  Given a smooth projective variety $X$ with a simple normal crossing divisor
$D:=D_1+D_2+...+D_n$, where $D_i\subset X$ are smooth, irreducible and nef. We
prove a mirror theorem for multi-root stacks $X_{D,\vec r}$ by constructing an
$I$-function, a slice of Givental's Lagrangian cone for Gromov--Witten theory
of multi-root stacks. We provide three applications: (1) We show that some
genus zero invariants of $X_{D,\vec r}$ stabilize for sufficiently large $\vec
r$. (2) We state a generalized local-log-orbifold principle conjecture and
prove a version of it. (3) We show that regularized quantum periods of Fano
varieties coincide with classical periods of the mirror Landau--Ginzburg
potentials using orbifold invariants of $X_{D,\vec r}$.

Algebraic Geometry