School of Science Department of Mathematics 53 Lattice-based Cryptography, Pseudorandom Functions and Algebraic Number Theory Supervisor: XIONG Maosheng / MATH Student: SAMIN Thanic Nur / MATH-IRE Course: UROP1000, Summer In this report, we present a summary of some topics discussed in “An Invitation to Arithmetic Geometry”[Lor97] by Dino Lorenzini. In particular, we talk about integral closures of rings, specifically the ring of integers and polynomials. We also discuss the curve Zf ( �), which is the set of zeroes of a polynomial f in an algebraically closed field and how it relates to the set of maximal ideals of � [x, y]/(f). We also discuss the concept of localization and local rings. Finally, we give a brief overview of how a post quantum safe algorithm was attacked. Lattice-based Cryptography, Pseudorandom Functions and Algebraic Number Theory Supervisor: XIONG Maosheng / MATH Student: YUEN Theodore / MATH-PM Course: UROP1000, Summer This report follows the main reference by Dino Lorenzini, by introducing the concept of integral closure of commutative rings in a finite field extension of their respective field of fraction, follow by an important characterisation of local Dedekind domain, method of determining the integral closeness of an integral extension to a Dedekind domain and how it relates to the property of unique factorisation of ideals. The case of �[x; y]/(f) will be discussed in more detail, where we can gain insight to some relations between algebraic properties and geometric properties, and we will also use this example to demonstrate some properties of a Dedekind domain shown in this report. Research in AI and Machine Learning Supervisor: ZHANG Tong / MATH Student: XIONG Yixin / MATH-SFM Course: UROP1100, Fall This report introduces some basic settings of the M5 Machine Learning competition, suggests some approaches to handle the data that appeared in the competition, analyze the top solution result and its advantages, tries the Lightgbm model and gives some generalizations and insights to further research and industry applications. This report is a study of the M5 Competition, mainly focusing on solving the problem of how to handle the M5 data set in a primary state. It contains a small proportion of realization and innovation, and more thoughts will be brought up in the future.

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