Reading Courses

Here is the list of books to bridge the gap between the “average undergraduate level” and the professional toolbox. Any comments/additions/objections are most welcome.

Algebra
Atiyah, McDonald, Introduction to Commutative Algebra
Shafarevich, Basic notions of algebra

Calculus on Manifolds, Differential Geometry
Boothby, An Introduction to Differential Manifolds and Riemannian Geometry
Morita, Geometry of Differential Forms
Dubrovin, Novikov, Fomenko, Modern Geometry
Milnor, Morse Theory

Complex Variables and Riemann Surfaces
Springer, Introduction to Riemann Surfaces
Forster, Riemann Surfaces
Miranda, Algebraic Curves and Riemann Surfaces
Titchmarsh, The theory of functions
Shabat, Introduction into Complex Analysis, Parts I (one variable), II (several variables)

Algebraic Geometry
Harris, Algebraic Geometry (A First Course)
Griffiths, Harris, Principles of Algebraic Geometry (Chapters 0,1,2)
Mumford, Algebraic geometry. I. Complex projective varieties.
Shafarevich, Basic Algebraic Geometry

Differential Equations
Arnold, Ordinary Differential Equations,
Arnold, Geometric Theory of Ordinary Differential Equations
Arnold, Mathematical Methods of Classical Mechanics
Ince, Ordinary Differential Equations

Singularities
Milnor, Singular Points of Complex Hypersurfaces
Arnold, Gussein-Zade, Varchenko, Singularities of Differential Maps, Vol. 1 (Ch. I), Vol. 2 (Ch. I & III)
Zoladek, The Monodromy Group