directions and conjectures of your work?
Make sure your abstract has adequate length. Very short abstracts may not have enough information to be useful, and very long abstracts may be very cumbersome to read and print. The average word count in the past conferences is about 115 words (unix word count), with most abstracts having between 60 and 200 words.