Low Dimensional Topology

September 4, 2009

Reader survey: graduate courses in topology and geometry

Filed under: Pedagogy — Nathan Dunfield @ 4:37 pm

In my department, we’re considering whether we have too many basic graduate courses, by which I mean courses with (mostly) fixed syllabi aimed at first and second year graduate students, as opposed to advanced topics courses which never cover the same thing twice. In geometry/topology, we have not less than 11 such one-semester courses, of which two arguably belong more to algebra:

  1. MATH 518 Differentiable Manifolds I
  2. MATH 519 Differentiable Manifolds II
  3. MATH 521 Riemannian Geometry
  4. MATH 522 Lie Groups and Lie Algebras I
  5. MATH 523 Lie Groups and Lie Algebras II
  6. MATH 524 Linear Analysis on Manifolds
  7. MATH 525 Topology (really, this is basic Algebraic Topology)
  8. MATH 526 Algebraic Topology (really, this is more advanced Algebraic Topology)
  9. MATH 527 Homotopy Theory
  10. MATH 533 Fiber Spaces and Char Classes
  11. MATH 535 General Topology

In contrast, my last university had just 4 or 5 such classes, and that was with the quarter system, so that’s only about 3 semester’s worth of such classes!

So I’d be curious to know how many such classes are there at your university (or any other university that you’re familiar with, e.g. where you went to grad school). Please post your data in the comments!

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6 Comments »

  1. MTH 868 Geometry and Topology I (differentiable manifolds)
    MTH 869 Geometry and Topology II (beginning algebraic topology)
    MTH 914 Lie Groups and Algebras I (Lie algebras)
    MTH 915 Lie Groups and Algebras II (Lie groups)
    MTH 916 Algebraic Geometry I
    MTH 917 Algebraic Geometry II
    MTH 930 Riemannian Geometry I
    MTH 931 Riemannian Geometry II
    MTH 935 Complex Manifolds I
    MTH 936 Complex Manifolds II
    MTH 960 Algebraic Topology I
    MTH 961 Algebraic Topology II

    There’s also a readings course with a more-or-less fixed syllabus covering gauge-theoretic invariants of 4-manifolds.

    Comment by Cooper — September 4, 2009 @ 5:50 pm | Reply

  2. Don’t know how broadly you want to interpret geometry/topology, but the following are Columbia’s non-analysis graduate courses:

    Algebraic Topology 1
    Algebraic Topology 2
    Modern Geometry 1
    Modern Geometry 2
    Lie Groups and Representations 1
    Lie Groups and Representations 2
    Commutative Algebra
    Algebraic Geometry
    Complex Analysis 1
    Complex Analysis 2 (almost always primarily complex manifolds)

    Upper level grad courses include Floer Homology and Symplectic Geometry, both of which are taught pretty regularly, but content varies. Occasionally we have an upper level 3-manifolds course too.

    Comment by Gilmore — September 4, 2009 @ 6:00 pm | Reply

    • Don’t know how broadly you want to interpret geometry/topology, but the following are Columbia’s non-analysis graduate courses.

      Pretty broadly, though maybe not quite including algebraic geometry, at least not the kind of course that follows e.g. Harshorne. Maybe that just makes me small-minded, though. We also have two algebraic geometry courses (MATH 510 Riemann Surf & Algebraic Curv and MATH 511 Algebraic Geometry) and MATH 524 is sorta our transcendental algebraic geometry course (e.g. Hodge theory, etc.).

      Comment by Nathan Dunfield — September 4, 2009 @ 9:24 pm | Reply

  3. You can just go to any department’s website and take a look at the course list. Personally, your list is about the same as at the University of Michigan and more than at Stony Brook or Rochester. That’s not surprising. Big universities have lots of students and professors, so they offer more courses. But, from personal experience, advanced topics courses should never ever be offered. First, they are almost always disorganized, terribly taught, with optional homework, where everyone gets an A. So none of the students ever learns, or has incentive to learn, anything in these courses. Second, the big problem as a graduate student is “focus”. When you’re stuck on your thesis it is just too easy in mathematics to wander off into some other topic. Offering these advanced courses just adds to a student’s distraction. In my experience the best students were always the one’s who took the fewest classes. Somehow they had a grasp for the breadth of the field, what interested them, and what they needed to know to write their thesis. All that without taking endless classes.

    Comment by Mayer A. Landau — September 4, 2009 @ 10:19 pm | Reply

    • Really? While I agree that people shouldn’t take every topics course offered, students really do learns things from topics courses. But then, I’m just basing that on my personal experience.

      Comment by Richard Kent — September 5, 2009 @ 9:05 am | Reply

      • Well… before everyone jumps in, I would say this. Stony Brook had topics courses that masqueraded as regular courses. By that I mean is that the bulletin only had a few regular courses. So the department offered topics courses that repeated every other year, and usually taught by the same instructor. So for example, algebraic geometry would be Topics in Algebra I, etc. These courses had a more or less set curriculum, with required homework. These courses are basically the advanced courses on the “basic courses” list that several people gave above. When I think of a topics course, I think of something like, “Perleman’s proof of Geometrization”. For sure it’s interesting. For sure, with a good instructor, you could learn a thing or two. But you’re really not going to learn very much or too deeply unless you really immerse yourself in the topic. And, if your thesis has zero intersection with that topic, you’re only delaying your graduation by doing so. If you look at the Congressional review of graduate programs, you’ll see that the average time to degree now is
        seven years plus. No wonder!

        Comment by Mayer A. Landau — September 5, 2009 @ 3:07 pm


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