Federal Govt – Singapore Airlines – private benefit from public decisions

February 25, 2006

The federal government has recently decided to not allow Singapore Airlines to compete with Qantas on the Sydney-Los Angeles air route.

It seems a bit odd that a private company can potentially benefit from a government decision, which seems, at face-value, to be limiting competition from another large carrier.

This raises another point – people often complain and ask for compensation when public bodies take actions that reduce the value of private property. But when public bodies make decisions that increase the value of private property, people take the increased value quite happily – which seems a little assymetrical.


International Congress of Mathematicians, 2006

February 12, 2006

The International Congress of Mathematicians is being held 22-30 August, 2006, in Madrid. It is the pre-eminent mathematical congress, and is put on every two years. Here’s a link: http://www.icm2006.org/

I’m planning on going to give a short talk on my thesis work and extensions to it – I’ll pay for it out of my own pocket and annual leave if work isn’t willing to. Lucky academics – they often get funding to go to conferences as its part of their job!


blog blog blogggggg…..

February 9, 2006

Part of the reason I started up this blog was to keep my mind on real maths – if I’m going to have a chance of staying in that game then I need to keep on thinking/researching mathematics.

It’s hard to find time to do this amongst everything there is to do in life, especially as full-time work tries to fill your mind with its concerns to the extent that it can.

I’ve obtained a little result about the Hecke algebra and spinor representations of the quantum superalgebra Uq(osp(1|2)) – I think I should write it up. 


Finishing a PhD

February 9, 2006

In January 2005 I started a full-time job writing math questions for tertiary entrance, and for secondary and primary school students – on reflection, though, it was a mistake to work full-time, as I was nearing the end of my PhD thesis. I submitted my thesis in April 2005, and in the two months before submission I was effectively doing two full-time jobs – it was insane! I was working in a new fairly full-on job during the day, and then going to uni and working till about midnight on average (a few days I worked till after 1am).

Don’t do it! While we all need money to live, you need free time to study, especially when you’re completing a thesis. At most, work part-time if at all possible.

 


Reaction to the Danish cartoons

February 9, 2006

The reaction to the Danish cartoons seems very strange to me – ok – let’s assume that governments are behind some of the demonstrations. To the rest of the demonstrators I say – they’re just cartoons.


Books and magazines I’m reading

February 9, 2006

At the moment I’m reading
1. The Ancestor’s Tale by Richard Dawkins – it’s very interesting, I havn’t read any of his work before, and
2. Climate Change, by Barrie Pittock (formerly of the CSIRO) – which also looks like a good read – but I’ll probably finish The Ancestor’s Tale before reading this one.

Late last year I read A Short History of Everything by Bill Bryson, which Bec gave me for my birthday. It’s fantastic – great popular science writing.

Last year I bought Roger Penroses’ The road to reality – a complete guide to the laws of the universe – a fantastic book, although tough going if you don’t have a background in physics! The title is a little cheeky, but hey, there isn’t enough humour in the world, so I’m all for it.

I also subscribe to National Geographic magazine and to Scientific American as well as well as an environmental magazine the CSIRO puts out.


Some other maths papers I’ve looked at

February 9, 2006

My friend Tel is near the end-game of his PhD (his supervisor is Mark Gould) – he is going to write it up soon – good luck Tel! And he’s published a paper with Mark (well, on the Archive):

Some Twisted Results: http://lanl.arxiv.org/abs/math.QA/0504184

The Australian Mathematical Society Gazette recently published the 3rd paper on election timing by D. Lesmono and Elliot Tonkes (up at Qld Uni), entitled A continuous time model for election timing http://www.austms.org.au/Publ/Gazette/2005/Nov05/ .

This paper appears to be an application of financial mathematical techniques and statistics to a toy model of Prime Ministers/Premiers calling an election in a political system such as the Australian House of Representatives.


Math papers I’ve recently looked at

February 9, 2006

I’ve been very busy with work in the last four weeks and so have only skimmed over a few maths papers, in addition to doing my own research (unsurprisingly related to Hecke algebras, as mentioned in the first post).

First of all, Nathan Geer, of the Georgia Institute of Tech, and whose web-page is http://www.math.gatech.edu/~geer/ , has recently published four papers (or put them on the archive: http://xxx.lanl.gov/ ) which seem quite interesting. Nathan works in low-dimensional topology (ie, knot theory) and Lie (super-)algebras and quantum (super-)algebras. I hadn’t known of Nathan or his work before seeing his papers on the archive – and I’d worked in similar areas for 5-6 years! Anyway, his papers:

Some remarks on quantized Lie superalgebras of Classical type, at http://arxiv.org/abs/math.QA/0508440 . I don’t fully understand the abstract, but a phrase I do understand is

“…we show that all highest weight modules of a Lie superalgebra of type A-G can be deformed to modules over the Drinfeld-Jimbo type superalgebra.”

 Ah ha! This tells us what some of the representations of the Drinfeld-Jimbo quantum superalgebras are like – they are “deformations” of the reps of the corresponding Lie superalgebras. I like this.

The Kontsevich integral and quantized Lie superalgebras, Algebraic and Geometric Topology 5 (2005), paper no. 45, pages 1111-1139. This paper also looks very interesting – it discusses the Links-Gould knot invariant that Jon Links and Mark Gould discovered in the early 1990s, up at the Maths Dept at Queensland Uni. Mark was one of my thesis examiners 🙂 

According to the abstract of this paper, “Le and Murakami showed that the quantum group knot invariants derived using R-matrices and the Kontsevich universal link invariant followed by the Lie algebra based weight system are the same”. In this paper, Nathan shows a a similar result of Lie superalgebras of type A-G and he investigates the Links-Gould invariant.I don’t know much about the Kontsevich universal link invariant (it’s one of the things I want to read up on some day) – but it’s interesting that this paper apparently shows more links between different areas of mathematics.

Louis Kauffman and David De Wit deserve a mention when discussing the Links-Gould polynomial – David did his PhD thesis on using Mathematica to automatically calculate Links-Gould “tangle-invariants” and has done some subsequent work on this – I understand that Louis Kauffman helped out as well.

Nathan also wrote these two papers:

Multivariable link invariants arising from sl(2,1) and the Alexander polynomial. (Joint work with Bertrand Paturear-Mirand.)  math.GT/0601291
Etingof-Kazhdan quantization of Lie superbialgebras.  To appear in Advances in Mathematics.  math.QA/0409563

I’m too tired to discuss these, so I won’t even begin!


RU486 and math papers I’ve recently looked at

February 8, 2006

Ok, I deliberately chose that title to spark controversy. If you’re not Australian, you might be unaware that the Australian Parliament is currently debating a proposed law that would transfer the Australian Health Minister’s effective control over the distribution of the drug RU486 to the Therapeutic Good Administration, which has this role for many other drugs.

The current situation seems very strange, but it can be understood in light of the influence that former Senator Brian Harradine had in 1996, when the existing laws were passed. My understanding is that Brian Harradine pushed for the amendment that brought about the current laws.

Of course, I don’t know a huge amount about this topic, but as a general approach it seems strange to me that RU486 is under the control of the Federal Health Minister, a most unusual arrangement – I don’t know what the Health Minister knows about the distribution of this drug unless the drug is so special that political leaders need control over it. Very strange. Mr. Abbott has suggested that people have discussed this Bill with reference to him being Catholic – which of course has nothing to do with anything! I can’t stand this kind of “political discourse” – absolute nonsense.

My prejudice would be to vote in favour of the Bill.

Now, it’s time for mathematics. But I’ll put that in the next post…


Hi

February 6, 2006

Hi,

I just set up this blog to record my thoughts on maths, science, social social, politics, and anything else that strikes my fancy. Why did I call it Sachi’s hyperbolic space? Well I’m a mathematician who’s studied 3-manifold invariants (hence the “hyperbolic”) and “Sachi” is my nickname.

I’m interested in all sorts of things including applying evidence-based enquiry to problems in society (eg to politics, education and economics), ecological economics, whole earth system studies, physics, pure mathematics, applied mathematics (eg how does turbulence arise from the Navier-Stokes equations, or does it even ???) and I’m attracted to the idea of studying m-dimensional objects in n-dimensional spaces (where m is less than n-1), just because it seems beautiful.

This blog might end up being just a forum for me, or it might be part of the “blogging community” – who knows – and perhaps B******may even write a post.

Here’s a taste of the mathematics I’m currently interested in. The Hecke algebra H(n,q) (also known as the Iwahori-Hecke algebra of type A(n-1)) taken over the complex field, is generated by the invertible elements {1, g(1), g(2), …, g(n-1)}, subject to the following relations (where q is a complex number):

  1. g(i)g(i+1)g(i) = g(i+1)g(i)g(i+1),  for i=1, 2, …, n-2;
  2. g(i)g(j) = g(j)g(i),  if |i-j| > 1;
  3. (g(i))^{2} = (q-1)g(i) + q,  for i=1, 2, … n-1.

Note that relations 1. and 2. are just the Braid group relations, so any representation of H(n,q) is automatically a representation of the Braid group B(n) on n strings.

Suppose now that you have a representation of the Braid group given by the algebra homomorphism:

p: g(i) -> R(i),

where each R(i) satisfies the equation

(R-a)(R-b)=0;

(suppressing the index i) for some non-zero complex scalars a, b where a and b are not equal. Then the algebra homomorphism defined by

g(i) -> -R(i)/a

yields a representation of H(n, -b/a) in the algebra generated by {R(i), i=1, 2, …, n-1} taken over the complex field. 

This is very cool, because I think that the Jones Polynomial (of knots and links) can be obtained from the Hecke algebra (or some representation of it) – then, by being sufficiently clever (making sure you have a Markov trace etc), you can obtain a link polynomial from your representation of H(n, -b/a), which, after some suitable reparametrisation, is just the Jones polynomial.

I have to check this, but I think it’s on the right track. Something similar happens with the Birman-Wenzl-Murakami algebra. If you have any representation of this algebra, you automatically have a link polynomial. Very nice.