讲座内容：

Let $\Omega$ be a bounded connected open subset in \mathbb{R}^n$with smooth boundary $\partial\Omega$. Suppose that we have  asystem of real smooth vector fields $X=(X_{1},X_{2},$$\cdots,X_{m})$ defined on a neighborhood of $\overline{\Omega}$that satisfies the H\"{o}rmander's condition.  For a self-adjoint sub-elliptic operator $\triangle_{X}=-\sum_{i=1}^{m}X_{i}^{*} X_i$ on $\Omega$, we denote its $k^{th}$ Dirichlet eigenvalue by $\lambda_k$. We will provide  an uniform upper bound for the sub-elliptic Dirichlet heat kernel and also give an explicit sharp lower bound estimate for $\lambda_{k}$,which has a polynomially growth in $k$ of the order related to the non-isotropical dimension. We will establish an explicit asymptotic formula (Weyl law)  of $\lambda_{k}$ that generalizes the M\'{e}tivier's results in 1976. Our asymptotic formula shows  that under a certain condition, our lower bound estimate for $\lambda_{k}$ is optimal in terms of the growth of $k$. Moreover, the upper bound estimate of the Dirichlet eigenvalues for sub-elliptic operators will  also be given, which, in a certain sense, has the optimal growth order.