Cohomology and geometry for Frobenius kernels of algebraic groups

This dissertation contains many results related to cohomology of Frobenius kernels for algebraic structures. It is divided into four themes. First, we compute the cohomology ring for the first Frobenius kernel of the unipotent radical of a simple algebraic group. New results on computing cohomology of the Frobenius kernels for parabolic subgroups are also obtained in this part. Next, we provide generalizations of all former results to the cohomology of small quantum groups at a root of unity. Third, we focus on cohomology computations for $SL_2$. Some observations about low degree cohomology for $B_r$ and $G_r$ are given beforehand. Finally, we look at the geometrical aspect of the cohomology for Frobenius kernels, namely nilpotent commuting varieties of $r$-tuple, and prove various properties in the case of low rank matrices.