One conference per answer. Explain why it was great (mathematically or otherwise), and preferably post a link to the conference website or to abstracts/proceedings.
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1$\begingroup$ Different conferences serve different needs for different people, and one person's "great conference" might be someone else's nightmare for a hundred different reasons. I imagine a similar but better-formed question could be appropriate for MO, but as stated this one really isn't. $\endgroup$– Harrison BrownCommented Nov 25, 2009 at 23:00
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5$\begingroup$ Community opinion (expressed by downvoting) seems to suggest that your question would be more appropriate on a different forum, so I'm voting to close your question as "off topic". Note that your question is interesting and I would like to answer it as well as read the answers -- but this interaction should happen in a different place, not on Stack Exchange-based platforms. $\endgroup$– Ilya NikokoshevCommented Nov 25, 2009 at 23:01
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$\begingroup$ And you're invited to join the thread at tea.mathoverflow.net/discussion/44/… $\endgroup$– Ilya NikokoshevCommented Nov 25, 2009 at 23:05
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$\begingroup$ Of course the notion of a "best" math conference is not absolute, which is why I'm interested to learn about what people like about a conference, and how conferences are run in different fields. If the question is downvoted to -7, or if there aren't many answers, I'll close it. $\endgroup$– Ian AgolCommented Nov 25, 2009 at 23:21
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$\begingroup$ If it's fine with the community, it's fine with me, sorry for the buzz :) $\endgroup$– Ilya NikokoshevCommented Nov 27, 2009 at 23:29
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6 Answers
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The famous AMS Summer Research Institute on Automorphic Representations and L-functions held at Corvallis in 1977. It was organized by Borel, Casselman, Deligne, Jacquet, Langlands, and Tate to give a coherent account of the state of the Langlands program. Beyond the organizers, speakers included Springer, Tits, Cartier, Lusztig, Piatetski-Shapiro and Arthur. Generally, speakers were assigned topics to talk on. The proceedings have been a wonderful resource during the thirty years since they were published. They are available on the AMS website (http://www.ams.org/online_bks/online_author.html#B).
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Now that this is turned into a community wiki post, I'll throw an answer out there: No, a non-answer!
And my non-answer is this: There are too many conferences, and the vast majority of them too specialized for most of us, to answer this question in a way that would be useful to most mathematicians. (Yeah, I am sticking my neck out here, speaking for “most mathematicians” when it's not altogether clear that I am competent to speak for myself.)
I tend to prefer narrowly focused conferences myself, in a field that I actually know something about. That way, I can at least feel appropriately ashamed when I can't follow a talk on a subject I had thought I understood. Which provides an incentive to do something about it. So if I may add to the original question, then? Why do people go to big general conferences? What do they get out of it? Okay, this may sound too negative. Breadth is good, and I wish I had more of it, but mathematics is a vast field and only a few of us can attain deep understanding in more than a couple of subfields, at best.
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8$\begingroup$ One reason to go to a big conference is that there are often talks by good speakers on subjects outside my area. Even if I don't plan to work on the subject area of the talk, it is nice to hear about interesting mathematics in general, and current developments in particular. $\endgroup$– S. Carnahan ♦Commented Nov 26, 2009 at 2:36
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The 1900's Paris ICM... just because of Hilbert's talk, in which he modelled the shape of a rather big amount of the mathematics that were to come; and not just by becoming fancy, but by identifying really difficult and interesting problems and, even more importantly, by pointing to logic and rigor so that others could see the necessity for a thorough cleaning of their mathematical building.
(Sorry, I could not locate any link to the proceedings ;P)
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The Talbot workshops are mostly aimed at graduate students and early professionals. The topics and plenary speakers vary year-to-year, but the ones I have attended were all very illuminating. The 2009 workshop, on the Fukaya category, was especially good, and there are notes from each talk on the web page linked above.
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I'll start. The New Zealand Institute of Mathematics holds a conference every year, a different location and topic each time. In 2006, at Taipa, the topic was geometric topology. Jeff Cheeger gave an excellent plenary series of talks on spaces of Einstein metrics. He and Tian proved certain collapsing results for these metrics, laying the framework for compactifying moduli spaces of Einstein metrics. There were two talks in the morning, and one in the evening after dinner, leaving the afternoon free for activities such as windsurfing. If you can make it, I recommend it. This year (January 2010) will be on quantum topology (although I have to say I won't be able to go).
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The Solvay Conference (see http://en.wikipedia.org/wiki/Solvay_Conference), although formally being a physics conference, was attended by Poincare and was fundamental to establishing quantum mechanics and general relativity as important topics in mathematical physics.