The cubic case of the main conjecture in Vinogradov's mean value theorem
  We apply a variant of the multigrade efficient congruencing method to estimate Vinogradov's integral of degree $3$ for moments of order $2s$, establishing strongly diagonal behaviour for $1\le s\le 6$. Consequently, the main conjecture is now known to hold for the first time in a case of degree exceeding $2$. 
Trevor D. Wooley
math.NT