Uniqueness theorems for dg enhancements

For more than twenty years there has been much interest in results describing which triangulated categories have unique enhancements, and results giving conditions sufficient to guarantee that certain functors must be Fourier-Mukai. We will begin the talk with a brief survey of this literature. Then we will move to recent work, joint with Alberto Canonaco and Paolo Stellari, which substantially improves on what was known about uniqueness of enhancements. After stating the main results we will sketch the key new ideas that go into the proofs.