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