B.S. in Mathematics, CUHK (expected graduation 2029)
E-mail: hstz@link.cuhk.edu.hk
Call me Tianze instead. Huang and Song are my parents' family names respectively.
This report reviews some basic definitions about Lie algebra and tensor products, constructs irreducible representations of $\mathfrak{sl}(2, \mathbb{F})$ with arbitrary positive integer dimension, and presents a proof of the Clebsch-Gordan formula.
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In life, his favorite was living.
I gathered some of the hardest words! Challenge me.
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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).
This paper focuses on a set of $n\times n$ matrices under modulo $m$, $\mathcal{S}_{n,m}$, and a group, $\mathfrak{T}_{n,m}$, generated by $\mathcal{S}_{n,m}$. $\mathcal{S}_{n,m}$ have the following properties: 1. the matrices are involutory; 2. any pair of matrices has order $m$; 3. any triplet of matrices has order $2$. This paper also explores the expression of elements in $\mathfrak{T}_{n,m}$ and the geometrical significance of the group.
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 \({U_1}^k{(U_1V_1)}^{a_1}{(U_1W_1)}^{a_2}{(U_1U_2)}^{a_3}\). This paper also determines that \(GT\) is isomorphic to \(\mathbb{Z}_{2} \ltimes ({\mathbb{Z}_{12}} \times \mathbb{Z}_{12} \times \mathbb{Z}_{12})\), 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.
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
My WeChat Official Account 勒贝格测度 (Chinese, may not display properly on PC or Mac), for my compositions and essays.
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Page created by Tianze. Updated Jun 24, 2026.