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:

- Kähler manifolds
- symmetric spaces
- classical groups
- invariant theory
- representation theory
- Young’s tableaux
- Fuschian groups
- Kleinian groups
- non-commutative geometry
- quantum groups
- dimension theory in algebraic geometry
- the Jones polynomial
- motivic cohomology
- floer cohomology
- modular forms
- symplectic topology
- analysis of several complex variables
- the fast fourier transform
- wavelets
- Martingales
- C* algebras
- von Neumann algebras
- maximal toral subgroups
- Hopf algebras
- categorification
- spinors
- quantum computers
- quiver representations
- syzygies
- JSJ decomposition
- spectral sequences
- stable homotopy groups
- J-holomorphic curves
- schemes
- symmetric functions
- cobordism
- stability of the solar system
- Gromov limit of metric spaces
- elimination theory
- resultants
- Lefschetz fixed point theorem
- Ricci flow
- Morse theory
- discrete Morse functions
- train tracks
- hyperbolic 3-manifolds
- spectrum of a group
- localization at a prime
- hearing the shape of a drum
- nonstandard analysis
- hodge theory
- group cohomology
- Killing forms
- Dynkin diagrams
- characteristic classes
- partial differential equations
- analysis
- affine buildings
- chamber complexes
- Chevalley groups

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This entry was posted on June 30, 2007 at 10:43 am and is filed under Jim. You can follow any responses to this entry through the RSS 2.0 feed.
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June 30, 2007 at 12:22 pm |

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

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)

July 6, 2007 at 6:56 pm |

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.

July 7, 2007 at 12:40 am |

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.

July 7, 2007 at 2:25 am |

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.

July 7, 2007 at 5:31 pm |

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

December 20, 2007 at 11:02 pm |

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!

December 20, 2007 at 11:04 pm |

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…

December 20, 2007 at 11:05 pm |

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

September 22, 2008 at 8:25 am |

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.

September 28, 2009 at 6:02 pm |

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!

October 18, 2009 at 8:25 pm |

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

November 30, 2009 at 7:46 pm |

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?

January 7, 2010 at 8:56 am |

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!

January 18, 2010 at 7:40 pm |

i want to br smart

January 4, 2013 at 5:59 pm |

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.