我们主要研究Hamitlon-Jacobi方程粘性解的奇点动力学，以及它在Mather理论及弱KAM理论中的应用，并以此解决Hamilton动力学的一些重要问题。在大于Mane临界值的能量面上，弱KAM不存在孤立奇点。奇点的动力学及其稳定性与Aubry类的拓扑结构、共轭点的产生与消失，粘性解以及障碍函数的正则性，局部极小轨道的构造有着密切的联系。这一领域目前来看是崭新的，是最优控制理论与Hamilton动力学的一些深刻问题的有机结合。我们拟解决以下问题：建立半凹函数的奇点传播的解析理论，解决粘性解奇点动力学的稳定性问题，建立奇点集拓扑结构与Aubry类的关系，给出奇点传播起点与终点与系统共轭点之间的关系，研究奇点与Lax-Oleinik半群收敛性之间的关系等。借此研究，我们还希望解决测地流系统的刚性，α函数可微性，局部极小不变集的存在性等问题。

We will study the sigularity dynamics and its stability problem of the viscosity solutions of the H-J equations with its applications to the Mather theory and the weak KAM theory, and its applications to the Hamiltonian dynamics.We know now that the sigularity of the barrier functions will propagate on the supercritical energy surface as well as that of the weak KAM solutions. This research area will be great application of the optimal control theory to some deep results in Hamiltonian dynamics. We hope to solve the following problem: building up an analytic setting for the propagation of singularities on the the weak KAM solutions and its stability problems; estabishing the relations between the topological structure of the Aubry class and that of the singular sets; charaterizing the conjugate points of the systems with the initial or ending points of the singular arcs; studing the singularities with the relation to the convergence of the L-O semigroup. Then we will apply them to the study of some concrete problems such as the rigidty problem on geodesic flows, the differentiability problem of the alpha-function, existence of the local minimal invariant sets, and so on.