Colour algebras and colour traces

First year pure mathematics students learn about the trace of a matrix or a linear transformation, and in dealing with Lie superalgebras and quantum superalgebras, one also deals with the supertrace of a linear transformation, which is a generalisation of the usual trace.

A number of people (e.g. David McAnally and Peter Jervis) have looked at colour algebras, which are generalisations of superalgebras. Recall that a superalgebra starts with a Z₂ graded vector space on which one defines a supertrace on linear transformations of the Z₂-graded vector space. With a colour algebra, the idea is to start with a Z_n-graded vector space and define a colour trace on linear transformations of the Z_n-graded vector space.

Colour algebras didn’t go very far as they weren’t seen as very useful - however, I’ve been wondering if they are useful in obtaining new knot invariants, and I’ve been starting to look at Z_n-graded vector spaces and colour traces again.

Colour traces are interesting and non-trival - even properly defining the phase factor one obtains when applying the permutation operation P to tensor products is not straightforward. If one fixes P² = id, one might obtain the usual supertrace (I’m not certain about this) and I’ve started investigating what happens if you don’t fix P² = id.

Final volume of the Proceedings of ICM 2006

I recently received the final volume of the Proceedings of ICM 2006. It’s on my work desk, open at the discussion of Grigory Perelman’s work on the Poincare conjecture. It’s very nice to be able to peruse it at work and very different to my work.

Question about the Kauffman knot polynomial

Let L be a link and F_L(r,s) the Kauffman knot polynomial for L where r and s are parameters in the Birman-Wenzl-Murakami algebra BWM_f(r,s).

Question: is F_L(-r,-s) = F_L(r,s) for all L?

If the answer is yes, then the link polynomials arising from the vector (fundamental) reprsentations of U_q(-q²ⁿ,q) and U_-q(q²ⁿ,-q) are the same for all L. The answer to the question is yes where L is a link with corresponding braid group element (s₁)^m for all integers m, but I don’t know the answer for arbitrary L.

Terence Tao - the cosmic distance ladder

Terence Tao has put up a nice post about calculating astronomical distances - the cosmic distance ladder. Download his presentation and enjoy.

Binomial trees and a closed form expression for the terms of a recurrence relation

Recently, while playing with an integral of a probabilistic function possibly inspired by the binomial trees I’ve been reading about in financial mathematics books, I was attempting to find a closed form expression for the following recurrence relation: 

Let c_k(n), k, n = 0, 1, 2, …, satisfy

c_k(n) = c_k-1(n) + c_k-1(n-k)

subject to the boundary conditions

c₀(0) = 1 and c₀(n) = 0 for all n=1, 2, …

Fun problem to tackle

Here’s a little maths problem to tackle:

Terence Tao’s blog

I’ve come across Terence Tao’s blog and put a link to it on my blogroll. Terry is of course one of the 2006 Fields Medallists. Enjoy reading his blog.

Movies on hyperbolic spaces and three-manifolds

I just came across a very interesting movie (in two parts) on mathematical hyperbolic spaces! (FYI they’re youtube files.) This movie should give some idea of what three-manifolds and hyperbolic spaces are - conveying the idea of a closed, connected, orientable three-manifold to a non-mathematical audience can be difficult (I always use two-dimensional analogues to illustrate the idea) but these kinds of movies should be able to help.

It’s kind-of mind blowing if you havn’t come across it before.

Paper submitted to J. Knot Theory and Its Ramifications

Tonight I submitted the following paper: paper.pdf to the Journal of Knot Theory and Its Ramifications - the paper hasn’t been refereed yet so I’m merely offering it here for the reader’s enjoyment.

ON THE UNCOLOURED QUANTUM LINK INVARIANTS
ARISING FROM U_q(osp(1|2n)) AND U_−q(so(2n + 1))

Is Our Universe Decaying at an Astronomical Rate?

This post is just a copy of a thread over at Cosmic Variance. I’m copying it here because it’s an example of some of the mind-bending things that you come across in physics and mathematics, and one of the reasons that people do physics (and math). I’ve only read the post, so reading the thread should be interesting. Apparently Lee Smolin has written some comments on the thread, so it should be interesting. Lee, of course, is at the Perimeter Institute and is one of the high profile people looking at loop quantum gravity.

The thread starts here.

OO’s and BB’s
John at 3:02 am, February 21st, 2007
One nice thing about being a scientist, or at least an academic one, is that occaisionally you get your mind blown without any drugs or anything. Someone comes along and just pulls the rug completely out from under you - a total Denial of Reality Attack - and then you are left on your own to pick up the pieces.

Today at UC Davis we had a seminar from Don Page of the University of Alberta. The title and abstract of this talk sounded like science fiction, so I reproduce it here:

Don Page, University of Alberta

Title: Is Our Universe Decaying at an Astronomical Rate?

Abstract: Unless our universe is decaying at an astronomical rate (i.e., on the present cosmological timescale of Gigayears, rather than on the quantum recurrence timescale of googolplexes), it would apparently produce an infinite number of observers per comoving volume by thermal or vacuum fluctuations (Boltzmann brains). If the number of ordinary observers per comoving volume is finite, this scenario seems to imply zero likelihood for us to be ordinary observers and minuscule likelihoods for our actual observations. Hence, our observations suggest that this scenario is incorrect and that perhaps our universe is decaying at an astronomical rate.

Boltzmann brains? WTF? Intrigued, I went. This is a well-respected, highly-cited cosmologist after all. A former student of Stephen Hawking, no less. The jargon in the abstract, though bizarre, had a certain je ne sais quoi…