Research Experience

My Math Research Experience

My academic journey is deeply rooted in mathematics, which fascinates me endlessly. I am also keenly interested in physics and chemistry, which complement my love for math and expand my understanding of the world. My academic path has been richly adorned with extensive research in mathematics. During my high school years, I've engaged in several projects Quaternion and Maxwell Equations, Prime Divisors of Perfect Number, Rubik's Cube Group, RSA Algorithm. These experiences not only honed my analytical and problem-solving skills but also deepened my appreciation for the elegance and complexity of mathematical theories.

Quaternions and Maxwell Equations

Introduction to this project: In this intriguing research project, I delve into the advantages of using quaternions for 3-dimensional geometric rotation over traditional Euler angles and matrices. My exploration begins with a historical overview, highlighting the significant contributions of mathematicians like Cardano and Hamilton in the development of complex numbers and quaternions. The paper then transitions to a technical exposition of quaternions, explaining their basic symbols and computation rules. I discuss Hamilton's perspective on quaternions and their potential impact on mathematics and physics, particularly in defining critical differential operators such as gradient, divergence, and curl. Furthermore, I investigate the geometric implications of quaternions, including their role in mapping between spaces and subspaces, and their application in understanding rotations within four-dimensional spaces. This part of my research emphasizes the importance of quaternions in projecting four-dimensional spaces into three-dimensional hyperplanes. Additionally, the project contrasts the use of Euler angles and rotation matrices with quaternion methods, offering insights into their respective advantages and limitations. The project concludes by highlighting the practical applications of quaternions in various fields, particularly in simplifying calculations involving rotations and transformations in three-dimensional space.

Link to the seminar slide of this project: Link

Prime Divisors of Perfect Number

Introduction to this project: In the research titled "Prime Divisors of Perfect Numbers," I examined the intricate characteristics of perfect numbers. Perfect numbers are intriguing in number theory, defined as integers that are the sum of their proper divisors. This project specifically investigates the properties and lower bounds of odd perfect numbers, a subset that has remained largely elusive in the mathematical community. The study begins with an overview of the historical context and basic concepts related to perfect numbers, tracing back to classical theories. The primary focus is on understanding the structure and components of odd perfect numbers. Through an examination of divisibility rules and prime number theory, the research seeks to establish a more concrete understanding of these numbers. A significant aspect of this study is the investigation into the number of distinct prime factors that an odd perfect number must possess. By exploring various mathematical theorems and employing analytical methods, the project aims to shed light on the lower bounds and structure of these rare numbers.

Link: https://drive.google.com/file/d/16XcEzUU2Xqlh2iT-ZfbouwQFEA1JCf2D/view?usp=drive_link

Rubik's Cube Group

Introduction to this project: In my research project titled "Rubik's Cube Group," I explored the mathematical intricacies of the Rubik’s Cube, particularly through the lens of group theory. This project is a deep dive into the structural and mathematical aspects of one of the most popular puzzles in the world. The research begins with a foundational understanding of group theory, introducing concepts such as permutations, symmetric groups, and the Rubik’s Cube group itself. The central idea revolves around the permutation group formed by the Rubik's Cube, denoted as G, where each element corresponds to a sequence of rotations on the cube's faces. I investigated the Rubik's Cube's permutations, focusing on the possible arrangements and movements within its structure. The study encompasses both "legal" and "illegal" states of the cube, providing a clear distinction between solvable and unsolvable configurations. This differentiation is crucial for understanding the puzzle's complexity and the mathematical principles governing its solutions. Throughout this project, I engaged with advanced mathematical concepts, including direct product groups, cyclic groups, and the application of group theory to solve real-life problems. The research not only deepens the understanding of the Rubik’s Cube as a mathematical model but also enhances problem-solving skills and analytical thinking.

Link: https://drive.google.com/file/d/14DvXasxGDX_dgGELws2b3pUIcQwFHrFn/view?usp=drive_link

RSA Algorithm and modulo arithmetic

Introduction to the project: In my research project, "RSA Algorithm and modulo arithmetic," I delved into the fascinating world of cryptography, focusing on the RSA algorithm - a cornerstone of modern digital security. This work is a comprehensive study of how mathematical principles, particularly those in number theory, underpin secure digital communication. At the outset, the project introduces the fundamental concepts of number theory and cryptography, setting the stage for a detailed exploration of the RSA algorithm. This exploration includes the mathematical process of encryption and decryption using RSA, emphasizing its reliance on the properties of prime numbers and modulo arithmetic. Central to the study is the RSA algorithm's use of a pair of keys - a public key for encryption and a private key for decryption. This asymmetric approach to cryptography is what makes RSA so powerful in securing digital communications. I explain how the RSA algorithm functions and demonstrate its application through practical examples, such as encrypting a simple message like "I LOVE MATH." Moreover, the project assesses the strengths and limitations of RSA. While it highlights RSA's robustness against various forms of cryptographic attacks, it also addresses its operational challenges, such as the computational complexity involved in generating key pairs and the slower processing speeds compared to symmetric cryptographic algorithms.

Link: https://drive.google.com/file/d/1mnu6vcKEXj65MGHqFDt0LzDtNKo17fOG/view?usp=drive_link