School of Science Department of Mathematics 38 Department of Mathematics Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: CHAU Yu Hei / DSCT Course: UROP1100, Spring UROP2100, Summer This document is a progress report that indicates an attempt in this summer to generalize the result of GuanLi which proves shows a variant of the Alexandrov- Fenchel inequality for star shaped domains. Some conditions are identified to yield results similar to Guan-Li's, however it would ultimately be shown that the conditions are not satisfied in some sense except for the known functions, so no new interesting results can be established. Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: CHUNG Tsun Ho Anson / MATH-PMA Course: UROP2100, Fall UROP3100, Spring UROP1000, Summer This progress report contains results derived during the last two months regarding general curve shortening flow. The first major result concerns when a translating soliton of the general curve shortening flow can be contained within a finite strip. This problem can be studied using elementary observations and ODE comparison principles. Essentially, we require the speed function to be bounded by the square root of curvature when curvature is close to 0. The second major result establishes a more general form of the Harnack inequality. Previous results considered only cases when the speed function is a monomial, or satisfies certain convexity requirements. Here we set out a condition that is satisfied by a large class of functions including polynomials and non-homogeneous functions. Geometric Flows Supervisor: FONG Tsz Ho / MATH Student: LAU Kwun Chai Michael / MATH-PM Course: UROP1100, Summer This report is to give a short summary on smooth manifolds and the motivation behind for developing this mathematical object. For example, definitions and some of the important result about regular hypersurface, manifolds with or without boundary, tangent spaces of manifolds, orientability and submanifolds will be given. Finally, using the result from smooth manifolds, such as immersion theorem and submersion theorem, we can show that the intersection curve between a regular hypersurface and plane parallel to tangent vector of the surface at certain point, is locally smooth and in fact a submanifold of the hypersurface.