In this talk, we first discuss the concepts of regularity and geometric regularity in commutative algebra. Next we present a published result:  If $I$ is a two-dimensional squarefree monomial ideal of a polynomial ring $S$, then $\mbox{g-reg}(S/I^n)=\reg(S/I^{(n)})$ for all $n\geq 1$. Finally, we talk about the recent progresses along this result.