October 28, 2010

MaBloWriMo II?

Posted by Akhil Mathew under MaBloWriMo | Tags: |
[9] Comments 

Last year I participated in the MaBloWriMo project of blogging every day for a month about a topic. I think I learned a bit of differential geometry as a result, and it was fun. But this semester, I have not really blogged very much. It’s partially been business and partially laziness. Given all the homework write-ups for my classes, the urge to blog is just subdued.

So my announcement that I am going to do MaBloWriMo again probably sounds rather silly. Nonetheless, I would like to give it a shot. I will probably not be able to post every day, but I’ll see if I can get at least fifteen posts up next month.

The plan is as follows. I will talk about commutative algebra, specifically the homological theory. Here are some of the topics I’d like to touch on:

  • Basic properties of depth and the analogy to codimension
  • The Auslander-Buchsbaum formula relating depth and projective dimension
  • Cohen-Macaulay rings and their basic properties
  • Properties of regular local rings (in particular, factoriality and the characterization in terms of finite global dimension)
  • Koszul homology and cohomology, and the application to the quasi-coherent cohomology of an affine scheme (as in EGA III)
  • How this all figures in Serre duality

That I think should be enough for several posts! Unlike last time, I will assume prior acquaintance with commutative algebra for these posts, in particular at the level of dimension theory.  We’ll see how well I keep my promises.

 

November 2, 2009

Parallelism determines the connection

Posted by Akhil Mathew under differential geometry, MaBloWriMo | Tags: , , , |
1 Comment 

I’m going to try participating in Charles Siegel’s MaBloWriMo project of writing a short post a day for a month.  In particular, I’m categorizing yesterday’s post that way too.  I’m making no promises about meeting that every day, but much of the material I talk about lends itself to bite-sized pieces anyway.

There is a nice way to tie together (dare I say connect?) the material yesterday on parallelism with the axiomatic scheme for a Koszul connection. In particular, it shows that connections can be recovered from parallelism.

So, let’s pick a nonzero tangent vector {Y \in T_p(M)}, where {M} is a smooth manifold endowed with a connection {\nabla}, and a vector field {X}. Then {\nabla_Y X \in T_p(M)} makes sense from the axiomatic definition. We want to make this look more like a normal derivative.

Now choose a curve {c: (-1,1) \rightarrow M} with {c(0)=p,c'(0) = Y}. Then I claim that

\displaystyle \nabla_Y X = \lim_{s \rightarrow 0} \frac{ \tau_{p, c(s)}^{-1} X(c(s)) - X(p) }{s}. (more…)