Bowen‘s Homepage

Welcome to my homepage!

Hello, I’m Bowen Liu, a PhD student in Yau Mathematical Sciences Center of Tsinghua University from Sept 2022, under the supervision of Chenglong Yu. My research interests focus on the non-abelian Hodge theory over algebraically closed field. Also, I’m interested in stabilities of vector bundle, Higgs bundle, and in derived category.

My email is liubw22 [at] mails [dot] tsinghua [dot] edu [dot] cn, and my workstation is S103 at C740 of Shuangqing Complex Building A. Please contact me by email or let’s have coffee at Shuangqing if you find any mistake in my notes, and all comments are welcomed.

DREAMS OF LOVE AND HOPE SHALL NEVER DIE!

Notes

Seminar Notes

• Seminar on Higgs bundles and related topics. Here is the seminar note written by Qiliang Luo, Bating Xie and myself, Spring 2025.

• Bridgeland stability and dHYM metric of line bundles on surface (note version). This is my final report for the course “topics in derived categories”, lectured by Yuwei Fan, 2024/11/28.

• Notes on Chern Inequalities. Based on a talk I gave, I discuss the Chern number inequalities of semistable torsion-free sheaves, 2024/05/12.

• Notes on a brief Introduction to Hodge Theory. Based on a talk I gave, I covered a brief introduction to Hodge theory, such as Hodge structure, variation of Hodge structures and differential geometry of Hodge bundles, 2024/04/09.

• Notes on Algebraic de Rham Cohomology. I discuss algebraic de Rham complexes, such as comparison theorems, E1 degenerations of Hodge to de Rham spectral sequences, Spring 2024.

• Seminar on toric variety, and here are some lecture notes: 09-23, 10-07, 10-28, 11-11, 11-18, 12-30, 1-07, Fall 2023.

• Curvatures of left-invariant metrics on Lie groups. Slides on curvature properties of left-invariant metrics on Lie groups, and the main reference is Milnor’s paper, 2023/09/22.

• Review of topology (note version) frequently used in geometry, 2023/06/27.

• Uniqueness of the K"ahler structure of complex projective space (note version). This is my final report for the course “advanced topics in complex geometry”, lectured by Xiaokui Yang, 2023/06/08.

• Principal bundle and its applications. Based on a talk I gave, In this lecture, firstly I introduce principal bundle and connections on it, and then I introduce its applications on Chern-Weil theory, Spring 2023.

• Spectral sequence and its applications. Based on a talk I gave in a seminar series about Bott-Tu’s GTM82, which is organized by Chenglong Yu, Fall 2022.

• Commutative algebra, Summer 2022.

Lecture Notes

• Abelian variety, lectured by Mao Sheng, Spring 2025.

• Linear algebra, lectured by Chenglong Yu, Spring 2025.

• Five lectures on Introduction to singularities, lectured by Ruijie Yang, and I typed the draft version of it, Summer 2024.

• Algebraic curve, lectured by Chenglong Yu, Fall 2023.

• Representation theory (Draft version), lectured by Emanuel Scheidegger, Spring 2022.

• Homology theory, lectured by Huabin Ge, Spring 2022.

Learning Notes

• Algebraic geometry. Notes I typed when I learned algebraic geometry from a course taught by Lei Fu, Fall 2023. (Last updated 2024/1/13)

• Symmetric space. Notes I typed about symmetric space, and the main reference is the textbook written by Sigurdur Helgason. (Last updated 2023/09/12)

• Complex geometry I (Draft version). Lecture notes about complex differential geometry. The initial draft of this note is taken in spring 2022. in the course about Hodge theory taught by Qizheng Yin, and that’s my first time to learn this beautiful theory. Later in Spring 2023 I added more useful techniques and advanced topics (such as Schwarz computations, Calabi-Yau theorem and so on) I learnt in the course “advanced topics in complex geometry” given by Xiaokui Yang. I also added some other materials from classical books to make this note self-contained.

• Partial differential equation, Notes I typed when I learned about PDE, and it mainly followed the textbook written by Evans, Fall 2022.

• Riemannian geometry. Notes I typed when I learned about Riemannian geometry from a course taught by Xiaokui Yang in 2022Fall, and from this course I learnt quite a lot, Fall 2022.

Teaching

Lecturer

• Review of Algebra1, a quick review for first year undergraduate students, Winter 2024,

Teaching Assistant

• Linear algebra, lectured by Chenglong Yu, Spring 2025.

• Riemann surfaces, lectured by Dan Xie, Fall 2024.

• Algebra2-H, lectured by Xiang He, solutions to homework, Spring 2024.

• Algebra1, lectured by Xiang He, solutions to homework, Fall 2023.

• Algebra2-H, lectured by Chenglong Yu, part solutions to homework 1/3/5/7, 9/11/13 and lecture note, Spring 2023.

• Abstract algebra, lectured by Xiang He, solutions to homework, Fall 2022.

• Algebra1-H, lectured by Chenglong Yu, lecture note, Fall 2022.

Travel (to workshops, summer schools and so on)

• Dec 22 - Dec 27, 2024, Workshop on algebraic geometry, TSIMF, Sanya, China.

• Aug 16 - Aug 23, 2024, Workshop on Hodge theory 2024, University of Science and Technology of China, Hefei, China.

• Jul 18 - Jul 24, 2024, International Summer School of Algebraic Geometry, Fudan University, Shanghai, China.

• July 8 - July 12, 2024, 2024 Tianyuan Summer School in Algebraic Geometry, TMRC, Kunming, China.

• June 24 - July 5, 2024, Integrable Systems and Algebraic Geometry, BIMSA, Beijing, China.

• Apr 22 - Apr 26, 2024, Complex Algebraic Geometry and Related Topics, MCM, Beijing, China.

• Jan 22 - Jan 26, 2024, Winter School on Homogeneous Dynamics, Fudan University, Shanghai, China.

• Jan 14 - Jan 19, 2024, Tsinghua-Tokyo workshop on Calabi-Yau, IPMU, Tokyo, Japan.

• Dec 22 - Dec 26, 2023, Workshop on algebraic geometry, TSIMF, Sanya, China.

• Aug 9 - Aug 13, 2023, Workshop for algebraic geometry and complex geometry, Wuhan University, Wuhan, China.

• Jul 10 - Jul 21, 2023, International Summer School of Algebraic Geometry, Fudan University, Shanghai, China.

• Jul 31 - Aug 5, 2022, ICCM2022, Tsinghua University and Southeast University, Nanjing, China.

• Jul 12 - Jul 22, 2021, Number theory summer school, Xi’an Jiaotong University, Xi’an, China.