# Research

My research can be divided into the following:

- Image compression. Image and video standards. After summer 2006, this has not been part of my research.
- Applications of random matrices and free probability theory in digital communication, finance and information retrieval. Current research lies here.

# What is free probability theory?

Free probability can be thought of as an analogue to classical probability,
in that random variables and their distributions, independence and expectations are treated as similar concepts.
The differences lie in the following:

- The probablility spaces differ.
In classical probability, random variables commute, since they can be interpreted as functions under multiplication.
In free probability, random variables may not commute. The theory can therefore be used on objects such as matrices.
The study of random matrices can in fact be thought of as a driving force for introducing free probability.
In recent years, new results in random matrix theory have been obtained by using results in free probaility theory.
- The definition of an expectation differs since the probability spaces differ.
The expectation on matrices in free probability is most often the (normalized) trace.
For random matrices is it most often the (classical) expectation of the (normalized) trace.
- The definition of independence in terms of expectations differ.
In classical probability, the rule is that the expectation of a product of independent random variables equals the product of the expectations.
In free probability, this rule is replaced with a more complex rule which also takes into account that the random variables do not commute.
This more complex rule is referred to as the
*freeness relation*,
with two random variables called *free* if they satisfy the free relation.
- Distributions of random variables are interpreted differently.
The distribution of a random matrix has an (asymptotic) interpretation as a distribution of eigenvalues.
- Convolution is interpreted differently.
For many random matrices, their (additive/multiplicative) free convolution has an interpretation as
the (asymptotic) distribution of the eigenvalues of the sum/product of the two random matrices.
Free convolution is therefore useful in predicting the spectrum of large random matrices.