Speaker : Maciej Cytowski Abstract: Numerical solutions and applications of Allen-Cahn equation. This talk will provide a brief introduction to dynamics and applications of the Allen-Cahn equation. Allen-Cahn equation is one of the simplest example of phase-field problems. It describes the evolution of a diffuse phase boundary, concentrated in a small region. It has been proved that this evolution approximates mean curvature flow. I will try to describe the dynamics of Allen-Cahn equation using the animations produced in my preliminary simulations. I will also show that some modifications of Allen-Cahn equation lead to applications in image segmentation. At the end of my talk I will formulate and describe the topic of my actual research, which is: "Numerical solutions of nonlinear diffusion processes over evolving geometries".