School of Science Department of Mathematics 39 Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: SHIH Jackson / QFIN Course: UROP1100, Fall UROP2100, Spring Continuing from the last semester, where we have acquired basic knowledge of Riemannian geometry, we started reading and have finished going through the classical paper in Ricci flow by Hamilton. After that, we started explore literature on our own to broaden our horizon and try to perhaps come up with some toy problems we could work on. As the starting point, I have scanned through a few papers on mean curvature flow. In the study of hypersurfaces evolving under (inverse) mean curvature flow, monotone quantities are often applied to proofs of some important results. In this report, I would first introduce the preliminary knowledge of geometry on hypersurfaces, mean curvature flow and present some results I have read through. Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: SU Zhaohao / DSCT Course: UROP1100, Summer Riemann manifold is one of the most important study objects in modern differential geometry and a basic topic in undergraduate mathematics study. In this report, we are going to review some interesting propositions and results during the study of calculus on manifold and Riemann geometry, such as the Cartan’s Magic formula, the local expression of the normal vector of a smooth hypersurface and relate it with the first fundamental form of Riemann manifold, and the orientability of real projective space with odd dimensions and try to give some brief proofs with basic computations to contribute to the better understanding of them.