## Monday, December 12, 2011

I've just updated the design of the blog. I think it looks negligibly better.

I'll also try to write more here rather than on my notebk's wiki.

I posted some exercises for Sean [pdf] which are fun calculus problems.

I suspect what I'll do next is write up my notes on differential geometry from Osserman's course I audited a couple years ago, as well as my notes on algebraic topology (from Dr Schwarz's courses). Then I'll work on spin geometry, "commutative geometry", analysis, and so on.

By "commutative geometry", I really mean spectral triples using commutative rings (Here I am sloppy, but meh I am a sloppy person!). It is also called Differential Calculus over Commutative Algebras, although there are no real texts on the subject...

I'll have to review the prime spectrum of the commutative ring $C(M)$ of continuous functions on a topological space $M$ and how it relates to the topology of $M$.

If we let $M$ be a smooth manifold, then we work with $C^{\infty}(M)$ — I am told there is a theorem due to Shields which says if $M$ and $N$ are smooth manifolds and $C^{\infty}(M)$ is isomorphic to $C^{\infty}(N)$ then $M$ and $N$ are diffeomorphic. How interesting! But I cannot find this theorem...

At any rate, vector bundles over $M$ may be considered by looking at the projective modules over $C^{\infty}(M)$.

We consider algebraic analogs for sections, vector fields, covector fields, and so on. It is really quite cute.

Noncommutative geometry is similar in setting up a dictionary between "algebraic stuff" and "geometric stuff", at least how Connes approaches it. It's just that the "geometric stuff" we work with is a smooth Riemannian manifold $M$ equipped with a spin structure, we consider spin bundles over it, and so on.

## MathJax

I am experimenting with MathJax on blogger, so bear with me people.

My reference for this subject is the thread at stackexchange on it.

Consider the "Harmonic Series" $\sum^{\infty}_{n=1}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots$ which diverges famously.

MathJax uses the $...$ or $...$ for "inline mathematics" and $...$ or $$...$$ for "display math", e.g., the mathematics produced above.

I don't know whether to keep it or not, because MathJax is sluggish on some computers. But it is the "way of the future", like blimps and autogyros.