This talk will present my recent results in certain directions within the realm of partial differential equations. On one hand, parabolic equations generate a semiflow in infinite dimensions, and hence we are interested in the dynamical question: what happens when time is large? We will explore the existence of a Lyapunov function, and the construction of global attractors (even when solutions grow unboundedly). On the other hand, elliptic equations with symmetric domains might give a priori symmetry of certain solutions themselves. We will treat spherical type symmetries. The main applications are the construction of black hole initial data for the Einstein equations, and the existence of vertices, spirals, and transition states for the Ginzburg-Landau equations.

Those results are mostly joint with Bernold Fiedler (Freie Universität Berlin), Juliana Pimentel (Universidade Federal do Rio de Janeiro) and Jia-Yuan Dai (National Taiwan University).