Someone above has already linked a fairly good discussion of career prospects. What I want to highlight is the difference in standard between a #10 and a #100 graduate program. We first begin with an analysis of the curriculum at George Washington University, ranked #100ish by US News: http://bulletin.gwu.edu/arts-sciences/m ... oursestextetianss wrote:

So I feel like you've looked into this more than I have: realistically, what path does one need to chart in grad school to become a research mathematician? Is it necessary to go to a school of this caliber? What's the difference between the #10 and the #100 school in terms of career prospects?

Something that I've been wondering recently.

My first observation is that Graduate Real Analysis 1 and 2 are nothing more than the analysis aspect of metric spaces (as seen in Munkres' Topology) and an undergrad differential topology course (no sard's or deRahm stuff). There is a Measure Theory course which covers chapters 1, 2, 3.1, 3.2, 6.1, 6.2 in Folland. They seem to have some more analysis courses but still no advanced courses beyond standard first year stuff except Ergodic Theory and Dynamical Systems.

My second observation is that the Grad Algebra courses are just undergrad algebra courses at top 10 schools. No Commutative Algebra or Algebraic Geometry course is offered.

Third observation is that a typical undergrad point-set topology course is equivalent to their graduate point-set class. Algebraic Topology covers only half the material of first semester Algebraic Topology courses at Northwestern, Berkeley (other top 10 schools cover more). No further courses in Algebraic Topology. No graduate manifolds course. Couple courses in knot theory but that's it.

Now for UC Berkeley, as their standards are easiest from among the top 10:

Graduate Real Analysis 1 covers both Point-set topology and chapters 1,2 of folland. The second semester covers 3-6 extremely thoroughly. The later classes are all offered and are much more difficult than the first year analysis sequence.

Grad Algebra 1 covers all of George Washington's two semester course plus representation theory and category theory. The second semester goes into deeper category theory, homological algebra, and the first half of Eisenbud. Two very difficult courses in Algebraic Geometry, multiple other algebra courses are offered.

While GWU's Algebraic Topology course only covers chapter 1 and part of chapter 2, Berkeley's covers all of chapters 1-3 with some more discussion of manifolds and homological algebra. Algebraic Topology 2 is homotopy theory and more knot theory stuff. Multiple courses in manifolds and beyond.

In general, the problems sets are much more difficult at UC Berkeley than at GWU. Moreover, students at UC Berkeley have to pass a comprehensive exam on honors undergrad material and pass oral qualifying exams by the end of their second year.