Goldbach Conjecture, what are new research methods after Chen's work?

Question

For Goldbach Conjecture, my understanding is that there are three major methods to attempt it:

1. Schnirelmann density
2. circle method
3. sieve method (Chen used two parameter sieve method to get his famous 1+2 results)

Those methods have been there for long time. I just wonder any new major methods or ideas in Goldbach conjecture research ?

Did Helfgott use or invent any major new methods ?

Thank you in advance.

Answer 1

For new methods on the binary Goldbach conjecture, Terry Tao's blog is a good reference (parity problem obstruction etc.); see also the MO question here, which explains the contributions of Helfgott. Also see his own explanation here. He himself said in talks that the binary Goldbach conjecture will require new methods, different from the ones used for the ternary Goldbach conjecture. There are several links which make this more precise (also see his summary on page $6$: "This, incidentally, is the point at which the basic approach breaks down for the binary Goldbach problem").