The classical Diophantine problem of determining  which integers can be written as a sum of two rational cubes has a long history; it includes works of Sylvester,  Selmer, Satgé, Leiman  and the recent work of Alpöge-Bhargava-Shnidman-Burungale-Skinner.  In this talk, we will  use  Selmer groups of elliptic curves and integral binary cubic forms to study some cases of the rational cube sum problem.  This talk is based on  joint works with D. Majumdar, P. Shingavekar and B. Sury.

[pre-talk at 1:20PM]