## Honours Sessions

### 01.015 SHARP Honours Session Term 1

From Probability Theory and Markov Chains to the Parrondo Paradox

Week 1 to 3
Instructor: Dr Cheong Kang Hao

Description:
Parrondo's paradox is the counter-intuitive result where mixing two or more losing games can surprisingly produce a winning outcome. These games are named after the Spanish physicist, J.M.R. Parrondo, who inspired the beginning of these games. We will build up the concepts of Markov chains and Monte Carlo simulation before going into the Parrondo games. Through this short course, it is hoped that students can replicate results, verify claims, and extend the analysis in unforeseen ways.

Number Theory and Cryptography

Week 4 to 6
Instructor: Dr Wong Wei Pin

Description:
Imagine how can two parties send secret messages through an unsafe channel without prior communications? This is the kind of problems that can be solved by cryptography. Cryptography is indispensable in this digital era. It plays the vital role of maintaining the confidentiality, integrity and authenticity of all kinds of information in all aspects of our lives. To the surprise of many, the foundation of cryptography is number theory. In this course, you will first learn some elementary number theory concepts such as Division Algorithm, Greatest Common Divisor, Euclidean Algorithm, Modular Arithmetic, Fermat’s Little Theorem, Euler’s phi-function and Euler’s Theorem, which serves as the pre-requisites to understand cryptosystems that will be covered in the course, which are Symmetric Ciphers, Asymmetric Ciphers, Diffie-Hellman Key Exchange, ElGamal Public Key Cryptosystem and RSA Public Key Cryptosystem.

Introduction to Complex Analysis

Week 8 to 10
Instructor: Dr Ali Godjali

Description:
In this module, we will give a basic and brief introduction to complex analysis: limit, differentiation, and integration of complex-valued functions with the aim towards computing certain complicated real integrals (that may occur in engineering applications) via complex integrals. Given the fast pace of the module, the students should be familiar with the arithmetic on complex numbers and should know the fundamentals of real (single) variable Calculus. Along the way, we will cover some key topics: Cauchy-Riemann equations, Cauchy’s integral formula, Taylor and Laurent series, residue integration, etc.

Computational Statistics with Locality Sensitive Hashing (LSH) Algorithms

Week 11 to 13
Instructor: Dr Keegan Kang

Description:
Suppose that you have a large data set X, and are given the task of using a machine learning algorithm on the dataset to do some prediction or classification. Intuitively, the time taken for the machine learning algorithm to run would be proportional to the size of the dataset.

However, if the machine learning algorithm uses distances between vectors in X to do prediction or classification, one can use a LSH scheme to reduce the dimension of these vectors, yet preserve the approximate distances between them. We thus achieve some speedup at the cost of some error.

This course will cover a brief overview of scientific programming, plotting scientific graphs, Monte Carlo simulation, and variance reduction methods, before moving on to certain LSH schemes (b-bit minwise hashing, random projections, conditional random sampling).

### 01.016 SHARP Honours Session Term 2

Introduction to Statistical Physics

Week 1 to 3
Instructor: Dr Dario Poletti

Description:
Students will embark on a journey from microscopic to macrosopic which will allow them to gain a deeper insight into thermodynamics. The students will also learn that the tools and approaches used can give insights outside of the realm of physics as, for example, in information theory and in finance.

Nonlinear and Chaotic Dynamics

Week 4 to 6
Instructor: Prof Ricky Ang

Description:
Students will learn about important concepts of nonlinear dynamics and chaos, focused on its applications to science and engineering.

Week 4: Linear stability analysis, Different types of Bifurcations
Week 5: Extension from 1 dimensional (1D) to 2D systems
Week 6: Lorentz equation, Chaos, Strange attractor

Introduction to Quantum Mechanics

Week 8 to 10
Instructor: Dr Tan Da Yang

Description:
This foundational short course seeks to build up the physical and mathematical understanding of matter that operates in the quantum regime. Students will be taught both the mathematical formalisms commonly used in describing the physics in the quantum world and various technological applications during the course, and be able to appreciate the symbiosis between theory and application. This course also aims to be a starter for interested students to acquire the relevant literacy to explore the fascinating quantum world independently.

Solid State Physics

Week 11 to 13
Instructor: Dr Ang Yee Sin

Description:
This module will introduce the physics of solid materials and will illustrate how solid-state physics is applied in modern electronic and optoelectronic devices. A quantum mechanical description of crystal lattice, electronic band structures, and the classification of various solid states will be introduced. An introduction of charge transport physics based on semiclassical transport formalism will also be given. The students will also learn about the physics of the emerging classes of novel materials, such as two-dimensional and topological materials.

### 01.017 SHARP Honours Session Term 3

Spectroscopy for Beginners

Week 1 to 3
Instructor: Dr Anariba Franklin

Description:
The short course will provide an introduction to Bohr’s atomic structure and atomic transitions, UV-Vis spectroscopy, including Beer-Lambert Law and types of instrumentation, and vibrational spectroscopy and instrumentation.

Organic Chemistry

Week 4 to 6
Instructor: Dr Tan Mei Xuan

Description:
Students will learn about different types of organic reactions, stereochemistry and isomerism of organic molecules, and structure determination methods such as Nuclear Magnetic Spectroscopy (NMR).

The Diabetes Pandemic: A Global Healthcare Problem of the 21st Century

Week 8 to 13
Instructors: Dr Leo Chen Huei and Dr Khoo Xiaojuan

Description:
Diabetes mellitus is a chronic medical condition that is associated with abnormally high levels of sugar in the blood. If left untreated, diabetes can cause many complications including ulcers, stroke, and even death. The number of people with diabetes is rising at an exponential rate and the global health economic impact is massive. According to the International Diabetes Federation, Singapore has the one of highest proportion of diabetics among developed nations. Specifically, at least one in nine Singaporeans have been diagnosed with diabetes and this number is projected to double by 2050. Rising rates of Type 2 diabetes may relate to increasing obesity, an aging population, changes to diet and adoption of more sedentary behaviours. There is an urgent need to increase public awareness about diabetes prevention and control.

This course aims to improve knowledge around how diabetes develops, its socio-economic impact, and associated disease complications. We will further examine current and emerging drug and engineering approaches to diabetes treatment and management.

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