Applications to join the department's thriving PhD programme are welcome from students with interests in analysis, probability theory or mathematical physics. The area in which I work, non-commutative probability, is an exciting combination of these three subjects. Knowledge of all of them is not necessary, but an interest to discover more is.
At present I'm particularly interested in the following topics.
- Quantum random walks. A classical random walk consists of repeatedly flipping a coin and moving left or right accordingly. This simple model illustrates many important ideas in probability theory. Its quantum generalisation corresponds to a system interacting with a sequence of identical particles; limit theorems have been obtained, but many interesting questions remain unanswered.
- Non-commutative stopping times. Stopping times are random times which, at any given moment, are known to have occurred or not. The time of the first rainfall this week is a stopping time; the time of the last rainfall is not. The theory of stopping times is vital for developing classical theories of stochastic integration. The proper non-commutative generalisation is known, but is yet to be exploited fully.
- Exotic forms of independence. The concept which separates probability from analysis is stochastic independence. Once one moves to the non-commutative world, more than one form of independence exists. Free independence was introduced by Voiculescu in the 1980s, and has important connections to random matrix theory, quantum information theory and representation theory. Connections for other forms of independence remain to be explored.
My research takes place within two fields: quantum probability, a branch of mathematics involving functional analysis, probability theory and quantum mechanics, and matrix positivity, an area of linear algebra with diverse applications.
