Structure of Hecke algebras modulo $p$

Recently, Joel Bellaiche and Chandrashekhar Khare studied the structure of local components
of the Hecke algebras acting on modular forms modulo primes $p \geq 5$ of level $1$ and proved
that they have Krull dimension at least 2, establishing the equality under certain assumptions.
In this talk, I will explain their method and show how to generalize their results to Hecke algebras
acting on modular forms modulo $p$ of level $\Gamma_0(N)$, $\Gamma_1(N)$ where $p \geq 5$ and $(p,N) =1$.