Supervisor: Dr Omer Bobrowski
Project description:
Initially motivated by the study of porous materials, percolation theory is one of the most active areas of research in theoretical and applied probability for more than six decades. Broadly speaking, it studies the emergence of large-scale structures such as infinite connected components, commonly in random graphs. The main goal of this project is to extend fundamental ideas in percolation theory from connected components to higher dimensional structures.
The approach we will take in this research is via the field of stochastic topology. This field has emerged over the past two decades, focusing on various topological (qualitative) properties of random structures. Many of the results in this field can be viewed as new higher-dimensional generalisations for classical results in probability theory and statistical physics. This field also provides the statistical backbone for Topological Data Analysis – the use of topological features in various data analytic tasks.
The notion of an infinite connected component is quite straightforward to conceptualise and define properly. The same, however, is not true for higher dimensional structures. Thus, the first question we wish to address in this project is: what is the higher dimensional analogue of an infinite component?Our approach will be to use the topological-algebraic language of homology (studying “holes”, “air-pockets”, etc.) to address this question. Once we develop a suitable generalisation, the first goal is to prove sharp phase transitions for the emergence of these new infinite structures. We plan to consider both discrete and continuous random models, generalising the commonly studied models in percolation theory. In addition to establishing a new line of exciting mathematical theory, we anticipate that the outcomes of this project will allow us to view classical percolation as a special case and provide novel insights about the existing theory.
Further information:
How to apply
Entry requirements
Fees and funding