Mathematical Reading Notes

This page records my mathematical reading program and note-taking structure. Public PDF notes will be linked here after they are uploaded and ready for release.

Stage I. Foundations and Mathematical Thinking

• Richard Courant and Herbert Robbins, What is Mathematics?
• G. Polya, How to Solve It
• Daniel J. Velleman, How to Prove It: A Structured Approach
• G. M. Fichtenholz, Course of Differential and Integral Calculus
• Sheldon Axler, Linear Algebra Done Right
• Paul R. Halmos, Naive Set Theory

Stage II. Undergraduate Core

• Walter Rudin, Principles of Mathematical Analysis
• David S. Dummit and Richard M. Foote, Abstract Algebra
• James R. Munkres, Topology
• Lars Ahlfors, Complex Analysis
• V. I. Arnold, Ordinary Differential Equations
• Michael Spivak, Calculus on Manifolds

Stage III. Advanced Undergraduate and Early Graduate Mathematics

• Walter Rudin, Real and Complex Analysis
• Serge Lang, Algebra
• Allen Hatcher, Algebraic Topology
• Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces
• John M. Lee, Introduction to Smooth Manifolds
• Jean-Pierre Serre, A Course in Arithmetic

Stage IV. Graduate-Level Analysis, Geometry, and Algebra

• Lawrence C. Evans, Partial Differential Equations
• Peter D. Lax, Functional Analysis
• Manfredo P. do Carmo, Riemannian Geometry
• Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology
• M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra
• Robin Hartshorne, Algebraic Geometry

Stage V. Research-Level Directions

• Hajime Urakawa, Calculus of Variations and Harmonic Maps
• John Milnor, Morse Theory
• John Milnor and James Stasheff, Characteristic Classes
• David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order
• Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics
• Simon Brendle, Ricci Flow and the Sphere Theorem