I gave the 4th and final lecture for my Cours Bachelier on bubbles last Friday, following the lecture that I reviewed here (where you can find links to the first two).
The central topic for this lecture was the local martingale approach of Jarrow, Protter and Shimbo (and previous references therein), where bubbles are characterized under the NFLVR (no free lunch with vanishing risk) condition. This is the current accepted way to express no arbitrage in modern mathematical finance language and is significantly weaker than an equilibrium condition, since it does not require any notion of optimality of market clearing. As such, the results for bubbles in this setting generalize those related to rational bubbles, which are typically done in equilibrium. On the other hand, it would be nice to use a similar mathematical framework (i.e semimartingales) to address the models for irrational bubbles that were discussed in the previous lectures. In my view, there are still many low hanging fruits for mathematicians to pick in this field.
The last part of the lecture (and the course) was dedicated to a brief review of the statistical tests proposed to detect bubbles in real data. Since I'm not a statistician, I mostly followed the this review, with a special emphasis on the work on volatility bounds.
I then concluded with a breakneck-speed tour of famous bubbles throughout history, including the tulipmania, which may of may not have been a bubble, the Mississippi and South Sea bubbles, which might have been just a huge (yet failed) macroeconomic experiment, and the crash of 1929, which almost certainly was a bubble, even under the most optimistic definitions of fundamental values.
All references (and possibly slides and lecture notes in the near future) can be found here.