Sophie F.A.Q 2.02b

This is a collection of questions that I keep being asked and find myself unable to answer well on the spot. Of course that's not to say the answers here will be that good either! This is the third draft, I mean to go through and revise them every now and then. I'm not sure that I can be bothered going through and making the long, be-comma-ed, badly grammered and spellt sentences less so, since thats all part of the Sophie Ambrose experience. (And it's the only way I know how to communicate.)

What is your Phd on?

The short version: Algorithms to deal with groups. (see next answer if you don't know what a group is!)
Long version: I'm doing a number of topics in algorithmic group theory. Divided vaguely into years below.

First year of my phd: Proportions of primitive prime divisor elements in classical groups.

(Jettisoned for being WAY too boring and hard)

Simple version:
In a lot of matrix algorithms there is a line somewhere which says "Choose a random matrix with a certain property". I was seeing how common these elements are, which makes it possible to analyse how long a step like this would take.

More detailed version:
Research into finding proportions of elements in GL(d,q) (the set of all dxd invertible matrices over a field of size q) whose order is divisible by a primitive prime divisor of q^d-1. (This is a special kind of prime that comes up a lot)

Second year: Black-box recognition algorithms.

Basics:
A "black box" group means that you can't really "see" the group elements. You can calculate what element you get if you multiply two together, or invert one etc. But unlike matrices, say, you can't just go "What is the entry in the top right corner". I have been developing a series of tests that allow you to figure out if a given black box group is actually just another group with all the elements renamed.
As an example, here is a dodgy, dodgy black box test for the group {0,1} under addition:

Choose random element x
If x+x=x 
  rename x to 0;
  choose another element y not equal to 0;
  if (y+y=0) 
     rename y to 1 and return TRUE;
  else return FALSE;
else 
  rename x to 1;
  set y=1+1; 
  if (y+y=y)
     rename y to 0; 
     return TRUE;
  else
     return FALSE;

Note that if the answer is "TRUE" you get a renaming, too. In my algorithms the map is always right, though sometimes it says "FALSE" when it shouldn't. (Unlike the one above, which does the opposite)

More detalied:
The two types of groups I've been recognising are simple sporadic groups and classical groups. A simple group is one with no nontrivial normal subgroups. Most of them can be classified into certain infinite families, and these have their own recognition algorithms. The ones that don't are called "sporadic" and each need their own recognition algorithm. Classical groups are groups of matrices which leave certain maps invariant. The most common example is the general linear group GL(d,q).

Third Year: Rank Profile

(Assumes linear algebra knowledge. And most of 3rd year was actually the stuff above)

Say you have a matrix and want to find a maximal (ie, as large as possible) subset of columns which is linearly independant. The "logical" way to do this is to start at the left and add each column you come to that is linearly independant of the columns you already have. The columns you get doing this is called the "rank profile" of the matrix.
(Well, actually, it's the numbers indexing the columns, ie if you picked columns 1,2 and 4 then the rank profile is {1,2,4})

I came across the rank profile trying to implement the Keller-Gehrig algorithm to find the charcateristic polynomial of a matrix. They used the fact (its fairly easy to prove) that the rank profile of a matrix is the same as that of it's row echelon form without ever saying so or even DEFINING the rank profile. They just said "Find the set of columns where each column is linearly indepedant of the previous columns of the matrix. This is done by calculating the row echelon form". I and my supervisor got VERY CONFUSED. We finally got it all sorted out and have written a simpler, much easier to understand version of the algorithm.

In trying to get to the bottom of the whole rank profile thing we also came across a way to adapt an algorithm to calculate the LSP decomposition (see below) of a matrix so that it simultaneously calculates the rank profile. Then realised it was just easier to find the row echelon form and calculate the rank profile from that. (The rank profile of a a matrix in row echelon form is the set of columns who contain the first nonzero entry of some row. Very easy to calculate.)

An LSP decomposition of a matrix A, by the way, is a set of matrices L,S, and P such that LSP=A and L is lower triangular with ones on the main diagonal, S is upper triangular plus some zero rows, and P is a permuation matrix. (Each row/column has exactly one 1 and the rest are zeroes)

And what's a group?

The most wonderful thing in the world!
Well, I like them. Basically they're things like matrices, numbers, rotations..anything that can be "added" or "multiplied".

More formally: suppose that we have a set S and a binary operation * on S. A binary operation is something like addition, or multiplication, where for any two elements s and t of S, we have s*t an element of S.
S and * form a group if we have: (I apologise for the dodgy maths-in-html. I need Latex.)

There is an identity e in S such that s*e=e*s=s for every element s of S.
For every s in S, there is an inverse (-s) such that s*(-s)=e.

So, for example, the integers under addition is a group, since 0 is the identity and -x is the inverse of x.
Invertible matrices are a group under multiplication with I the identity and inverses are inverses :)
An example of a finite group is {0,1} under addition, where 0+1=1+0=1 and 0+0=1+1=0. Zero is the identity and both numbers are their own inverses.

Now you might ask how is this useful? Well, if we discover something about groups theoretically, then we can apply it to any group. Theorems that were originally proved for permutations can be applied to matrices and used for all the things matrices are useful for. (You know- codes, graphics, that sort of thing) Also, see Symettry, symettry for an example of applying group theory to jewellry.

A field is like a group under "addition" which also has "multiplication". For every element x except the identity of addition (ie zero) we can divide by x. (This is a quick and dodgy definition!)
The most familiar example is, well, numbers (Including fractions) with 1 the identity for multiplication, and 1/x the inverse of x under multiplication.
An example of a finite field is again {0,1}, with 0*0=1*0=0*1=0 and 1*1=1/1=1.

How can you like maths?

Because it's so cool! But I didn't always feel that way. I found maths pretty dull (with the odd vaguely interesting idea) until year twelve, where we did some cool things in calculus. But even so I only continued with it at uni because I was good at it and it was useful towards physics. It was only only in third year that we really got into the interesting stuff.
The problem with maths is that most people who study it are doing so because it's required for something else, so all the things you get taught are useful but boring.

Of course not everyone would like maths even if they did what I consider to be the interesting stuff. There a lot of areas of maths that I find difficult and/or boring, even in pure maths which is my area. It depends a lot on the way you think. Also, if you're really not good at something, it's hard to enjoy it. (Hence me and sport)

To me, pure maths is this beautiful construction of elegant ideas that exist independantly of this confusing, inconsistent world. I used to feel that way about physics, but because there people are trying to understand rules that are created by the universe, not by people, all you can get is approximations and flawed models. Maths is a purely human construction based on ideas that we comprehend intuitively, and so can be understood more completely.
I am aware of Russels paradox and the implication that maths has holes. I just try not to think about it too much :)

Oh- and if by maths, you mean statistics... well, I don't understand how anyone can like that either! Although I'm willing to believe it's possible.

Whats with the whole atheism thing?

First I should say that strictly speaking I'm an agnostic, since I'm open to the idea of some sort of vague, non-interventionist god. But there are certain ideas I simply cannot conceive of being true which precludes my acceptance of pretty much every religion I've come across. (except the really vague ones like Deism)

These ideas include: Objective good and evil, a soul, reincarnation, and the idea that just cos a being created you it deserves worship (Respect and obescience maybe. But not worship)

Also, I seem to be incapable of religious faith, which really takes the fun out of religion!

So what DO you believe?

The world is too complicated for our feeble primate brains to understand. Instead we take in some of the details and use pattern-matching to try to work out whats going on and why. We all think differently, so the conclusions we come to will be different.
The way I see things, there is no God. The world makes a lot more sense if everything is just a matter of chance and chaos theory (so to speak) than if there is some benificent being up there controlling things. I have just as much of a problem with reincarnation. Every religion I've come across feels like an obviously constructed story to make an unfair and random world just and livable.

However, having talked with religious people who are just as intelligent and observant as me, to them the world makes a lot more sense if there is a God (or goddess or whatever). Maybe they're right and I'm just not seeing the signs, theres really no way to tell. However, I'm not going to go through life ignoring what feels right and true on the off chance I'm wrong. I do make an effort to respect other peoples beliefs, though. Some of them are quite enlightening- even if religions are wrong about some things, they still offer tried and tested spiritual guidance, which I can't scoff at.

Have you even tried religion?

I was quite a devout christian until the age of 15. I read the old testament till I couldn't take any more genocide. I read the new testament and took to heart all that stuff about the narrow path, so did my best to do what Jesus would want. I just ended up feeling guilty all the time, for things I really couldn't do much about, like the fact I couldn't bring myself to like everybody.
Although I tried "opening my heart to God" I never felt for a moment that he was watching or protecting me. I did experience a sense of a malevolent presence judging me but am inclined to put that down to an overactive and somewhat morbidly superstitious imagination. The moment I let myself entertain the idea that God didn't exist it was like the scales had fallen from my eyes and everything made sense. I became a happier person and was able to make moral decisions more easily, since I had found myself disagreeing more and more with even the pared down liberal version of Christianity I had adhered to.

My parents are intelligent and openminded Anglicans, and I think the reason I stayed religious for so long is that all the atheists I met were illogical and just doing it to spite their parents and/or because they'd had bad experiences with religious people.

I still sort of miss the whole "Peace be with you" thing, though. That was fun.

Do you think religious people are deluded fools?

No. Or at least no more than the rest of us :) Unlike many atheists, I don't think that religious people are deluded and I have seen through to the stark truth with my Super Atheist Brain. I don't think anyone is capable at staring at the stark truth, it's just that my dodgy human ways of dealing with it happen not to involve God.

I am occaisionally a little scared to think that perhaps the only reason I can be a (fairly) happy optimistic atheist is that nothing really bad has happened to me, and that if i was ever to face the kind of unendurable hardship that seems to be most peoples eventual lot, I'd break down and possibly turn to religion. Or just go crazy...

I don't deny that many otherwise quite sane people have had intense undeniable spiritual/religious experiences. I don't entirely discount the idea that there is a genuine Something Out There people connect with. I disagree with every theory I've heard as to how it works, though, and see no personal benefit to being anything other than an open-minded sceptic. I feel that the boundaries between spiritual experience, intuition, subconscious desire and delusion are blurry and hard to define. There are plenty of examples of people who thought they were having valid religious experiences who objectively were just crazy ("God told me to kill them!") or projecting their own desires ("God told me to make lots of money in unscrupulous ways!") Personally I find I'm better off erring on the side of caution. This has a lot to do with my own experiences.

If I worry about things too much, I can experience them, generally in a hypocondriac I-have-a-toothache way but sometimes in a scary there-are-wasps-buzzing-in-the-walls way. My subconscious tends to give me unpleasant and erronious information like "Every time you look forward to something it goes wrong" or "Today you will die!!". Also I have a number of mildly mentally ill relatives. All in all, while I don't like to make generalisations about other people I think its not implausible to assume any "supernatural" experience I have is probably all in my head. By making this assumption I am able to ignore the crazy crap my brain sometimes spits out at me. I still enjoy the positive crazy/"spiritual" stuff, I just don't take it seriously.

That said, my mum has almost exactly the same attitude as me, but is a devout christian. As she puts it, even if it is a delusion it keeps her otherwise sane better than being an atheist did, and I don't have a problem with that. It just didn't work that way for me.

What are you and Cam's food issues?

EDIT: I (Sophie) am now being SUPER careful about gluten, dairy and soy. This makes me very hard to buy food for, best to avoid it unless you're sure it's ok.

I have a bunch of allergies as well as reflux. Cam has a few allergies. These are all fairly minor, we won't die or anything but will get indigestion if there's more than trace amounts of any of these foods in something we eat. Theres also some flavours we Really Don't Like.

Me (intolerances): Milk fat (esp cheese), lactose, coconut, cocoa butter, soy ~~(small amounts are ok, but not say a whole meal based on tofu)~~, leek, artificial sweetners. Not sure about goats milk etc. Possibly pine nuts.
Me (reflux): Acid unless neutralised by a base (this includes fruits like plums, most fruit juices, "real" yogurt, salad dressing, soft drink, and vinegar), alcohol (unless entirely cooked away), peppermint, paprika, watercress, too much fat (ie deepfried stuff), significant quantities of mustard
Me (dislike): Fruit creams, anything too spicy
Cam (allergies): cheddar cheese in large quanitites, cumin
Cam (dislike): Mint. Anything spicy ie pepper, ginger, chilli, wasabi... Stuff like Teriyaki where it adds to the tatse but there's no spiciness is fine.

Also I'm Very Sensitive to caffiene, so tend to avoid even mildly caffienated stuff like chocolate cake past 4pm-ish unless it looks really yummy and I don't have work the next day :)

What do you want for your birthday?

I'm not going to put anything that would be silly to have two of, which cuts out a few things. Ask me in person!

You can't go wrong with:
Book vouchers.
Socks, size 8. Nice ones :)

Colourwise I like black, rich purples, blues and reds. Not pink.

Do NOT buy me:(No offense to anyone who has bought me this stuff. You might have been an exception, and anyway its the thought...)
Clothes.
Teddy bears, unless they're interesting in some way. And even then...
Jewelry, unless you're really sure I'll like it, my taste is odd. Although pendants are a little safer.
Perfume. It's nice, just I get given it at a rate much higher than I use it!

Food:
SEE PREVIOUS SECTION. This is probably a bad idea. I do like nice nougat, but make sure it's free of gluten etc.

Old info: I like medium to good quality chocolate with roast almonds, praline or just a nice selection. Not fruit creams.
I would prefer a small amount of nice choclate to a large box of Red Tulip. I don't like Guy Lian enough given how much it costs!
I have a milk fat/cocoa butter intolerance, but don't like SuperDark unflavoured stuff, which leaves the flavoured dark Lindt stuff or "Old Jamaica" etc

If you still want to know more about how I tick...

Why not have an amble though the scenic route?