Intro to Smooth Manifolds, Lee -- sweeping intro to geometry with minimal prereqs, great at balancing the nitty gritty details with conveying intuition
A Course In Arithmetic, Serre -- classically terse and elegant intro to algebraic and analytic number theory. Goes from quadratic forms to Dirichlet's theorem to modular forms in a mere 100 pgs!
Princeton Lectures in Analysis, Stein & Shakarchi -- 4 books covering much of classical/modern analysis, they really shine in their discussion of applications
The large scale structure of space-time, Hawking & Ellis -- The most mathematically satisfying treatment of general relativity I've found. High points include proof of singularity theorems!
Spin Geometry, Lawson & Michelson -- Deep dive into the enigmatic "spin groups" and their applications in geometry. Also the only good (book) reference I could find on the index theorem
Lee is a good book but Loring Tu's An Introduction to Manifolds is a masterpiece in clarity, conciseness, and notation. I'd heavily recommend it over Lee, which can be a bit meandering, for a first course, with a follow-up with Lee and others for material Tu leaves out. Tu's followups in differential geometry and algebraic topology also follow smoothly from it.
Intro to Smooth Manifolds, Lee -- sweeping intro to geometry with minimal prereqs, great at balancing the nitty gritty details with conveying intuition
A Course In Arithmetic, Serre -- classically terse and elegant intro to algebraic and analytic number theory. Goes from quadratic forms to Dirichlet's theorem to modular forms in a mere 100 pgs!
Princeton Lectures in Analysis, Stein & Shakarchi -- 4 books covering much of classical/modern analysis, they really shine in their discussion of applications
The large scale structure of space-time, Hawking & Ellis -- The most mathematically satisfying treatment of general relativity I've found. High points include proof of singularity theorems!
Spin Geometry, Lawson & Michelson -- Deep dive into the enigmatic "spin groups" and their applications in geometry. Also the only good (book) reference I could find on the index theorem