In the first part of this talk, I want to discuss heat kernels for a class of degenerate elliptic operators which are closely related to {\it Grushin operator}: \begin{equation*} L_G=\frac{1}{2} (X_1^2 + X_2^2)=\frac1{2}\Big(\frac{\partial}{\partial x}\Big)^2+ \frac{x^{2m}}{2}\Big(\frac{\partial}{\partial y}\Big)^2 \end{equation*} where $X_1=\frac{\partial}{\partial x}$ and $X_2=x^m\frac{\partial}{\partial x}$. It is easy to see this operator is elliptic except when $\{x=0\}$. We refer it as the {\it missing directions}. Given any two points in the Grushin plane, we shall use Hamilton formalism to find all geodesics connecting these two points. Unlike Riemannian geometry, we see that there are points $P$ in the Grushin plane that arbitrarily near $P$ which are connected to $P$ by an infinite number of geodesics. This strange phenomena brings up a new geometry that is so-called {\it subRiemannian geometry.} Next, we may find a modified complex action function which is essentially play the same role as $d^2(x,y)$ in Riemannian geometry. Finally, we may solve a transport equation to find the volume element to obtain heat kernels. In the second part of this talk, I will also discuss trace asymptotics for the heat kernel of the operator $L_G$.