Definitions and proofs in extended abstract(mathematics) I have to submit an extended abstract of my paper for a competition. The extended abstract must be about 2~4 pages, and the competition requires it to include all the methods and results. Of course, specific definitions and proofs are included in my original paper(about 20 pages), but I can't just summarize them all into 2~4 pages without omitting some ideas or methods. So my questions are


I have some concepts that are not frequently used. Should I write them in my abstract?
Do I have to write precise definitions?
How should I summarize my proof? Can I skip the specific processes and only state main ideas? Can I omit proofs of lemmas?


(My advisor said that he has no idea of how to write an extended abstract, so I hope I can get some help from here)
Before you start writing, try to create a 5-minute elevator pitch talk about your paper. Imagine you're describing the results to your friend (or actually invite a friend) and talk through the main idea of the paper. Practice it several times and note which things you keep in, and which you decide to leave out. This will give you an idea of what to include in your extended abstract.

It is simply not possible to squeeze everything from 20 pages into a 4-pages without losing something, so stop trying. Inevitably you will have to omit some results, or simply name them without going into details. You will have to drop some technicalities (which can contain some useful ideas as well) and so on. The aim of the extended abstract is to communicate the core ideas, not fully elaborated proofs and  methods. The elevator pitch technique above helps you to tell what's central and what's not.