"The fourth dimension and beyond - the paradox of working in unimaginable worlds"
Emeritus Scientia Professor Ian Sloan AO FRSN
Date: Wednesday 3 September 2014
Professor Ian Sloan is not content to work in an environment of four dimensions – he is quite at home in space with many more dimensions than most of us are accustomed to. Many mathematical problems can be considered as problems and multidimensional space – the question is how do we imagine these environments? The dimensions of a space can be considered to be the number of directions that you can go from any single point within it. For example, in our four-dimensional world, from any point we can go in three spatial directions, plus time. If we are in a six-dimensional environment, we can go in six directions from any given point and mathematically we don't need more than six variables to describe this environment. But why would we be interested in multidimensional spaces?
Many problems are best analysed in multi-dimensions. For example, the shop may have 250 stock items. This can be thought of as a single point in 250-dimensional space. Each stock item has a price, so there are another 250 dimensions to consider. One area where this approach has a major application is evaluating certain types of financial transactions such as derivatives.
An investor may want to analyse a potential investment in Woolworth shares, for example. The payoff might be thought of as the closing share-price over a period of 250 trading days in, say, $10 increments. Using multidimensional mathematics, the investor can calculate the expected payoff at a certain trading day on the basis of what the closing share price might be. Such calculations soon become extremely complex, in fact, too complex to be evaluated (this is known as "the curse of dimensionality" a term coined by Richard Bellman, a noted researcher in this field). And what if the payoff can take any value, rather than being in $10 increments – does this make it even harder? Well, fortunately it does not.
Using a statistical approach known as the Monte Carlo method, these highly complex functions can be evaluated quite accurately. The Monte Carlo method may be thought of as a technique whereby one randomly throws points at a target and evaluates whether the points fall on the target or outside it. If the target can be mathematically described, the functions can be evaluated with considerable accuracy after no more than a few thousand random "throws". This enables highly-complex derivative functions to be evaluated quite accurately. The problem then lies in the assumptions underlying the mathematical model used to define the "target". Flaws in the underlying assumptions have resulted in many a lost fortune!
Despite the accuracy of the Monte Carlo method, with highly complex functions the number of random throws can become very large and the question arises, can we do better than generating the throws randomly? For some problems, using a pattern, for example, a lattice has been shown to converge much more quickly.
Multidimensional mathematics is one of our most powerful tools in solving problems from financial derivatives, to metadata analysis, to cosmology. Professor Sloan provided a particularly clear insight into a highly complex and very powerful mathematical technique.