摘 要： The Witt algebra $g$ was the first non-classical simple Lie algebra discovered by E. Witt in the 1930's. Chang(张和瑞) first classified simple $g$-modules over an algebraically closed field $k$ of characteristic $p>2$. In this talk ,we exploit some geometric structure on the Witt algebra $g$. More precisely, let $G=Aut(g)$ be the automorphism group of $g$ with $Lie(G)=g_0$. We determine the structure of the nilpotent commuting varieties of $g$ and $g_0$. We show that they are reducible and equidimensional. Irreducible components and their dimension are precisely given. As an application of a result by Suslin-Friedlander-Bendel, we determine the spectrum of maximal ideals of the Yoneda algebra $\bigoplus_{i\geq 0} H^{2i}(G_2, k)$ of the second Frobenius kernel $G_2$ of $G$. For the case $k=\bar{F}_q$, we further determine the stable nilpotent orbits in $g$ under the Frobenius morphism. Consequently, we obtain the number of $F_q$-rational points in the nilpotent variety. This is a joint work with Hao Chang.