A Typical Day at Work

Dear Dr. Polchinski

Your book is very, very confusing. I do not understand it at all. I look forward to a day that may come, where I might refer back to it, finally understanding everything you are saying. That day is not now. That day is not anytime soon.

For Shame, CNN

Browsed tonight on tv: Ashlee Simpson being interviewed on Larry King Live. If that weren't bad enough, the sit in for Larry was Ryan Seacrest. The world is ending.

Two Cool Things about Topology

1. On the earth's surface, there are at least two points with the same temperature and air pressure.

2. If I make a sandwich with a piece of ham, a piece of cheese, and two slices of bread, no matter where I place them in the sandwich, and how big the pieces are, I can always cut each piece in two with one straight wack of the knife.

HIlarious!

From the Onion:


Philandering String Theorist Can Explain Everything
October 12, 2005 Issue 41•41
BATAVIA, IL—Fermi National Accelerator Laboratory physicist Laird Karmann, a noted string theorist and accused philanderer, said Monday that he can "explain everything" if his wife Elizabeth will just give him a chance. "Surely, anyone can see that, mathematically, the universe is composed of Riemann surfaces, having positive-definite metrics, across which the attached 'loops' or free 'strings' have a (1+1) dynamic topology," Karmann said. "But string behaviors are Lorentzian, meaning that they—like me—need an intense dual-phase Wick rotation now and then just to stay in rational space. I mean, it was just a blowjob." Elizabeth refused to accept her husband's theory, suggesting that he study the transformational loop dynamics implicit in her hurled wedding ring.

MIT Mutants

I'm having quite a tough time with this algebraic topology class, it dominates my schedule. Yeserday, as usual, I was working on the problem set (which remained very incomplete) until about two in the morning. Working with me were a college freshman, two sophmores, and a college senior. What's more, they were way better at math then I was. Two of them had effectively skipped high school, and one of them claimed that he had seen quotient groups in the fourth grade. I don't know that I had seen quotients in the fourth grade. These people are mutants.

Sometimes, walking around MIT's concrete excuse for a campus, and witnessing the complete lack of regard for humanities around here, and the absurdly long hours required for undergraduates, I wonder "Who in their right mind would come to such a place for college?" It's these mutant people. They are a scary, scary army of wunderkind dorks. I want to go back to where there are just regular smart people.

Exotic R^4

I thought it was very cool when, about a year ago, I learned that Milnor proved there were 7-spheres which are not diffeomorphic to the standard 7-sphere.

Let me try to explain briefly what this means. Two spaces are said to be homeomorphic if they are topologically equivalent. You can think of two spaces being "topologically equivalent," as being able to morph one space into the other without cutting or tearing the space. For example, a 2-sphere (the sphere you're used to) is homeomorphic to those Mickey Mouse ballons you can get at Disneyworld. You can get his ears by molding the sphere, but you don't have to tear anything. The two sphere, however, would not be homeomorphic to the surface of a donut. You'd have to cut a hole in the sphere to get something like the donut.

In order to do calculus (and hence, physics) on such spaces, we introduce something known as a differentiable structure on a space, making it into what's known as a differentiable manifold. The concept of a homeomorphism can then be applied to these spaces, and refined to a diffeomorphism--that is a homeomorphism that is somehow nicely differentiable.

What is surprising (at least to me) is that there are spaces that you can put a differentiable structure on, which are homoeomorphic, but not diffeomorphic. It's hard to picture what this means, as there are no such spaces in two or three dimensions.

Anyway, I thought it was cool enough when I learned that there were smooth seven dimensional spheres which are not diffeomorphic (they are homeomorphic, though, which is what allows me to call them all spheres in the first place), but yesterday I learned that there are actually homeomorphic to four dimensional Euclidean space (R^4), but not diffeomorphic to the standard R^4. These spaces are called "fake" or "exotic" R^4's, and appearently there is a countably infinite number of them. Cooler still, is that this is that 4 is the only n for which there are exotic R^n''s. Four also happens to be the number of dimensions that we live in. Interesting.

This raises the possibility (at least to me), that you could define gauge theories (like the standard model) on exotic R^4's and perhaps get some physical effects that are not part of the standard model. I have no idea what this would even mean, or if anybody does research on this (if so, I'm sure it was Witten at some point). Doing some quick searches on the archive and the internet, I only find one paper related to this idea, specifically on GR with a statement that locally physics on these spaces would be the same, but that there could be some global (but non-topological) effects. Does anybody know anything about this? Has this work been done (Dr. Z, I'm looking in your direction for comments)?