What is mathematical modelling?

To understand my research it is necessary first to introduce a few concepts - quantitative models, mathematical models, symmetries and anomalies.

What is a quantative model? A model is very much like an analogy except that in good models we seek to represent quantitative (measurable) information. A model with only qualitative information would be Shrek's claim that: "Ogres are like onions, they have many layers". It might tell us something about Ogres, that they are creatures with deep inner lives, but it doesn't tell us anything we could actually measure. For example, what is the average height of an Ogre, how fast can they run, how many friends do they have...

A good example of a quantitative model which most of us are familiar with is a map of the Earth. A map can give us information about certain features of the terrain such as: where there are lakes, roads and villages relative to the compass and each other. A map can also include extra details such as contour lines to represent how high a particular piece of land is, without having to make the map some three dimensional miniature of the landscape.

A good model is chosen to have just the right amount of information for its purpose. Considering maps once more, the map of the London Underground is an excellent and simplistic model if you want to know how to get from Charing Cross to Piccadilly but don't care how far apart they are.

An example of a not-so-great underground map, in my opinion, is that used by the New York subway system. 

It tries to show where subway stations are relative to the actual geography of New York and is consequently much more confusing.

What is a mathematical model? Unlike the map a mathematical model seeks to describe systems in terms of equations. For example, let's say we have a train that travels at 10cm per second around a track. A simple model determining how far the train has traveled would be

This model tells us one fact about the train, how far it has traveled in a given time. It did not require us to know what shape the train was, what colour it was or any other irrelevant details. The train could have more complicated behaviour, for example it could be accelerating, and we would need a different mathematical model to describe it. 

There is one very important feature of our train model that is necessary for it to be describing something we call "physics" - that is it is predictive. It can tell us something about the future of the train. Similar situations to our train model were important in the early history of science, such as determining how far a cannonball would go if we shot it upwards at such and such an angle or why objects fell towards the earth with the same acceleration. Nowadays we are interested in much rarer systems: the universe a few seconds after the big bang, gases cooled down to temperatures that can only be achieved in the laboratory or black holes. Nonetheless the principle is the same - write down equations that have just enough information and see whether they predict the future behaviour of the system. If they don't; try a different mathematical model until we get it right.

Ok, so what do you do? In my recent research I have been trying to write down systematically describe all mathematical models describing particles in a box with scale symmetry. Often it is the case that the scale symmetry is broken when I solve the equations and therefore the symmetry is anomalous. With my collaborators I have managed to completely classify all these scale invariant theories (there are infinitely many) and figure out which ones are anomalous. Moreover I have been able to show how anomalies affect the observed behaviour of these systems.