I wish I were smart.

by

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

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:

Greg:

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.

Michiexile:

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.