MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

2 deleted 1 characters in body

If you intend to study on your own the best approach is to follow a structured sequence just like in an ordinary Math degree.... but nevertheless not forgetting that everything is interconnected and prerequisites and applications are highly nonlinear among different subjects (like remarked in some comments above). A more detailed list could be this one (each column to be learned simultaneously within the rest of topics):

Calculus (one variable) -> Vector Calculus -> Functions of One Complex variable -> Measure Theory

--------------------------------> Ordinary Diff. Eq. -> Partial Diff. Eq. -> Variational Calculus -> Integral Eq.

Linear & Multilinear Algebra -> Group Theory -> Rings & Modules -> Intro to Representation Theory

Combinatorics & Graph Theory->

Elementary Number Theory

Affine & Euclidean Geometry -> Projective Geometry -> Differential Curves & Surfaces

----------------------------------------> Point Set Topology ----------------------------------------------> > Introduction to Elementary Algebraic Topology

Elementary Statistics --> Elementary Probability -> Advanced Statistical Methods

Real Analysis -> Functional Analysis -> Complex Analysis (several variables)

Dynamical Systems (and Chaos)

Partial Differential Equations (general theory)

Commutative Algebra -> Homological Algebra -> Category Theory

Lie Algebras -> Representation Theory

Smooth Manifolds -> Algebraic Topology

--------------------------> Differential Topology

--------------------------> Algebraic Geometry

--------------------------> Riemannian Geometry -> Complex Geometry -> Symplectic Geometry

I do not know about advanced statistics and probability, and graduate number theory should should deal with analytic number theory and algebraic number theory with class field theory up to diophantine and arithmetic geometry.

May be you could make your own list according to your tastes looking up some course sequences and syllabus offered by good universities.

1

If you intend to study on your own the best approach is to follow a structured sequence just like in an ordinary Math degree.... but nevertheless not forgetting that everything is interconnected and prerequisites and applications are highly nonlinear among different subjects (like remarked in some comments above). A more detailed list could be this one (each column to be learned simultaneously within the rest of topics):

Calculus (one variable) -> Vector Calculus -> Functions of One Complex variable -> Measure Theory

--------------------------------> Ordinary Diff. Eq. -> Partial Diff. Eq. -> Variational Calculus -> Integral Eq.

Linear & Multilinear Algebra -> Group Theory -> Rings & Modules -> Intro to Representation Theory Combinatorics & Graph Theory -> Elementary Number Theory

Affine & Euclidean Geometry -> Projective Geometry -> Differential Curves & Surfaces

Point Set Topology ----------------------------------------------> Introduction to Elementary Algebraic Topology

Elementary Statistics --> Elementary Probability -> Advanced Statistical Methods

Real Analysis -> Functional Analysis -> Complex Analysis (several variables)

Dynamical Systems (and Chaos)

Partial Differential Equations (general theory)

Commutative Algebra -> Homological Algebra -> Category Theory

Lie Algebras -> Representation Theory