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This is a translation of Roland Barthes' Mythologies, which studies the evolution of social symbols.
Soap-powders and Detergents discussed how new chemicals helped reform our perception towards washing, and how psychoanalysis symbols change.
Text in Simplified Chinese.

Disclaimer:
The Following projects are finished by myself in high school, some notations might not be standard, some expressions and methods may not be ideal. Conceptual mistakes may appear. However, I tried my best in these projects anyway (using my knowledge back then).

On a Linear Group Isomorphic to $C_2⋉C_m^{n-1}$

This paper focuses on a set of $n\times n$ matrices under modulo $m$, ${𝒮}_{n,m}$, and a group, ${𝔗}_{n,m}$, generated by ${𝒮}_{n,m}$. ${𝒮}_{n,m}$ have the following properties: 1. every matrix is involutory; 2. any pair of matrices has order $m$; 3. any triplet of matrices is involutory. This paper also explores the expression of elements in ${𝔗}_{n,m}$ and the geometrical significance of the group.

• This project received the Second Prize in S.-T. Yau High School Science Awards 2024.

On a Linear Group of Seventh Chord Transformation

Seventh chords are widely used in music. What are the relationships between different seventh chords? Can these relationships be explained mathematically? This paper classifies transformations between seventh chords into six types using Euler's principle, namely $U_1$, $V_1$, $W_1$, $U_2$, $V_2$, and $W_2$. Each of these transformations is in matrix form. This paper constructs a group generated by these transformations and names this group the General Transformation Group ($GT$). This paper claims that every element in $GT$ can be expressed as ${U_1}^k{(U_1{V}_1)}^{a_1}{(U_1{W}_1)}^{a_2}{(U_1{U}_2)}^{a_3}$. This paper also determines that $GT$ is isomorphic to {\mathbb{Z}}_2⋉({\mathbb{Z}}_{12}\times {\mathbb{Z}}_{12}\times {\mathbb{Z}}_{12}), and its center is isomorphic to $C_2\times C_2\times C_2$. By using this group, we can examine the transformation between dominant sevenths and half-diminished sevenths from a generalized 4-chord perspective. This group also gives musicologists a theoretical guideline to seventh chords and their relationships, and provides musicians and auto accompaniment with a clear picture of the usage of seventh chords. This paper encourages further research into transformations of chords beyond seventh chords, aiming to broaden the applicability of the $GT$ group's principles.

• An additional interactive 3-D model of the chord transformation structure (Tonnez) is available here (take about 10 seconds to load).
• This project received the Fourth Prize in the International Science and Engineering Fair 2024 and Honorable Mention by AMS.

Mathematics Scholarship
Department of Mathematics, CUHK
November 2025

Second Prize
S.-T. Yau High School Science Awards 2024 (Mathematics)
November 2024

Fourth Place | Info
2024 Regeneron International Science and Engineering Fair, Society for Science
May 2024

Honorable Mention | Info
2024 Regeneron International Science and Engineering Fair, American Mathematical Society
May 2024

Outstanding Student | Info (Chinese)
National Science Talent Program, P.R.C. Ministry of Education, China Association of Science and Technology
December 2023

Second Prize
S.-T. Yau High School Science Awards 2023 (Mathematics)
November 2023

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