I am a PhD student in the School of Mathematical Sciences of Fudan University advised by Prof. Quanshui Wu. My current research interests lie primarily in noncommutative algebra, Poisson algebra and deformation theory. I am also interested in Rubik’s Cube, table tennis and badminton. Here is my homepage on the website of the World Cube Association.

Previously, I received my bachelor’s degree in Mathematics and Applied Mathematics at Shanghai University, supervised by Prof. Zhuo-Heng He.

## Education

B.S. in Mathematics and Applied Mathematics, Shanghai University Thesis: | 2017-2021 |

## Publications

**The equivalence canonical form of five quaternion matrices with applications to imaging and Sylvester-type equations**

Shao-Wen Yu, Zhuo-Heng He, __Tian-Cheng Qi__, and Xiang-Xiang Wang

Journal of Computational and Applied Mathematics, 2021. paper

## Teaching Assistant

Modern Algebra II(H), Syllabus, Fudan University, China | 2023.9–2023.12 |

## Study Notes

I am a person who is keen on using $\LaTeX$ to write some notes on algebraic topics. The following are some of my notes. There might be some typos and mistakes in these notes. Please let me know if you see one. Thanks! I should mention that we always adopt the following assumptions in these notes.

**Set-theoretical Assumptions** The axiom system used in my mathematical world is the von Neumann-Bernays-Gödel axiom system (abbreviated as NBG system, BG system or GB system) and I recognize the axiom of global choice. The concept of class was introduced by John von Neumann in 1925, P.I. Bernays introduced the BG system in 1937, and was improved by K.F. Gödel in 1940. J.R. Shoenfield published in 1954 the following **theorem**: Any statement that only contains set variables in the NBG system, if it can be proved to be true in the NBG system, then it is also true in the ZF system (Zermelo-Fraenkel axiom system). Similarly, if the statement containing only set variables in the BGC system (the axiom of choice is added on the basis of the BG system), and if it can be proved to be true in the GBC system, then it is also true in the ZFC system. For a class, we can also define an equivalence relation on it. Then the notion of equivalence classes arises. A basic question is: is it then possible to select an element from each equivalence class (possibly a proper class)? In this case the axiom of global choice does not seem to work. But, it is proved by D.S. Scott, an American logician, that this can always be down!(See Scott’s trick) That is, for a nonempty class, suppose there is an equivalence relation on it, then we can pick an element from each equivalence class!

Readers may refer to this book for some basic materials about set theory.

- Commutative Algebra II (Chinese), last revised Aug. 2024.
- Dedekind Domain (Chinese), last revised May. 2024.
- Zariski Tangent Space (Chinese), last revised July. 2024.
- From Affine Varieties to Affine Schemes (Chinese), last revised July. 2024.
- Goldie’s Theorem (Chinese), last revised Apr. 2024.
- Polynomial Identity Algebras (Chinese), last revised June. 2024.
- Category of Algebras with Trace (Chinese), last revised Apr. 2024.
- Notes on Freyd–Mitchell Embedding Theorem (Chinese), last revised Oct. 2023.
- Formal Deformation of Associative Algebras (Chinese), last revised Nov. 2023.
- Structure Theory for Finitely Generated Modules over P.I.D. (Chinese), last revised July. 2024.
- Morita Equivalence (Chinese), last revised Dec. 2023.
- Classical Isomorphisms about Finitely Generated Projective Modules (Chinese), last revised Mar. 2024.
- Brenner-ButlerTheorem (Chinese), last revised Aug. 2023.
- Classical Representation Theory of Finite Groups (Chinese), last revised Dec. 2023.
- Lie Algebra (Chinese), last revised July. 2024.