Noise sensitivity and inhomogeneous branching processes

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The project consists of two related strands. The first concerns noise sensitivity, which is a new and exciting branch of probability theory with strong links to computer science. One of the primary examples used in the development of the theory of noise sensitivity is the Majority function, which is the indicator that a particular random walk is positive after a large time. This function turns out not to be noise sensitive, and the first goal of this project is to expand on that result by considering other random walks and other events related to random walks. More ambitiously, further down the line it may be fruitful to consider a dynamic random walk and ask about its recurrence/transience properties.The second strand involves inhomogeneous branching processes. These structures appear in many applications, and recently tools have emerged that allow us to understand their behaviour. The aim of this part of the project is to extend these tools to boundary cases that have so far not been investigated, such as branching Brownian motion in a quadratic breeding potential.Once I have an appreciation of the tools and techniques from each of these two areas, I will be able to consider the noise sensitivity of branching structures. It is well known that, on a heuristic level, the early behaviour of branching processes determines the dominant forces that shape their later evolution. Noise sensitivity will be a way of putting this understanding on a rigorous footing.