As a mathematics student, I have always held myself to high standards, ensuring that every course I take has a solid foundation, rigorous logic, and a reasonable system framework. For each course, I use LaTeX to write my notes, and I also refer to many books by other authors. I integrate the more concise and clear theorem proof processes, more rigorous definitions, innovative course structures, and interesting ways of thinking into my notes.

When I was studying the course on Mathematical Analysis, I read “Mathematical Analysis” by Vladimir A. Zorich, “A Course of Mathematical Analysis” by Grigorii Fikhtengol’ts, and “Principles of Mathematical Analysis” by Walter Rudin. Among these, I independently finished the entire “Mathematical Analysis” by Vladimir A. Zorich, carefully reading and attempting to derive every theorem and proof. This book also helped me get started with Point Set Topology and Elementary Differential Geometry.

When I studied Advanced Algebra, I read “An Introduction to Algebra” by Alexei Kostrikin, “Advanced Algebra” from Peking University, “Algebra” by Roger Godement, and “Linear Algebra Done Right” by Sheldon Axler. Each book has its unique features. In “An Introduction to Algebra” by Alexei Kostrikin, I self-studied abstract algebra. In “Algebra” by Roger Godement, I learned new approaches to matrix simplification and finding Jordan forms. Sheldon Axler’s “Linear Algebra Done Right” took a different path, showcasing its style in finite-dimensional linear functionals and dual spaces. My interest in algebra grew steadily, and I later joined a research group focusing on the Casac-Alvero conjecture within the fields of Commutative Algebra and Algebraic Geometry, during which I studied “Commutative Algebra” by Michael Atiyah. I also read through some chapters of GTM 211 later on.

When studying Probability Theory and Mathematical Statistics, in addition to reviewing relevant lecture notes from Moscow University, I also read “A Probability Path” by Sidney Resnick. The latter rewrites probability theory using the language of measure theory and real analysis, and all theorems are given rigorous and detailed proofs. Of course, I have also read Andrey Kolmogorov’s probability theory from the Moscow Mathematical School.

When I took the course on Ordinary Differential Equations, I referred to many Chinese textbooks, but the one that left the deepest impression on me was “Ordinary Differential Equations” by Vladimir Arnold. While proving Lyapunov stability, he introduced many visual images, which made a strong impression on me. Additionally, the arrangement of his chapters was quite reasonable.

In my studies of real analysis, complex analysis, and functional analysis, I have been reading books by Elias M. Stein, who skillfully integrates the four segments of analysis, showcasing the connections between different analytical fields and emphasizing that analysis is a unified whole. Real analysis focuses on issues of measurability, functional analysis is concerned with the properties of operators, and complex analysis investigates holomorphic functions in the complex plane. Meanwhile, Fourier analysis acts like a main thread, being repeatedly mentioned across all segments of analysis, with its own theoretical development within each area.

For every course offered by the university, I dedicate my free time to reading textbooks by other authors, not just limiting myself to the school’s lectures and materials. Although this has consumed a considerable amount of my time and involved going through a lot of repetitive content, different authors possess distinct logical approaches and unique insights. They have provided me with various ways of thinking and emphasis on understanding new knowledge. In the proof of some theorems, Walter Rudin’s proofs are more innovative, placing greater importance on thought processes; Vladimir A. Zorich’s proofs are more systematic; Sidney Resnick’s proofs are exceptionally comprehensive; Elias M. Stein’s proofs are more concise and clear, with a very definite purpose. The diversity in theorem proofing by these authors has also influenced my mathematical thinking and approach, guiding me on when to follow conventional methods, when to engage in critical thinking, and when to be more creative, such as in constructing appropriate variable relationships.

I have also self-studied a significant amount of extracurricular knowledge. I have learned point-set topology and some content on algebraic topology, including homology theory. To engage in research, I taught myself commutative algebra and parts of algebraic geometry. I have also encountered Galois theory.