**Copyright
George Christos 2003**

sorry for the scant (or lack of any) details in some places. that's because i no longer (or do not have much time to) work in some of these areas

**Chaos theory and
fractals**

**A
fractal is an object which looks self-similar on all scales.
If we examine a fractal under a magnifying glass (of any strength) we
essentially see the same structure, with the same amount of “roughness”.
In the picture above the 'eyes' repeat themselves inside the picture ad
infinitum. Examples of fractals in reality are the intricate network of blood vessels in our bodies,
the shape of clouds, the shape of a pine tree. in fact most things are
fractal as opposed to geometric. One
of the most interesting features of a fractal is that it seems to have a
non-integral or fractional dimension (hence its name).
Although a single segment of a blood vessel has dimension one
(corresponding to a line), the network seems to uniformly protrude into a
3-dimensional space. [Interestingly,
there is no cell within the human body which is more than a few cells away from
a blood vessel.] The effective
dimension of a network of blood vessels is somewhere between one and three. **

**Chaos
theory is the understanding that some things we cannot understand, even if the
underlying mathematics is understood. This is especially true of iterative
systems. One
of the most interesting features of chaos (or pseudo-random behaviour) is that
if we start with two almost identical states then the difference between these
states increases exponentially fast until the two almost identical initial
states have evolved into two entirely different states.
Examples of chaotic systems include the weather, and mixing cement or cake mix.
You
may observe this phenomenon the next time you mix a cake or cement. Notice
how two nearby particles of cake mix drift apart rapidly, but then almost come back to the
same spot (notion called a 'strange attractor').
Chaos and fractal go hand in hand. There is lots of stuff on the Internet
on chaos theory, so I won't say much more here. **

**Genetic Algorithms **

**Nature has solved many complicated problems in the course of
evolution. We can use these same ideas mathematically to solve complicated
optimisation problems. I am looking at applying this technique to
solve some real world problems. An example is setting the football fixtures
(see below). I supervised a PhD student in this area.**

**Memetic Algorithms **

**These
are like genetic algorithms except that information is processed somehow before
it is transmitted down the line. **

** self-organizing
and adaptive
systems (other than the brain)**

**This
subject is of interest to me but haven't really done much in this area as yet.**

**elementary particle physics
and quantum field theory - this is what i did
my Phd (I mean DPhil) on at Oxford University**

**lots
of my own ideas**

**I won't go into this stuff
here, such a long time ago, too much to say and I have really moved on.
One of my main achievements was being invited to write a Physics Report on
"Chiral Symmetry and the U(1) Problem". This received a lot of
attention after Nobel Laureate Gerard 't Hooft wrote a whole physics report to
counter one little negative thing I said about his work. I claim that
instantons do not solve the U(1) problem (a famous problem in particle physics)
because people where using the wrong chiral symmetry charge operator. In
his article 't Hooft refers to people by name except that I am referred to as
reference 7, even though I appear a few times on every page. I would have to say
that we still agree to disagree.**

"You're right from your
side,

I'm right from mine."

Bob Dylan, __One Too Many
Mornings__, 1964

- quantum chromodynamics (theory of strong sub-nuclear interactions between quarks and gluons)

- chiral symmetry and current algebra

- U(1) problem, and anomalies

- large N (quark colour) expansion

- loop space

- bound on the number of leptons and quark flavours

- quantum field theory and gauge theories (and quantum mechanics)

** Monte Carlo simulations of
large polymer chains**

**We
simulated a polymer chain with 320 beads on a slow VAX computer, while someone
in Canada on a Cray was only able to simulate a polymer with 32 beads.
Later I was able to simulate 64 polymers in a box. I did the programming
and thought of most of the fast algorithms, even though when I was hired I had
no programming experience. I recall in those days that we sue to get a
bill every month for the amount of CPU and disc space we used on the
mainframe. Most people would receive a bill of around $20. After a
few months in the job, I was running up bills of hundreds of thousands of
dollars and soon millions of dollars each month (don't worry it was just funny
money). In the end the computing centre had to abandon its billing system
because they could only handle 6 digit figures.**

**theoretical
chemistry, electron charge and spin density configurations**

**boring
for me**

**Mathematics in Sport
**

** Over the last few years I have become interested in applying
mathematics to sport (just for fun - actually everything i do is just for
fun). Here are some of the things I have thought about in my
leisure. **