School of Science Department of Mathematics 42 Geometric Flows Supervisor: FONG, Tsz Ho / MATH Student: SU, Zhaohao / DSCT Course: UROP2100, Fall 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 study Hamilton’s paper, THREEMANIFOLDS WITH POSITIVE RICCI CURVATURE, which was published in 1982 and was considered to be a very important research result in the field of differential geometry. We introduced some notations and some basic results of this paper here, and show more details about the evolution of the curvature, most of them are the computations based on the covariant derivative on the normal geodesic coordinate to help understand this paper better. Geometric Flows Supervisor: FONG, Tsz Ho / MATH Student: WU, Sen Yuk / MATH-IRE Course: UROP1000, Summer This is a review of “Three manifold with positive Ricci curvature” by Richard Hamilton in 1982. The result given by the author in the paper is more or less the same as a weaken version of Poincare conjecture, which gives extra conditions in dimension and compactness of the manifold. A Machine Learning Approach to Study the Relationship Between Urban Morphology and Urban Heat Island Supervisor: FUNG, Jimmy Chi Hung / MATH Student: MOK, Wan Hin / MATH-IRE Course: UROP4100, Fall UROP4100, Spring Urban morphology plays a significant role in the urban heat island effect. Previous semesters, we built various Random Forest models to predict land surface temperatures (LSTs) based on urban morphology and geographical factors. This semester, we made minor changes to the models and focused on evaluating their performance in greater detail. Specifically, we studied the models' ability to predict LSTs beyond the training period and examined the relationship between model performance and the amount of cloud cover. Additionally, we attempted to utilize the model to simulate the urban heat island effect for future development sites.
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