I wish I were smart.


I’m Jim. I graduated from Cornell three years ago, and I’m currently a postdoc at Texas A&M. I’m nominally a specialist in geometric group theory, but I like anything geometry related: group theory, topology, graph theory, differential geometry, discrete geometry and combinatorics, algebraic geometry, etc. I also really enjoy teaching, and I’m likely to post all sorts of entries about lower-level mathematics.

I thought I’d start off with an odd sort of post. A few months ago, I wrote down a list of all the things I would know more about if I were smarter. The goal is to learn a little something about each of these, and then cross it off the list. (In reality, the list seems to grow longer every time I learn something new.)

In no particular order:

  1. Kähler manifolds
  2. symmetric spaces
  3. classical groups
  4. invariant theory
  5. representation theory
  6. Young’s tableaux
  7. Fuschian groups
  8. Kleinian groups
  9. non-commutative geometry
  10. quantum groups
  11. dimension theory in algebraic geometry
  12. the Jones polynomial
  13. motivic cohomology
  14. floer cohomology
  15. modular forms
  16. symplectic topology
  17. analysis of several complex variables
  18. the fast fourier transform
  19. wavelets
  20. Martingales
  21. C* algebras
  22. von Neumann algebras
  23. maximal toral subgroups
  24. Hopf algebras
  25. categorification
  26. spinors
  27. quantum computers
  28. quiver representations
  29. syzygies
  30. JSJ decomposition
  31. spectral sequences
  32. stable homotopy groups
  33. J-holomorphic curves
  34. schemes
  35. symmetric functions
  36. cobordism
  37. stability of the solar system
  38. Gromov limit of metric spaces
  39. elimination theory
  40. resultants
  41. Lefschetz fixed point theorem
  42. Ricci flow
  43. Morse theory
  44. discrete Morse functions
  45. train tracks
  46. hyperbolic 3-manifolds
  47. spectrum of a group
  48. localization at a prime
  49. hearing the shape of a drum
  50. nonstandard analysis
  51. hodge theory
  52. group cohomology
  53. Killing forms
  54. Dynkin diagrams
  55. characteristic classes
  56. partial differential equations
  57. analysis
  58. affine buildings
  59. chamber complexes
  60. Chevalley groups
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15 Responses to “I wish I were smart.”

  1. morilac Says:

    Other random things to think about:
    Integrable Systems
    Using Grobner bases for things that people care about
    Gromov-Witten invariants
    Mirror symmetry
    Arithmetic Geometry
    Affine Lie Algebras
    Vertex Algebras
    Monsterous Moonshine
    Minimal Models
    Geometric Langlands Reciprocity
    Pretty much anything Langlands would be nice
    Atiyah-Singer Index theorem
    Exotic \mathbb{R}^4
    and my personal goal for the present:
    Derived Algebraic Geometry
    Also, I can make your list shorter by one: Hopf Algebras and Quantum groups are the same thing (some people might define them slightly differently, but theres no real consensus)

  2. michiexile Says:

    Hey cool – I actually know a little about several of those subjects. Depending on how little, I’d list 5, 24, 25, 29, 31, 40, 43, 44, 48, 52 and 53 to the things I know anything at all about.

    52 is my specialty, so I end up knowing quite a bit about it, and 43, 44 belong to things I’ve even taken a summer school on. For 29 and 40 I recommend the Cox, Little, O’shea book on algorithmic algebraic geometry, that do a pretty readable starting treatment of it. It also gives a tiny first taste of 11, and deals a LOT with Gröbner bases and why you care.

  3. Jim Belk Says:


    It’s cool that Hopf algebras and quantum groups are the same — I feel smarter already.

    I should definitely add some of those other things, especially Gromov-Witten invariants, Monstrous Moonshine, the word “Langlands”, and the Atiyah-Singer Index theorem. Some of those others are beyond my horizon, e.g. I don’t think I’ve ever heard of vertex algebras or derived algebraic geometry.


    Cool, and thanks for the tip on Cox, Little, O’Shea. I remember reading part of that book a few years ago and enjoying it immensely. Probably I should take a look at it again.

  4. John Armstrong Says:

    About a week ago I started in on a series of posts on categorification you might like. That first one is a general overview, and since then I’ve been working through categorifying the notion of a monoid.

  5. Peter Says:

    At least you are smart enough to correctly use the subjunctive in your title.

  6. random Says:

    I hear ya man. I have a Chemical Engineering Degree from a top School, an MBA from a top school. I make good $$, but some times I feel so dumb. I know that I am not and I know that I am on the right tract, proof when I think of something that has been already accomplished but I didn’t know it was. Unfortunately I always feel two steps behind. That is not to say that I am not happy with my life, but I just wish I know more and be more effective in what I do and in my decisions.

    ps. I grew up in Horseheads about 30min south of route 13.. oh yeah!

  7. random Says:

    if you dont mind I would like to post that I grew up 30 min south of Route 13 in Horseheads NY. I bet if we played the name game we may know the same people…

  8. random Says:

    I meant Ithaca… yeah it’s one of those nights

  9. Ed Dunkle Says:

    Dood, you’re smart! Try to enjoy it a little bit. I never got past Stokes Theorem in freshman calculus and have kind have felt like a dummy ever since.

  10. Holly Says:

    This is awesome! I’m going to start one of these! Unfortunately, I’m rather young and don’t really know what most of this means, so I don’t really can’t discuss any of this. But I’m going to start a list like this!

  11. Steven Heilman Says:

    Hey- if you are curious about 49, I would read Kac’s article (1966) or Gordon, Webb and Wolpert (1992). I actually just wrote about some related results of Steve Zelditch, for a general audience, here: http://bfrank.org/overeducated/?p=77

  12. melon Says:

    You’re plenty smart. Heavens know how I’m going to get past linear algebra and diffy Q this semester. So much for my “easy” first year of college…
    Speaking of Cornell, is the food really as good as it’s hyped up to be?

  13. Dawn Says:

    Whoa… I wish I were smarter too, but that’s quite a list. I barely passed basic math. For me I wish I were smarter in dealing with people!

  14. chimmy Says:

    i want to br smart

  15. lovedthe Says:

    Smart: can figure shit out quick.
    Knowledgeable: Know shit.
    Really smart – can combine the two.
    Genius: Only does this at weekends – because even if you do find the answer to everything, you still need time for Mario. Or Girls. Or Boys. But mainly Mario.

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