Mathematics

Among geometric evolution problems the motion of a surface accordingto its mean curvatureis the best known and it has been widely studied in the last fourdecades. Since singularities mayappear during the evolution, several weak formulations have beenproposed to describe the longtime behaviour of surfaces. One of the possibilities is to representthe initial surface as the level setof an auxiliary (initial) function and then to let evolve all thelevel sets of such a function accordingto the same geometric law. This procedure transforms the originalgeometric evolution probleminto an initial value problem for a suitable degenerate parabolic PDE,which can be treated usingthe machinery of viscosity solutions. This is the so-called level setapproach and the course willmainly focus on it (and on the special case of the mean curvatureflow), by introducing also themain notions and results concerning viscosity solutions needed for the theory.As time permits, in the last part of the course we will also presentmore recent developments.Topics may include: minimizing movements, distributional formulationvia distance functions, crystalline and  generalized (possiblynonlocal) curvature motions.