The main objective of the project is to study geometric flows, more precisely equations in which one evolves a geometric object (for example a metric or a surface in the Euclidean space) in time, expecting it will improve its properties in time, for example become more symmetric, starting resembling familiar objects, such as spheres or cylinders. These geometric equations usually develop singularities in a finite amount of time, after which one cannot expect to have a nice solution to the considered geometric flow. One would like to understand more closely what happens at those singular times, and what the singularities look like. This should help one find a way to define a solution past the singularities. After repeating this finitely many times, one should get in the end very familiar geometric objects, and this could help, for example, to understand and classify all possible topologies of the initial geometric object. The PI expects that new students and postdocs, besides current ones will be trained, and that they will benefit from the research activity. The PI also plans to co-organize workshops in various topics in Geometric Analysis. This project is to study singularity formation in asymptotic sense, and to classify singularities in nonlinear parabolic equations which come from differential geometry problems, such as the evolution of a hypersurface in the Euclidean space by functions of its principal curvatures, and the Ricci flow. The first part of the project is