||Please visit "https://sites.google.com/view/choiks/seminar" for Zoom ID. Ancient flows have been intensively studied in the mean curvature flow, a higher dimensional version of the curve-shortening flow. In particular, ancient mean curvature flows are useful to investigate singularities. In this talk, I will be talking about our study of the ancient solution of alpha-curve-shortening flow in R^2. Daskalopoulos, Hamilton, and Sesum classify ancient solutions for alpha=1 case, however, for alpha<1, very few are known and especially for small ones. Along this direction, we first construct a family of non-homothetic ancient flows whose entropy is finite. We determine the Morse indices and kernels of the linearized operator of self-shrinkers to the flows. Conversely, we are able to classify all the ancient solutions with finite entropy. It turns out all ancient solutions have the same asymptotic as the ones we have constructed. This work is joint with Keysongsu Choi.