In this talk, I would like to attempt to give a brief glimpse at and around the recent developments related to the Cap Sets problem:
Recently, there has been a string of breakthrough results which started with a paper of Croot-Lev-Pach, related to bounds on progression free sets in $Z_4^n$. This was followed up by a great improvement to a longstanding problem on the bound on Cap Sets (progression free sets in $Z_3^n$) by Ellenberg-Gijswijt. The proof is surprisingly simple, and has found several applications, most notably in proving the Erdos-Szemeredi Sunflower conjecture, improvements in bounds on tri-coloured sum free sets, great improvements in removal lemmas in $Z_2^n$ etc..
