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I have been to a number of advanced lecture courses (of between 3 and 10 lectures) over the years, given (in principle) by experts to graduate students and experts in neighbouring fields. Examples of this are summer schools.

My impression is that this usually devolves into lectures given by experts in an area aimed at other experts in the same area. Graduate students are usually left by the wayside in lecture 1.

Is the whole endeavour futile? or do people have principles/suggestions to maximize the usefulness of advanced lecture series
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I'm not sure if this topic will survive here, but I'd sure like to hear suggestions and ideas. –  Todd Eisworth Nov 19 '11 at 2:13
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It is not futile, and there are suggestions aplenty. However, to say "maximize the usefulness" is too vague to make the question, um, useful. Give some particular metrics or goals, and I can then tell you which of the principles and suggestions I have may apply. Gerhard "Ask Me About System Design" Paseman, 2011.11.18 –  Gerhard Paseman Nov 19 '11 at 4:15
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Is your question aimed at a person giving such lectures or attending them? In the first case, an obvious suggestion is to aim your lectures at the actual audience you were told to address, ignoring any more experienced people sitting in the room. And consider preparing exercises for your lectures to keep people engaged with the material outside of the lecture itself. In the second case, check that the advertised lecturer actually pays attention to who the audience is supposed to be instead of who the speaker wants the audience to be. –  KConrad Nov 19 '11 at 4:48
    
I can't decide if I find this question too vague or too specific: I find it too vague, because it asks for suggestions without specifying what aspect these suggestions should address (choice of speaker? nature of lectures? choice of subject?). At the same time, I find it too specific because I can't help feeling that most suggestions for advanced lecture series would also apply to single lectures, or weekly seminars, etc. –  Thierry Zell Nov 19 '11 at 18:57
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5 Answers

Here are my 2 cents:

1) Assume that the audience knows nothing about the topic you want to present and next to nothing about any other things.

2) Choose one result you'll be aiming at and develop "the general theory" just enough to be able to do the proof. Cut all special terminology to the bare minimum.

3) Do one little thing in each lecture with clear beginning and a clear logical point at the end, so that even if someone comes to just one lecture in the middle of the series, he'll be able to learn something. Never stop in the middle of a tangled proof.

4) Wake up the audience every 5 minutes by stopping and asking "obvious questions". Wait for an answer. If none comes, you've lost everybody.

5) Run question-answer sessions (you can devote first 10-15 minutes of each lecture to that or you can spend a couple of classes exclusively on that, it is your choice)

6) If you are not lazy, make notes for each lecture and distribute them a day before the lecture. If you are lazy, designate somebody to take notes of your lectures and look at what he/she has written in the end of each lecture.

7) Write on the blackboard clearly as you speak and use high-tech (projectors, movies, etc.) only to show pictures. Remember: the speed of absorbing new information never exceeds the speed at which you can write it in full sentences on the board.

8) Above all, remember that your task is to teach other people new things, not to show how much you know or how great your subject is or to impress them with anything else.

That should do it if you are giving a lecture.

1) Read a bit about the topic that is going to be presented before coming to the lectures

2) Sit in the front row and ask the questions as soon as they arise. Don't be afraid to interrupt the lecturer. Believe it or not, but if you don't understand something, 50% of other people in the audience don't understand it either.

3) Make notes. Read the notes for each lecture before you go to the next one. Catch the lecturer before the lecture and ask any questions you have from the previous one.

4) If the lecturer is out of reach and you have a question, ask anybody in the audience.

5) If you have friends in the audience, run "discussion sessions" after each lecture with them. Sometimes 5-10 minutes are enough to make the difference between "being lost" and "understanding everything"

6) Above all, remember that you came to learn, not to admire the greatness of the lecturer and the subject.

That should do it if you are attending a lecture.

This, of course, applies to "lecture series with proofs". There are also "purely expository series" but I would advise to just avoid them on both sides.

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When I was a grad student, the math departments of Harvard and MIT occasionally had a joint lecture for two hours in the evening. The speakers were sometimes local faculty and sometimes visitors. The speakers were allowed to talk about any topic they wanted, but they were not allowed to even mention any of their own theorems. With no chance to "show off," they produced some great talks. I particularly remember a lecture by Hartley Rogers on recursion theory, starting from zero and ending, two hours later, with a perfectly understandable presentation of a finite-injury priority argument.

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What was the reason you weren't allowed to mention your theorems? –  Woett Nov 20 '11 at 5:40
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@Woett: This rule was to make sure the talks remained accessible for grad students and other non-experts. Some people, if permitted to mention their own theorems, will feel obligated to mention them and even to emphasize them, so their talks become explanations of how important, deep, and otherwise wonderful their own work is. Such talks tend not to be useful for grad students --- who are perfectly willing to take on faith that the speaker's work is important, deep and wonderful, but who would really like to understand something. –  Andreas Blass Nov 20 '11 at 21:26
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Some years ago (2004) I participated in the excellent session organised at the IMA in Minneapolis by Peter May and John Baez. My impression was that everyone thought highly of the lectures (I may be biased as I did give some of them.)

One principle used was that even when the 'big name' who had done the original work was in the audience, it was rather someone who had worked with and used that work who gave the talks. (I.e someone who had had to learn the stuff!)

The lectures were timed within the fortnight so that on one topic perhaps four hours were planned (but perhaps 6 to 8 hours were available!) We were encouraged to say... "can we have a bit more background on such and such?" (I noted several very interesting reactions where someone was a supervisor of some of the students who were there. The supervisor suggested that someone might expand on a topic for the sake of the students. Several times I noted that (i) the students looked very pleased at this, and (ii) lots of other people also looked pleased!) There would be a call for volunteers to give the extra sessions. They might be full sessions or just in the bar in the evening with a few people around someone discussing the topic.

The bane of lecture course by experts may be that they plan for four hours(or whatever0 and have a very intense idea of what need to be said. They therefore force feed the audience.. and , continuing the metaphor, the audience find it hard to digest. I did not feel that that happened at all in that IMA workshop. (If Peter or John see this may once again say that it was very instructive, very useful and very enjoyable. Thank you again.)

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As a student—albeit an undergraduate student—who's been to talks that were well over his head and as a student who has presented math that was intended for a general audience but ended up going over people's heads, I'm really interested in this question. For what it's worth:

I'm going to the JMM in Boston this year and I'm entertaining the idea of emailing people who's talks I'd like to go to and ask them to email me their paper so that I can get a preview of coming attractions. I'm thinking that reading the paper will be as helpful to a deeper understanding of presentations as reading the chapter does before it's covered in class. If I were in your position, I would try to get my attendees access to some literature ahead of time. I don't know how realistic this is, but the idea is that this is one way for people to find out what tools are going to be used in the talk. If they need to make a trip to the library (or Wikipedia) before understanding the material, they can do that before the talk.

I also like the idea of recorded lectures. That way if I don't understand something, I can go back, rewatch it, pause, work out the computation, whatever and start it going again.

And I think the link in the first answer has an extra character in it and it should be: http://www.ams.org/programs/research-communities/mrc-12

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Not to discourage you from emailing speakers beforehand, but do check the arXiv for their papers first... –  Ben Webster Nov 19 '11 at 17:42
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If possible, it might be a good idea to organize a problem session, with problems/exercises given to participants in advance (say, a day before) and the session devoted to presenting the solution. The problems needn't be difficult, but have to solve actual exercises forces you to understand the definitions and concepts in enough detail to be able to use them, not only on vague "big picture" level. I've attended schools/workshops where this idea was implemented and the feedback was overwhelmingly positive.

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