School of Science Department of Mathematics 52 Efficient Numerical Methods for Dynamic Interface Supervisor: LEUNG Shing Yu / MATH Student: ZHAO Yubo / SSCI Course: UROP1100, Summer This paper will study and discuss several methods for testing and comparing the accuracy of symplectic integrators representing a Hamiltonian system. Symplectic integration of Hamiltonian dynamics is already a developed method, and high-order (say 4th or 5th) explicit methods are established for separable Hamiltonian with quadratic kinetic energy. However, for general, nonlinear Hamiltonians, no explicit representation is found. Since symplectic integrators are numerical methods specialized for the Hamilton system and computation is excessively time-consuming, we have to balance accuracy and computing speed. Starting from the framework, we investigate methods for Hamiltonian of different orders, then compute and investigate their accuracy. Some Aspects of High-dimensional Probability Supervisor: WANG Ke / MATH Student: WU Zhiang / COSC Course: UROP1100, Summer The theory of random graph is related to both combinatorics and probability theory. It describes certain probability distributions over a set of graphs. Erdös and Rényi proposed the first model of random graphs, which has numerous variations and applications in engineering. In this report, we study (n,m) and (n,p) models and the evolution of a random graph. Some Aspects of High-dimensional Probability Supervisor: WANG Ke / MATH Student: XU Yitao / MATH-SF Course: UROP1100, Summer The whole report introduces some basic knowledge about concentration inequalities. We first introduce some tail probabilities of a single random variable and the sum of independent random variables, such as Markov’s inequality and Chernoff bounds. Then we show the proof of sharper Hoeffding bounds for bounded random variables. Next is the introduction to a martingale-based method which can be used to get the concentration of functions of independent random variables. Lastly, we introduce the ∅-entropy and its application to concentration inequalities.