Abelian Grapes

Hey all! Exciting news, I've been attending a weekly differential geometry seminar at ASU, and I finally understood a lecture (about conjugacy limits in Cartan subgroups of SL(n, R), looking specifically at n=3, and then applying use of the hyperreal numbers). I don't know much about Cartan subgroups or Lie groups in general, but I could follow along the lecture and parse, understand, and reflect on examples and terms and such. Cool stuff! I guess I am learning math, who knew?

My favorite part about math is the terminology--for example, take the hyperreal numbers, *R, (equivalence classes of sequences of real numbers constructed by taking all real series and denoting their convergent point a hyperreal number--and apparently if it doesn't converge like (0,1,0,1,0...) then you just assign it to either 0 or 1? I don't know all the ins and outs of them). The hyperreal numbers *R are cool because they're non-Archimedian. Numbers like the reals are Archimedian because if there's a number a < 1/n, where n is any natural number, then a as to be 0. But the hyperreals have infinitesimals like ϵ= (1, 1/2, 1/4, ...), which is smaller than everything, but bigger than 0. This also means the hyperreals have infinitely large numbers, like 1/ϵ--which should remind you of how the Riemannian sphere acts with infinity and 0. And then *R has appreciable numbers, which are just real numbers plus infinitesimals α = a+ ϵ, a in R. Pretty cool stuff. Turns out the hyperreals are a much easier way to conjugate by a series of matrices, which is needed to do to find conjugacy limits of things. But the reason I bring this up is because some of the terms are pretty cool. If you have an appreciable number, α = a+ ϵ, and you want to go back to the reals, you just "take the shadow of α," which, though it just means to take the real part, a, of α, makes it sound really cool, like it's from some children's TV show about children's card games. And another cool term, galaxy, is represented Galϵ(x) = {y in *R, | |x- y| <= kϵ}, k in R. During the lecture, when she introduced "galaxies," I turned to the guy to my right and said "I love math," and he nodded in agreement.

Anyway, this post should be fairly short and easy to grasp, especially compared to the other posts, and the posts about to come. This will just introduce the concept of a group.

A group is a set G with a binary operation, * : G x G -> G, such that the following three properties

1. For g,h,i in G, (g * h) * i = g * (h * i), ie, * is associative
2. There exists an identity element, e, in G such that, g in G, g * e = e * g = g
3. For each g in G, there exists an inverse g-1 in G such that g * g-1  = g-1 * g = e

But remember that because * is a binary operation, then for G = {a,b,c} to be a group, a*b has to also be in G as well. This property is called closure and is essential to bear in mind.

Furthermore, a group G is abelian if its binary operation is commutative, that is, gh=hg, g,h in G.        

Common groups include the integers under addition represented as (Z, +) or referred to Z mod n, consisting of the elements {0,1,2,...,n-1}. To form the multiplicative group (Z, •), one must remove the 0 element, as to satisfy the inverse requirement of a group--g in Z, 0•g = 0 =/= 1.

Symmetry involves and necessitates group theory. Given a finite set A, its permutations (bijections from A->A) result in interesting properties, with application to geometry. If A = {1,2,...,n}, the group of possible permutations of A is called Sn, the symmetry group of n.

If n=3, then we may model geometrically the elements of S3 with the vertices of a triangle. 

from Troy University

Here, various permutations of Sn yield different transpositions. For example, mapping 1->2, 2->3, and 3->1 results in a rotation by 120°, but the mapping 1->1, 2->3, 3->2 results in a reflection across the angle bisector of vertex 1.

Also, groups are key for complex analysis. Not only are there many groups involving C, but the big one, which this project is titled for, is called the Linear Fractional Group, ie all the Mobius Transformations together. Matrices are a great way to represent groups, and in the next post about Mobius Transformations, I will probably also introduce the analogous matrices.

EDIT== By the way, the title is a joke based on