Will these courses will be good enough preparation for a pure math PhD program?

[inknit]

I'm currently a freshman and these are all the math courses I plan on taking in 4 years.

Undergrad:
- Calc II (taken), Calc III, Linear Algebra I, Linear Algebra II, Real Analysis I, Real Analysis II, Ordinary Differential Equations, Intro to Abstract Algebra, Complex Variables, Survey of Algebra, Number Theory, General Topology, Differential Geometry, Advanced Multivariate Calculus

Graduate:
-Algebraic Topology I, Complex Analysis I, Homological Algebra, Measure Theory, Algebraic Topology II, Complex Analysis II....3-6 more graduate math classes

Also, do you think it's necessary to take some physics, computer science classes? I already took programming.

 

[doodle_sack]

My first advice for you would be "take courses that you think you will enjoy", because in the long term perspective how much you have learnt will matter very less as opposed to how much you have enjoyed. Also, learning and enjoying what you learn have very good correlation, one can easily affect the other. Sorry for this boring advice, just my 2 cents!

The simple answer to your question is yes. I would add Functional Analysis to your list of "courses to take". Regarding Physics & Computer Science, if you can take courses from theoretical computer science that will be so cool, both the subject and the math you apply to them. I don't know much about Physics courses.

 

[mathwonk]

More important than how many courses you take, is what you understand well. I would say especially important is advanced calculus (multivariable), i.e. roughly the content of spivak's little book, calculus on manifolds, although it might be better to read that after having a more traditional course on the material.

E.g absolutely basic results include Green's theorem and the implicit function theorem of two variables, and yet it is rare (in some places) to find even an advanced graduate class in which most students know these theorems well.