Serre’s modularity conjecture and $\mathrm{mod}~p$ Langlands program

Serre made a remarkable conjecture in his 1987 paper relating mod p representations of $Gal(\bar{Q}/Q)$ to $\mathrm{mod}~p$ modular forms. In this talk, I will review Serre’s conjecture (by numerical examples) and explain its relation to $\mathrm{mod}~p$ Langlands program. I will also discuss some recent progress on the subject.