School of Science Department of Mathematics 51 Numerical Methods for Solving PDEs on Surfaces Supervisor: LEUNG, Shing Yu / MATH Student: WONG, Chi Sum / MATH-AM Course: UROP1000, Summer This project aims to introduce numerical methods for computing the equilibrium state of the Allen-Cahn Equation, considering a given initial condition and a homogeneous Neumann boundary condition. Rather than employing an explicit scheme that necessitates surface triangulation, this project utilizes the level set method for computation. The initial approach involves utilizing the forward Euler method, while subsequently introducing an enhanced scheme based on the operator splitting method and the CrankNicolson scheme. The implementation of these methods will be carried out in MATLAB, and the algorithm has been provided. However, it is important to note that the project does not include numerical experiments, nor does it provide a proof of accuracy and stability. Some Aspects of High-dimensional Probability Supervisor: WANG, Ke / MATH Student: XIONG, Yixin / MATH-SF Course: UROP1100, Fall UROP2100, Spring UROP3100, Summer The study of high-dimensional statistics has witnessed the emergence of a prominent trend known as matrix concentration, which serves as a crucial link between probability theory and theoretical computer science. This research project aims to explore both historical and newly emerging topics that contribute to a deeper understanding of concentration inequalities. Beginning with fundamental results in linear algebra, this report delves into the matrix Laplace transform method and the matrix Bernstein inequality, shedding light on their significance and applicability. Some Aspects of High-dimensional Probability Supervisor: WANG, Ke / MATH Student: XU, Yitao / MATH-SF Course: UROP2100, Fall The project explores two different methods to derive concentration inequalities, including the geometric method and the entropy method. We first study the concentration function and some of its applications which is the base of the geometric method. Then, we mainly studied the entropy method, which is inspired by Logarithmic Sobolev inequalities to derive concentration inequalities. We learned the usage of this method to improve Talagrand’s deviation inequalities and obtain some other inequalities, which have different applications.

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