Nonlinear evolution equations are important tools in differential geometry and mathematical physics. Major examples considered in this course are the mean curvature flow, Hamilton's Ricciflow, Harmonic map heat flow and certain reaction-diffusion systems.
In general, solutions to these equations develop singularities in finite time. The theme of this lecture series is the study of the behaviour of solutions near such singularities. Topics covered will include maximum principle arguments, local smoothness estimates, rescaling techniques, compactness theorems, monotonicity/entropy formulas as well as concepts and ideas from minimal surface theory and geometric measure theory. Applications to problems in differential geometry will also be discussed.