Hi

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.

10 Responses to Hi

  1. Steve Edney says:

    Good to see you have your own blog sacha. I must say I’m pleased someone is blogging pure maths even if it makes a poor ex-theoretical physicist’s head spin.

  2. Steve Edney says:

    Do you allow comments?

  3. blumensacha says:

    Hey Steve, I certainly do! I’m just learning how to use this blog, so hopefully it’ll all be going sweetly very soon!

    So you’re a ex-theoretical physicist – how interesting! What area did you study or were you interested in? And were the career/emotional/psychological prospects outside physics better than inside? I can understand this if this was what you were thinking. The career prospects for maths PhDs is not absolutely fantastic.

  4. Yobbo says:

    I like this template. I’m planning on stealing it.

  5. Sacha says:

    I like it too. It’s the best one from the ones provided.

  6. Sinclair Davidson says:

    Hi Sacha

    i’ll keep an eye on the non-math content…

  7. Steve Edney says:

    Ah ha. Comments are appearing.

    Sacha, I was studying solar flares, a couple of different topics with in that, but probalby the more interesting part was looking a whether we can use the theory and models of self-organized criticality to explain the observed power law distribution of flare size.

    As for career prospects, I left after finishing my PhD because I was a bit disillusioned with the research area. It was interesting enough, but I can’t say I was devoted enough to work on it for academic wages. Not like others who really lived and breathed their research areas. So a friend suggested I go for a job in quantitative finance, and here I am not 5 years down the track.

    Basically we work on developing and implementing pricing models for derivatives contracts, (options, futures, interest rate swaps etc) , and also the risk models for the banks. While we don’t do the hard core maths behind this (mostly because its already done), we do have to be able to understand and adapt it as there is always demand for new products which are variations on the standard model and mean you need to understand the theory to tweak it for the new situations.

    I’ve worked for a couple of different domestic banks. The money is good, a multiple of academic salaries, the work is fairly interesting although at parts can be a bit boring at times but you get that. It requires a fair bit of technical maths knowledge even if you are not using it all the time. It also requires you to be fairly adept at coding. The finance side can be learnt pretty quickly on the job, unlike the mathematical side – this is the reason they hire maths and physics PhDs. Oh and once you have some experience you should never have trouble getting a job, particularly good if you are interested in living in Tokyo, London or Singapore.

    Send me an email if you are potentially interested in this type of thing. I don’t know of any jobs currently but if you were interested then let me know as I have some contacts.

  8. blumensacha says:

    Hey Steve,

    Thanks for this – it’s very interesting.

    When I finished my PhD I was so sick of it – at times the only thing that kept me going was the idea that I wasn’t going to let anyone stop me finishing – but those were the dark days. I looked at a math/physics book only two months after submitting the thesis, much to my surprise, and now I’m actually enjoying doing maths – and I’d really like a maths research job.

    I’ve been working in the school education sector for 12 months now, and I now know that I don’t like working in an office and that the science thing of finding things out agrees with me more than fulfilling contracts does! While the money might not be as good in the research world, all the other aspects of it make up for it, especially as you’re doing something you want to.

    Maybe this is just the perspective of someone outside the research world wanting to get in, but there you have it.

    Perhaps working in the financial sector is a good strategy for being employable – the math research sector is not very large! A friend of mine, Freaky, was into mathematical physics and did his PhD in wave functions of electrons on twisted one-dimensional waveguides (finished about ’97), then he did a Masters in Financial math and is now happily living in London – he did it for employability reasons and he’d be doing maths in the work.

    A friend of mine who I went through Honours with, and whom I shared an office with for a few years in the PhD, is becoming depressed about his chances of work – he has been doing physics postdocs in Nth America for 5 years and he thinks that he has poor career prospects (permanent physics jobs are hard to come by in America). He’s too negative.

  9. Steve Edney says:

    Sacha,

    The majority of my friends who I went through honours and PhD’s with have left research. A few in finance, a couple to become patent attorneys, at some into IT, and another to school teaching. About 4 are doing some form of research but at least two of them have been on a string of short term jobs and would expect them to drift away soon enough as well.

    For most of them I don’t think they have regretted the switch out of academia, even though they enjoyed their time doing PhDs.

    Personally the only think I miss is teaching which was fun.

  10. blumensacha says:

    I was just thinking that all bar ~4 of my friends who went through honours and phd land have left research.

    I’ve always been interested in international relations – not that I’ve formally studied or been involved with it – it seems like a potentially very worthwhile thing to do – although there is the “being a government’s mouthpiece” thing, but there you have it.

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