Has there been any progress toward the Birch and Swinnerton-Dyer conjecture after
The current status of the Birch & Swinnerton-Dyer Conjecture
Has there been any progress toward the Birch and Swinnerton-Dyer conjecture after
The current status of the Birch & Swinnerton-Dyer Conjecture
No, the conjecture is still wide open for rank $r\geq 2$.
The closest thing to progress is the work of Bhargava and Shankar that quantifies the rank $0$ case and shows that BSD holds for a positive proportion of elliptic curves. You can find it here:
For subsequent developments, see for example
Benedict Gross recently gave a series of lectures here at the University of Virginia on things related to the Birch and Swinnerton-Dyer Conjecture. One of the recent notable developments he mentioned is the work of Yun and Zhang. It is about the function field analogue but they obtain information about the full Taylor series of the $L$-function. The paper has recently been accepted at Annals of Mathematics.
Searching `Birch and Swinnerton-Dyer conjecture' by Google, there are two short preprints on this theme:Yongxiong Li, Yu Liu, Ye Tian and K.Morita.