First of all, I have two comments about the blog. One, the CIE chromaticity chart has passed the first axis and angle of rotation post as the number one post on the blog, all time. Second, the post about the minimum number of triangles in a triangulation has moved slightly ahead of the tint-tone-shade color post. I suspect, however, that those two posts will keep changing places… we’ll see.
Last weekend was mostly regression and abstract algebra… I looked at several more data sets… and I moved a little further into Dummit & Foote’s “Abstract Algebra”.
I spent yesterday evening looking at two things: Internal Model Control in process control theory, and the stellar interior equations. I’m still more mystified than not about both.
For the stellar interior equations, I have two distinct sets… one from my undergraduate text, and one from the classical reference by Clayton… I get to reconcile the two sets of equations. For IMC, there seem to be four different ways of looking at the same thing… but I haven’t convinced myself that all four are, in fact, the same thing.
What I mostly found time for during the week was “The Shape of Inner Space” by Shing-Tung Yao (and Steve Nadis) 978-0-465-02023-2. Yao, in case you don’t know, is a Fields medalist, and the discoverer of what are now called Calabi-Yau manifolds. And those are used in string theory… in addition to being relatively recent mathematics in the differential geometry in complex spaces. (I’ve only taken a few small steps out of real spaces.)
I think I can safely say that one form of string theory posits that the universe has 10 dimensions, of which we see 4, the spacetime of relativity. Where are the other 6 dimensions? They are a compact (no, that’s not just a pun) 6-dimensional space. Very likely a Calabi-Yau space.
For now, if you want to know a little more, head out to Mathworld and start searching on “Calabi-Yau”.
Oh, let me summarize Yau’s description. Calabi wondered if there was a compact, Kähler manifold, with vanishing first Chern class, that could have a Ricci-flat metric. Yau proved there was, and won the Fields medal.
Yau actually spends a chapter explaining, in general terms, what all those words mean. For that alone, the book is useful reading before one embarks on the study of, say, Kähler manifolds.
There are other chapters which are more about physics than about math, such as the one where he talks about the possibility that the 6 compact dimensions are wound tight like a spring… and could go sprong!… turning the universe fully 10-dimensional, and thereby destroying it as far as we’re concerned. (We would go sprong, too.)
In a nutshell, the book is worth reading if you are even mildly interested in the mathematics of string theory — not for the math per se, but as a preface to the mathematics.
Now let me get back to some more mundane mathematics… Internal Model Control, I think… and, later, I should start drafting the regression post for this Monday evening.