C is for Coding

Coding Theory is the area of maths that I'm researching and writing my doctoral thesis on. Essentially it's all about achieving efficient information transmission (or storage) by adding or removing redundancy.

C is also for Ceol - the Irish word for music. Each year as part of National Irish Week (Seachtain na Gaeilge) there is an album released of popular music with Irish language lyrics instead. The 96cambridge channel on youtube has some of the songs from these.

For a mathematician we have Cauchy- one of the most prolific mathematicians of all time with almost 800 papers.His name appears frequently and in many seemingly disparate branches of maths.

As mentioned before C is for Commutative. In maths we say an operation * is commutative whenever a * b = b * a. Addition and multiplication are commutative but subtraction and division are not.

Honourable mentions go to Crochet, Cryptography and Calves that ache after dancing my socks off at the wedding!

Writing Scared, Sick and Tired

First thing this morning an email arrived. The final version of my paper has been received and compiled. Whew. Soon I will be excited about going to Paris to present it. Right now I'm exhausted.

Back in January I realised the submission date for the International Workshop on Coding and Cryptography was approaching fast. By the time I'd agreed to submit a paper with my supervisor I had five days to write up my work as a paper. At which time I came down with a viral infection. Thankfully I had the research complete and most of it loosely typed up. Unfortunately it's the words in between the maths, the history and conclusions that I find the hardest of all. Years of practice studying pre-tonsillectomy, lots of determination and a HUGE amount of support from G and Suey meant I made it. My first paper submitted to an International conference.

A few days recovering from the illness were ok but I still had seminar to write and present, a talk to prepare and give and on top of all that a more general theorem to prove. When March 1st came along I found out my paper was accepted! Delighted! There were a lot of corrections to do before final submission though, not surprising given my feverish, analgesic addled state writing it. Finally got it all adapted, edited and typeset perfectly last night, just in the nick of time.

While it is joint work with other authors, I would like to secretly dedicate "Characteristics of Invariant Weights Related to Code Equivalence over Rings" to writing 'even though' scared (even though you're scared you won't be able to do it, scared it won't be good enough, scared you'll make a mess of it, scared you'll miss the deadline), sick (trying to plan your painkillers so you're not too fuzzy-headed to think, or too in pain to concentrate, even though you should be in bed, ignoring how your body feels so you can work) and tired (tired of writing, tired of having to work all the time, tired of being awake for too long) and the amazing magical people that encourage and support us to do it.

Now back to writing the Thesis.

Projective Geometry

Learning a lot about Projective Geometry at the moment. Even gave a talk on it to the maths society. It's really interesting and fun, unfortunately I think geometry is a little too literal minded for my abstract take on everything. The idea is to generalise to geometries on rings and then use these to create low density parity check codes in an analogous way to how they are created using geometries over finite fields. Watch this space! (but not too enthusiastically)

Scrabble in Research

Have finally found a way to describe my research area to 8 year olds. If they know about Scrabble that is. Essentially I work on proving extension theorems about weights on ring linear codes. If one really simplifies this you can think of it as follows. Think of the words we make in scrabble as being codewords, out of the set of possible combinations of the letters only some of them are actually words. Every letter has a score assosciated with it, usually on the letter tile. If we call this the weight of the letter and the total score (neglecting any triple-letter-score bonuses etc) we call the weight of the word. Then the possible scores that we can attatch to the letters and the repercussions of these changes are essentially what I work on. Not much help in real scrabble though! Of course the alphabets I use are different, the rules for making 'valid' words must have certain properties and there are infinitely many of each letter in the imaginary scrabble bag.

Mean Maths!

I had a nice formula for something. Then I couldn't prove its validity when I went to type up my results. Wrote up things the awkward formula-free way. Asked my supervisor for help because I'd found a flaw in the formula. So he gave me a helpful idea. Which was great, I wrote up a nice new proof of the formula. Except I then realised I still have the same little flaw. Have now eventually found a work-around, which is a pain, and can continue. Continue and completely re-write and re-latex all my major work that is. So relieved and dejected. An interesting combination, added to which we have the nerves about the poster I'm trying to design. Lots of working late this week will probably be continuing for some weeks.

Getting Places

Had a very exciting day yesterday, mathematically. I think the insomnia gave me lots of time to think things over so that I came up with ways to solve the problems I was working on. Then proceeded to show all sorts of wonderful correspondances between the ring and the algebra I'm working with. Hooray! Plodding throught the details today and hopefully there'll not be too many fiddly bits. One of the best meetings with my supervisor I've had in a while. He had vision and I had an idea of how these things connected. Great Stuff! If I've it all written up my friday I can have a fun weekend of LaTeX :)

Radical excitement

Ok, I think I have the radical of the algebra in terms of the radical of the underlying ring. Hooray! A tiny step along the way but it might be as far as it's possible to go so I'm pleased, provided it's all correct of course and I don't wake up in the middle of the night realising I've made a mistake somewhere along the way....*fingers crossed*

Now just the invisible theorem and the tricky review to finish...

Radical Times

Ok, at the moment I'm trying to characterise the Jacobson radical of the factor algebra of the algebra of complex functions from a ring under multiplicative convolution. If you think that sounds convoluted I probably shouldn't add that this was the simplest explanation I could come up with. I've typed up a rough sketch of some of the background work I've done on this algebra, what it is and how it works. The Jacobson radical however is proving elusive, despite all my best efforts. Essentially I've a big set of objects with some structure and I'm trying to find a little very structured bit inside it that has special properties. Tricky stuff :) Hopefully once I've found out what it looks like I'll be able to use it to prove a theorem that another researcher in the field has already proved a different way. Made especially difficult since I can't seem to find his proof or any mention of it.

Also, I've been aksed to review an article on an area related to mine for a recognised journal. Unfortunately I've not much experience with this area which means an awful lot of background reading. I don't think my review will be up to much although I think I've found a neater way to define one of the objects they use. My lack of global vision on the topic precludes any idea of the overall relevence of the work.

Frustration

It looks like most of the work I've been doing for the past while on weights over Frobenius rings has already been done/published by someone else in a series of 3 papers. He went about it a different way and the three papers overlap heavily, it's hard telling which order they were written in, so hopefully the alternate approach will still be worth something. Probably not. It's difficult even figuring out exactly what he has proved and where it leaves me.

semigroup algebras

Think i might be one the verge of a breakthrough in proving a certain isomorphism i've been working on :) hooray! i just need to try get the formulation of the map right. Could be tricky though. I'll keep working at it nonetheless!

Ring Linear Coding

I am giving a short talk on ring-linear coding today. I have put the pdf of my talk on my website for anyone who might be interested in learning about rings, codes and Mac Williams' Equivalence Theorem extended to codes over Frobenius rings. Wish me luck!

Mnnn Monomial Transformations....

Found a lovely paper from the 1940's detailing how monomials are made up of normal cycles and then some papers by Ralph B. Crouch(1) based on that work. Not directly related to my current work but could be underlying Ioana Constantinescu's Monomialsatz paper. Also cool and interesting :)


(1) Wonder if his middle name is Bartemius