The so-called RoCK (or Rouquier) blocks play an important role
in representation theory of symmetric groups over a finite field of
characteristic $p$, as well as of Hecke algebras at roots of unity.
Turner has conjectured that a certain idempotent truncation of a RoCK
block is Morita equivalent to the principal block $B_0$ of the wreath
product $S_p\wr S_d$ of symmetric groups, where $d$ is the "weight" of
the block. The talk will outline a proof of this conjecture, which
generalizes a result of Chuang-Kessar proved for $d < p$. The proof uses
an isomorphism between a Hecke algebra at a root of unity and a
cyclotomic Khovanov-Lauda-Rouquier algebra, the resulting grading on the
Hecke algebra and the ideas behind a construction of R-matrices for
modules over KLR algebras due to Kang-Kashiwara-Kim.