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could you rank these classes in terms of necessity to take if I intend to pursue a Ph.D in pure math? I don't know if I can fit everything, but I want to make sure I take the most important ones:

Ordinary Differential Equations
Geometries
Applied Regression Analysis
Chaotic Dynamics
Probability
Introduction to Statistical Inference
Set Theory
Elementary Theory of Numbers
Real Analysis I
Real Analysis II
Advanced Linear Algebra
Combinatorial Theory
Fourier Series and Boundary Value Problems
Abstract Algebra I
Abstract Algebra II
Differential Geometry
Functions of a Complex Variable
Topology
Applied Analysis

Thank you in advance~~

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I would suggest talking to an advisor. What areas of mathematics do you enjoy most? That is a pretty important question you should answer first.

Usually,

  • Real Analysis
  • Abstract Algebra
  • Functions of a Complex Variable
  • Linear Algebra
  • Topology

would be considered "core" courses for pure mathematics. The next ones depend moreso on your personal interests (take them all if you have time, they are interesting and useful!):

  • Probability
  • Set Theory
  • Elementary Theory of Numbers
  • Combinatorial Theory
  • Fourier Series and Boundary Value Problems
  • Differential Geometry

The following courses are more in the applied direction.

  • Ordinary Differential Equations
  • Applied Regression Analysis
  • Chaotic Dynamics
  • Introduction to Statistical Inference
  • Applied Analysis

Edit: I am not actually sure what area "Geometries" is referring to.

Edit 2: After some thought, I put Topology into the first category.

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  • $\begingroup$ Thank you Eric. Geometries covers Euclidean, Hyperbolic, and other Geometries. $\endgroup$ – HelpMath May 24 '11 at 0:44
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    $\begingroup$ I agree with almost everything, but I would definitely say that topology is a core course for somebody who wants to do a PhD in pure maths. $\endgroup$ – Alex B. May 24 '11 at 0:46
  • $\begingroup$ I agree with Alex and add that Set Theory should also be highly valued, though you will pick up many of the core concepts of Set Theory in other courses (particularly in Topology). $\endgroup$ – Austin Mohr May 24 '11 at 1:11
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I'd add Ordinary Differential Equations to the list. Imho it is a must for anybody who wants to become a mathematician. DE crop here there and everywhere, sort of most basic ideology in many parts of maths. Besides to go to many other courses like Fourier Series and Boundary Value Problems, Differential Geometry and Chaotic Dynamics seems to me strange without at least some preliminary knowledge in DE.

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This is the core you =need=, and do it in this order.

Real Analysis I

Advanced Linear Algebra

Real Analysis II

Functions of a Complex Variable

Ordinary Differential Equations

Abstract Algebra I

Abstract Algebra II

Elementary Theory of Numbers

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