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<p>1st year: Honors Calculus
2nd year: Analysis or Honors Analysis
3rd/4th year: Algebra or Honors Algebra, (Topology), (Functional Analysis), (Complex Analysis), (Differentiable Manifolds), (Probability), (Discrete Math), etc.</p>
<p>You’ll end up having to take a lot of the ones in the parentheses, although they’re not all required and they’re not part of any sequence, just individual one-quarter classes.</p>
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<p>Graduate schools really couldn’t care less about the ‘name brand’ of your school. They care about the education you’ve received prior to applying to grad school, and so if you transfer to Chicago having the same math background as other 2nd or 3rd years, then you should be in good shape. Otherwise, you’ll probably have some catching up to do, but if you extend your residence at Chicago to an extra year or so, you should be in good shape.</p>
<p>If two people are applying from Chicago, and one is a year ahead math-wise than the other, that person of course has the advantage. It’s the same scenario. It all depends on what level you enter Chicago with and how successfully and quickly you progress.</p>