At TTI yesterday, K. V. Subrahmanyam gave a talk giving one of the better overviews of Ketan Mulmuley's Geometric Complexity Theory approach to separating complexity classes. This approach reduces various problems including P ≠ NP to hard problems in algebraic geometry.
Afterwards at lunch, Ketan made it clear that he believes
- GCT will eventually lead to proving P versus NP. In fact, any proof that NP is different than P must go via GCT.
- Such a proof will not happen in my lifetime.
Ketan is not even giving me that opportunity. Consider a huge mountain and you want to reach the mountaintop. Ketan comes along and says he'll teach you how to create the tools needed to climb the mountain. It will take a hard month of study and actually these tools aren't good enough to climb the mountain. They need to be improved and these improvements won't happen in your lifetime. But don't you want to learn how others will climb the mountain centuries from now?
If you want to spend the month, go here and start reading. Let me know when you've been enlightened.