Learning Linear Algebra: I learned most of my linear algebra from functional analysis and module theory, but having seen generally the context and exposition of Linear Algebra Done Right by Axler, I would suggest this book to undergrads. Relearning Linear Algebra: Advanced Linear Algebra by Roman.
Learning Calculus: I have no idea how I learned calculus.
Learning ODE: God help me if I ever need to solve one of these.
Learning PDE: God help me if I ever need to solve one of these.
Set Theory and Foundations: Analysis I by Tao.
Learning Group Theory: Topics in Algebra by Herstein, Relearning Group Theory: Basic Algebra I by Jacobson.
Learning Ring Theory: Basic Algebra I by Jacobson.
Learning Galois Theory: Basic Algebra I by Jacobson and Abstract Algebra by Dummit and Foote, Learning More Galois Theory: Algebra by Lang.
Learning Basic Module Theory (Structure Theory over a PID): Advanced Linear Algebra by Roman.
Learning Commutative Algebra: Introduction to Commutative Algebra by Atiyah and Macdonald.
Learning Basic Analysis: Elementary Analysis by Ross and Analysis I by Tao, Learning Metric Space Topology: Principles of Mathematical Analysis By Rudin, Learning Basic Fourier Analysis: Fourier Analysis: An Introduction by Stein and Shakarchi.
Learning Functional Analysis: Real Analysis: Modern Techniques and Their Applications by Folland, as well as Some Fantastic Course Notes by my professor Murugan.
Learning Measure Theory: I have no idea why I learned measure theory.
Learning Probability: Probability: Theory and Examples by Durrett.
Learning (a very small amount in my case) Markov Chains: Markov Chains and Mixing Times by Levin and Peres.
Learning Basic Complex Analysis: Fundamentals of Complex Analysis with Applications to Engineering and Science by Saff and Snider, Learning (a very small amount in my case) Complex Geometry: Complex Geometry by Huybrechts.
Learning Basic (Algebraic?) Topology: A Basic Course in Algebraic Topology by Massey, Learning Algebraic Topology: Some Fantastic Course Notes by Williams, if a book is needed, Algebraic Topology by Hatcher - I am not particularly fond of this book but it seems to be the lesser of all evils.
Learning Differential Topology: Differential Topology by Guillemin and Pollack, Differential Forms in Algebraic Topology by Bott and Tu, as well as Some Fantastic Course Notes by my professor Kupers Reference for Differential Topology: Smooth Manifolds by Lee, I would not try to learn from this book, but it has many useful facts.
Learning Representation Theory for Finite Groups: I like This Book by Webb, the standard reference Linear Representations of Finite Groups by Serre is also good.
Learning Basic Category Theory: Category Theory in Context by Riehl, being comfortable with categorical language and methodology is still a work in progress for me.
Reference For Basic Lie Theory: Lie Groups, Lie Algebras and Representations: An Elementary Introduction by Hall is simple rigorous and concise.
Learning Elementary Number Theory: An Introduction To - The Theory of Numbers by Niven.
