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**

- Commutative Algebra II (Chinese), last revised Apr. 2024.
- Bass Number (Chinese), last revised Apr. 2024.
- Homological Smoothness of Commutative Algebras (Chinese), last revised Oct. 2023.
- Affine Hypersurfaces (Chinese), last revised Feb. 2024.
- Hochschild-Kostant-Rosenberg Theorem (Chinese), last revised Nov. 2023.
- Quotient Variety of Affine Varieties (Chinese), last revised Dec. 2023.
- Discrete Valuation Rings and Smoothness of Affine Curves (Chinese), last revised Nov. 2023.
- Cohen–Seidenberg Theory (Chinese), last revised Nov. 2023.
- Rank of Projective Modules (Chinese), last revised Nov. 2023.
- Perfect Fields (Chinese), last revised Dec. 2023.
- Dedekind Domain (Chinese), last revised Jan. 2024.
- Zariski Tangent Space (Chinese), last revised Jan. 2024.
- Sheaf of Regular Functions on Affine Varieties (Chinese), last revised Jan. 2024.
- Product Varieties and Affine Algebraic Groups (Chinese), last revised Nov. 2023.
- Affine Poisson Varieties (Chinese), last revised Dec. 2023.
- Basic Notions in the Ideal Theory of Poisson Algebras (Chinese), last revised Mar. 2024.
- Basic Galois Theory (Chinese), last revised Apr. 2024.
- Some Topological Properties of the Maximal Spectrum (Chinese), last revised Dec. 2023.
- Geometric Vector Bundles over Quasi-affine Varieties (Chinese), last revised Jan. 2024.
- Vector Fields on Affine Varieties (Chinese), last revised Jan. 2024.
- Kaplansky’s Theorem for U.F.D.s (Chinese), last revised Mar. 2024.
- Quadratic Integer Rings (Chinese), last revised Jan. 2024.
- Discriminant of an Algebraic Number Field (Chinese), last revised Jan. 2024.
- Birational Equivalence of Irreducible Affine Varieties (Chinese), last revised Jan. 2024.
- Rings of Smooth Functions of Smooth Manifolds (Chinese), last revised Jan. 2024.
- Module of Derivations (Chinese), last revised Jan. 2024.
- Covector Fields and Kähler Differentials (Chinese), last revised Feb. 2024.
- Alternating Tensors and Kähler Differentials of Higher Degree (Chinese), last revised Feb. 2024.
- Interior Products, Koszul Differentials and Contraction Maps (Chinese), last revised Feb. 2024.
- Canonical Poisson Structure on Symplectic Manifolds (Chinese), last revised Feb. 2024.
- Multivector Fields on Manifolds (Chinese), last revised Feb. 2024.
- From Affine Varieties to Affine Schemes (Chinese), last revised Jan. 2024.

**Noncommutative Algebra**

- Goldie’s Theorem (Chinese), last revised Apr. 2024.
- Polynomial Identity Algebras (Chinese), last revised Apr. 2024.
- Rings of Differential Operators, last revised Mar. 2024.
- Dixmier-Moeglin Equivalence (Chinese), last revised Dec. 2023.
- Graded Algebras (Chinese), last revised Dec. 2023.
- An Example of a Centrally Infinite Division Ring (Chinese), last revised Jun. 2023.
- Endomorphism Ring of an $\aleph_{0}$-Dimensional Vector Space (Chinese), last revised Nov. 2023.
- Differential Core of Prime Ideals, last revised Dec. 2023.
- Trivial Extension of Rings (Chinese), last revised Dec. 2023.
- Wedderburn Principal Theorem (Chinese), last revised Jan. 2024.
- Frobenius Algebras (Chinese), last revised Mar. 2024.
- Quantum Affine Spaces (Chinese), last revised Mar. 2024.

**Homological Algebra**

- Notes on Freyd–Mitchell Embedding Theorem (Chinese), last revised Oct. 2023.
- Van den Bergh Duality in Hochschild (Co)homology (Chinese), last revised Sep. 2023.
- Formal Deformation of Associative Algebras (Chinese), last revised Nov. 2023.
- Tor Functor Commutes with Direct Limits (Chinese), last revised Nov. 2023.

**Group Theory**

- Dihedral Group (Chinese), last revised Sep. 2023.
- Grothendieck Group (Chinese), last revised Jul. 2023.
- Simplicity of $A_{n}$ for $n \geq 5$ (Chinese), last revised Oct. 2023.
- Sylow’s Theorem (Chinese), last revised Oct. 2023.
- Free Product of Groups (Chinese), last revised Nov. 2023.
- Pushout in the Category of Groups (Chinese), last revised Nov. 2023.
- Cyclic Decomposition of Abelian Groups with Finite Exponent (Chinese), last revised Dec. 2023.

**Module Theory**

- Structure Theory for Finitely Generated Modules over P.I.D. (Chinese), last revised Mar. 2024.
- Morita Equivalence (Chinese), last revised Dec. 2023.
- Injective Hull of a Module (Chinese), last revised Apr. 2024.
- Projective Cover of a Module (Chinese), last revised Oct. 2023.
- Socle of a Module (Chinese), last revised Mar. 2023.
- Classical Isomorphisms about Finitely Generated Projective Modules (Chinese), last revised Mar. 2024.
- Tensor Fields and Multlinear Functions (Chinese), last revised Feb. 2024.
- An Equivalent Definition of Alternating Tensors (Chinese), last revised Feb. 2024.

**Category Theory**

- Quotient Category (Chinese), last revised Aug. 2023.
- Adjoint Functor (Chinese), last revised Aug. 2023.
- Functor Category (Chinese), last revised Aug. 2023.

**Representation Theory**

- Brenner-ButlerTheorem (Chinese), last revised Aug. 2023.
- Classical Representation Theory of Finite Groups (Chinese), last revised Dec. 2023.
- Central Characters (Chinese), last revised Mar. 2024.

**Nonassociative Algebra**

- Polynomial Poisson Algebra (Chinese), last revised Oct. 2023.
- Differential Graded Lie Algebra (Chinese), last revised Nov. 2023.
- Lie Derivatives of Differential Forms (Chinese), last revised Feb. 2024.