This was written building upon advice from my doctoral advisor and years of experience speaking at competitive seminars where the audience members engage with the speaker in an aggressive way to try to derail the talk. Much of the advice is also useful in a less aggressive setting. Any additional suggestions are welcome and I may well add them to the list. - Christina Sormani
How to give a talk on your own research (not a survey talk)
0. Decide what to present in consulation with the person who invited you to speak.
0a. What is the audience? Do you have a result that would be most useful to that audience? Your goal is for people to apply your results.
0b. Who is in the audience? Be sure to research what they did so that you can mention them by name.
0c. Only present work that is already on the arxiv both to allow people to immediately read more details and to avoid theft. Feel free to vaguely mention working on newer results but do not give details unless it is a closed meetingvof close trusted collaborators. Do not mention anything without permission of your coauthors (even over lunch)
0d. If possible arrange to arrive the day before the talk and give yourself at least an hour after the talk to allow discussion.
0e. If possible find out the layout of the blackboards so you can plan how each will be used and when each will be erased.
0f. If you have a collaborator attending the talk, ask him or her not to answer questions on your behalf. You might also suggest that they could skip the talk if they wish.
1. Possibly before even being announced or immediately after:
1a Write the title on the board.
1b Write “joint work with” and the full names of all collaborators and their affiliations , possibly include arxiv numbers
1c: Avoid using initials to ref to coauthors and avoid acronyms throughout the talk
1d. Smile even if you become nevous or angry. A smile displays confidence and control.
1e. Remember you are the expert on your work! Prepare to answer standard questions with short answers. (If you don’t know the answer, it’s fine, just mention something relevant and suggest a discussion afterwards for more detail.)
2. Provide the setting (first 15 minutes)
2a. provide key definitions and important theorems absolutely needed to state your main theorems including graphics that makes the notions more clear. Key examples are also helpful. (Answer questions that clarify things but do so quickly and do not let anyone slow your talk down. “I’ll explain that in a few minutes” is fine even if you have no intention of doing so. Some questions are deliberately disruptive.)
2b. Be sure to include names of those responsible for these definitions and theorems and years: [Gromov 83]. When citing someone with a common last name include their initials. (If someone mentions you forgot to cite somebody, apologize that you didn’t remember to mention them or had been about to mention them. Avoid getting caught in politics. You can bring up the politics after your talk if you feel it is important).
2c. When citing yourself write the first initial of your last name and a squiggle [S~~] or [S~~-Wenger]
2d. Mention “Other work in this area has been done by...”. and just list names. This is where you are sure to mention people in the audience, famous people (to raise the value of your work), and less well known people (to be fair).
3. State your key theorem/s at the 15 minute mark
3a. Write “Main Theorem” or “Theorem A” “Thm B” and cite yourself and all coauthors on this theorem. Put the one you plan to prove on a far left or right board so that you can refer to it for the rest of the talk. (If someone interupts to ask if the theorem isn’t just some other person’s theorem, explain the difference between your theorem and the one they are confused about. You know your theorem is new. Usually they are confused.)
3b. Provide further background and related theorems of others as in 2 (If someone asks why you didn’t prove some other statement respond either that the other statement was already done and by whom, or that the other statement is false and why, or say that would be interesting to discuss after the talk.)
3c. Pause to ensure your statement is understood. Possibly draw a picture or provide an example which helps one understand the theorem starement. (This is a good point to answer clarifying questions. You might be asked why an hypothesis is needed. Be prepared to provide either an example or conjecture that it might not be necessary. Explain how you employed the hypothesis in your proof.
3d. Provide an intuitive outline of the proof including graphics if it is possible to do so very quickly. Something that fits in one board. Otherwise wait. Do write “proof uses work of” and cite appropriate papers if you will not get to this later.
3e. List applications including names of collaborators who did those applications with you.
4. Present the proof of one of the theorems (state which) (have various points where you may cut off naturally at the end of the time so it looks as though you presented everything you wanted to even if you didn’t)
4a. Avoid erasing the statement of the theorem you are proving and key background if possible
4b. Be very clear to write out when a part is a proof by contradiction or by induction or by the method of continuity... Be sure to write “thus” and implication symbols as needed and “by hypothesis”. Do not count on people hearing everything that you say.
4c. Always attribute the right people when applying their results or techniques “by Gromov’s Compactness Thm” or “by [Gromov83]” or “as in [...]” including your own prior work being careful to list coauthors as in 2b
5. When time is nearly up close up your talk and volunteer to discuss the rest of the details afterwards. Always make it appear as if you presented everything you were planning to present and that you timed everything perfectly.
5a. Be prepared to speak 5-10 more minutes if they have time to stay later.
5b. Get the names and emails of everyone who asks you a question at the end. And then email them a link to the paper/s and ask if they wish to know more.
5c. Send an email thanking everyone who invited you to speak and/or attended your talk.