Tim Gowers is going to write about his idea for circuit lower bounds in the form of a seven-part dialog between three characters.
The first installment is out, and it already features a gentle and beautiful introduction to the models of boolean circuits and boolean formulas, and to the natural proof impossibility result. He also brings up the very interesting meta-question (which sounds rather new to me) of whether a “random easy function is pseudorandom.”
(The title of the post is how a character in the dialog refers to impossibility results such as natural proofs.)
You may remember the polymath1 project, in which Tim Gowers envisioned, and realized, a “massively collaborative” approach to solving an open question in mathematics. The project succeeded, and in fact exceeded its goal. Various subsequent polymath projects are under way, suggested by Gil Kalai and Terry Tao. Notably, Terry Tao has launched a project to find an efficient deterministic algorithm to construct large primes.
Meanwhile, Gowers has been thinking about his next polymath proposal, and he has recently written about the possible projects that he has in mind. One of the projects involves a new approach to proving circuit lower bounds. No detail of this approach is given; nor it will, unless it becomes a polymath project. Gowers would like people to comment on his post indicating which projects they are most interested in.