Nothing makes blogging stop dead like the start of the quarter. Well, that combined with the fact that I sort of reached the logical end to the topic I was posting on. There are lots of neat things we could do from here, but they are all rather technical (like sliding handles around and cancelling them to make our manifold simpler) but have nice simple geometrical interpretation. So I don’t really feel like going into that, since the bulk of the powerhouse tools of Morse theory that the everyday person needs were covered (I guess that isn’t totally true since we’re missing a few quick things for Lefschetz).

There are a few brief corollaries and applications left over that I’m contemplating. The most interesting to me is the one that says if a manifolds admits a smooth function to with exactly two critical points which are both non-degenerate then it is homeomorphic to a sphere. Yes I said homeomorphic! That is what is so interesting. We only have a diffeo if it is dimension less than or equal to 6. There are “exotic ’s”.

I feel ready to move on in general. The next logical step is to develop some homology, but this is a hard question. I certainly do not have the motivation or patience to build this from the ground up starting with definitions. So I’m not sure how to proceed. I also may give up on Lefschetz for a few months and do things more related to things I’m doing for classes.

In other news, I went to a neat talk on harmonic measure theory. Normally I’m not very fond of measure theory, but this was pretty cool. There was even a result that I don’t remember now that had to do with the set on which the measure was full was an algebraic variety or something.

Anyway, figured I’d update at least once this week.

I’m done! Well, for now. I’m pretty sure I didn’t pass all three, so I’m still not done with these darned tests.

Now I have to decide what I’m going to talk about. I decided I was going to do no math for a week after these tests were done. But I don’t feel that way now. I actually feel sort of motivated to try to look at some things I don’t have time to look at when school is in session (or when I’m studying for prelims).

From my point of view, my options seem to be Morse theory (something I’ve been threatening to do for probably 6 months now). If I did this, I’d probably just try to get to a few of the results of the form: If M is a manifold that admits a smooth real valued function with precisely two critical points, then it is homeomorphic to a sphere. Or something like that, I haven’t looked at it for awhile, so it might not actually be that. But I think it is really awesome that you can somehow get at topological facts, based purely on what real-valued functions it can admit.

I could work through Topology from the Differentiable Viewpoint and learn some things about cobordism (a term I hear thrown around a lot, but only vaguely know that it is in reference to what manifolds are boundaries of other ones or something).

I could pick up where I left off on the algebraic geometry, although I’ll probably do that during school since I’ll be taking algebraic geometry, so it might not be the best choice for right now.

I could do some of Bott and Tu’s book. I’ve only actually read the first part, and am quit curious as to what is in the rest of it.

I could try for the third time to read Zwiebach’s A First Course in String Theory, because I’m darned determined to learn what string theory is. Although, I suspect it will go even worse this time than last time considering its been well over a year since I took quantum mechanics.

Or I could switch gears and do some posts on books I’ve read (which are quite a few since my last book post) and movies I’ve seen. It is still the case that every day my Lost in the Funhouse post has the most hits. Darn you “survey of modern american lit classes” for causing so much confusion.

Or you could suggest something, and I might ignore it or actually do it.

Well, things can get ultra busy around mid-terms. I don’t think I’ve posted in two weeks. What I really wanted to do next was to post some category theory basics. I’m not sure if I should, though, since so many math blogs have already done this. I then wanted to go on to define the fundamental group purely in categorical language. It turns out to be a really nice construction compared to the tedious typical way.

Instead, I’ve recently become quite interested in rings. There also seems to be a very large lack of “pure” ring theory in the blogosphere. Sure rings pop up and are needed by people doing things with algebraic geometry, say, but using ring theory isn’t the same as developing it.

I’m going to cover the basics quite quickly with the assumption of previous exposure, since I really want to get to some of the more interesting constructions (i.e. localization), then I’ll slow it down.

Ring: We have a set with two operations, we’ll call them addition and multiplication. The addition part forms an abelian group, and the multiplication…well, it puts you back in the set and is associative. We need a way to relate these operations, so we also require that and , i.e. there is a distributive law in effect. Note that there is not required in general to be a multiplicative identity, multiplicative inverses, or commuting of the multiplication.

NOTE: Until I say otherwise I will assume the ring is commutative (meaning multiplication) with 1 (meaning having a mult identity). will always denote this.

Subring: A subset of that is itself a ring.

Ideal: A subring that “swallows” multiplication. So is an ideal if for any we have that for all .

Prime ideal: An ideal is prime if for element we have that either or .

Principal ideal: An ideal is principal if it is generated by a single element. So an ideal is generated by “a” if .

Maximal ideal: A proper ideal such that there is no other ideal with the property (where all containments are proper).

Domain: A ring in which the cancellation law holds. i.e. if and , then . Note that no element can divide zero, so if , then either or .

We can quotient in the natural way: is the set of cosets of I where our operations are and . We get the nice result that any ideal of is of the form where is an ideal of (and ).

I think that may lay down all the terminology I’ll need to get started. I’m not sure if I’ll really use any of these terms for awhile, though.

Common rings: . Note that we can get from by taking “quotients.” This can be made precise for any domain. It is called the fraction field of denoted .

Let me take some time to explain this, since it is the motivation for localization. We want to form a field that contains as a subring such that the elements of , say have the form where . Note that this “looks” like division, and in fact is division in the case of .

To make this process precise takes a bit of work, though. Set up . Define (a,b)~ (c,d) iff . This is done since we want our relation to look like fractions, a/b , so a/b=c/d if we can cross-multiply and get the same thing. It is straightforward to check that this defines an equivalence relation. Now we let be the set of equivalence classes.

Our operations on should mimic those of fractions, so our addition is [a,b]+[c,d]=[ad+bc, bd] and [a,b][c,d]=[ac,bd]. These are well-defined and it is just computation to check that the axioms of a field are satisfied. (If you want a hint: the zero is [0,1] and the 1 is [1,1], the additive inverse of [a,b] is [-a,b] and the mult inverse is [b,a]).

Before finishing up, I want to point out how restrictive we had to be. We want a more general way of doing this. We don’t want to require that R be a domain, and we don’t want to have to take fractions with every single element in R. It turns out this general process is extremely useful and it is called localization.