Supervisor: Dr Natasha Blitvic
Project Details :
In combinatorics, we like to count things. In probability, we like to draw them at random.
There are many interfaces between combinatorics and probability theory, including but not limited to sampling from discrete sample spaces with some form of combinatorial structure. In fact, in many branches of probability, combinatorics can be seen to play a structural role. For instance, in noncommutative probability theories (such as free probability or random matrix theory), combinatorial objects often give expressions for expectations of products of random variables. In integrable probability, combinatorial objects are often seen to encode probabilities of stochastic processes, such as interacting particle systems. Yet other times, combinatorial objects appear to possess properties (namely, a form of positivity) that indicate that they could play such a probabilistic role, without a known probabilistic model for these. This project will start by considering several open questions at the aforementioned points of this vast interface, for example understanding via the moment method certain types operators on Hilbert spaces or, conversely, developing an operator model for an important class of combinatorial sequences, before giving you a chance to decide which link between probability and combinatorics interests (or puzzles) you the most.
Further information: How to apply Entry requirements Fees and funding