No particular textbook.
PDE. Knowing differential geometry is also recommended but not required. No knowledge on particular flows is needed, I will introduce all what I need during the course.
I will talk about singularity formation in geometric flows, with the emphasis on the mean curvature flow and the Ricci flow. Ancient solutions, which are the solutions that exist for all times from negative infinity appear to be singularity models in above mentioned flows. Therefore, understanding and classifying those is very important in understanding the singularity formation. I will start with known results about ancient solutions in the semilinear heat equation and will talk about known classification results of ancient solutions in the Ricci flow and the mean curvature flow.