We were often reminded that absolute Galois groups of p-adic fields admit automorphisms not arising from field theory, but we were never told in a clear manner why the existence of such exotic automorphisms is relevant to the task of proving Szpiro’s Conjecture; perhaps the reason is a simple one, but it was never clearly explained despite multiple requests. (Sometimes we were told it would become clearer later, but that never happened either.)
After a certain amount of this, we were told (much to general surprise) variations of “you have been given examples.” (Really? Interesting ones? Where?) It felt like taking a course in linear algebra in which one is repeatedly told “Consider a pair of isomorphic vector spaces” but is never given an interesting example (of which there are many) despite repeated requests and eventually one is told “you have been given examples.”
Persistent questions from the audience didn’t help to remove the cloud of fog that overcame many lectures in the final two days. The audience kept asking for examples (in some instructive sense, even if entirely about mathematical structures), but nothing satisfactory to much of the audience along such lines was provided.
For instance, we were shown (at high speed) the definition of a rather elaborate notion called a “Hodge theater,” but were never told in clear succinct terms why such an elaborate structure is entirely needed. (Perhaps this was said at some point, but nobody I spoke with during the breaks caught it.) Much as it turns out that the very general theory of Frobenioids is ultimately unnecessary for the purpose of proving Szpiro’s Conjecture, it was natural to wonder if the same might be true of the huge amount of data involved in the general definition of Hodge theaters; being told in clearer terms what the point is and what goes wrong if one drops part of the structure would have clarified many matters immensely.
The fact that the audience was interrupting with so many basic questions caused the lectures to fall behind schedule, which caused some talks to go even faster to try to catch up with the intended schedule, leading to a feedback loop of even more audience confusion, but it was the initial “too much information” problem that caused the many basic questions to arise in the first place. Lectures should be aimed at the audience that is present.
Statt durch Fragen in Vorträgen sollen die Teilnehmer den Stoff wohl lieber von solchen Animationen lernen 🙂