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 interest lies in the intersection of complex differential geometry and complex algebraic geometry.

My email is liubw22 [at] mails.tsinghua.edu.cn, and my workstation is S103 at C740 of Shuangqing Complex Building A. Please contact me if you find any mistake in my notes, and comments are welcomed. Thanks in advance for your attention.

Seminars

• (2024/05/12) It’s a lecture note of a talk I gave in a seminar. In this talk, we discuss the Chern number inequalities of semistable torsion-free sheaves, and here is the reference.
• (2024/04/09) It’s a lecture note of a talk I gave in a seminar. In this talk I give a brief introduction to Hodge theory (Hodge structure, variation of Hodge structures and differential geometry of Hodge bundles) as preliminaries for the paper.
• (2024Spring) It’s a lecture note about algebraic de Rham complex (such as comparison theorems, E1 degenerations of Hodge to de Rham spectral sequences and so on)
• (2023Fall) A seminar about toric variety, and here are some lecture notes: 09-23, 10-07, 10-28, 11-11, 11-18, 12-30, 1-07.
• (2023/09/22) Curvatures of left-invariant metrics on Lie groups. It’s a slide about curvature properties of left-invariant metrics on Lie groups, and the main reference is Milnor’s paper.
• (2023/06/27) Here are note and slides for a review of topology frequently used in geometry.
• (2023/06/08) Here are note and slides for my final report of course “advanced topics in complex geometry”, lectured by Xiaokui Yang, and topic of the final report is the uniqueness of the K"ahler structure of complex projective space.
• (2023Spring) It’s lecture note about principal bundle and its applications. In this lecture, firstly we introduce principal bundle and connections on it, and then we introduce its applications on Chern-Weil theory.
• (2022Fall) It’s a lecture note of talks (about spectral sequence and its applications) I gave in a seminar about Bott-Tu’s GTM82, which is organized by Chenglong Yu.
• (2022Summer, last updated 2023/01/30) Here is the draft of lecture note I typed for a seminar about commutative algebra.

Notes

• (Last updated 2024/1/13) Algebraic geometry. It’s a note I typed when I’m learning algebraic geometry from course taught by Lei Fu in 2023Fall.
• (2023Fall) Algebraic curve, lectured by Chenglong Yu.
• (Last updated 2023/09/12) Symmetric space. It’s a note I typed about symmetric space, and the main reference is the textbook written by Sigurdur Helgason.
• (Last updated 2023/10/13) Complex geometry I. It’s a lecture note about complex differential geometry. The initial draft of this note is taken in 2022Spring, in the course about Hodge theory taught by Qizheng Yin, and that’s my first time to learn this beautiful theory. Later in 2023Spring 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.
• (2022Fall) Partial differential equation, it’s a note I typed when I’m learning PDE, and it’s mainly followed the textbook written by Evans.
• (2022Fall, last updated 2023/07/27) Riemannian geometry, lectured by Xiaokui Yang, I also added some related materials.
• (2022Spring, BICMR) Representation theory, lectured by Emanuel Scheidegger.
• (2022Spring, BICMR) Homology theory, lectured by Huabin Ge.

TA

• (2024Spring, last updated 2024/04/24) Solutions to homework of Algebra2-H, lectured by Xiang He.
• (2024Winter) A review of Algebra1
• (2023Fall) Solutions to homework of Algebra1, lectured by Xiang He.
• (2023Spring, last updated 2023/08/18) Lecture notes of Algebra2-H, lectured by Chenglong Yu, and here are solutions to homework 1/3/5/7, 9/11/13.
• (2022Fall) Solutions to homework of Abstract algebra, lectured by Xiang He.
• (2022Fall, last updated 2023/01/23) Lecture notes of Algebra1-H, lectured by Chenglong Yu.

Travel

• 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.