Project description: Rémy (1985) introduced a random tree-growth process for uniform binary leaf-labelled trees with n leaves at step n. Aldous (2000) and Schweinsberg (2002) studied a treevalued up-down Markov chain with uniform stationary distribution, by combining, for each step, the uniform removal of a leaf (down-step) by Rémy's growth step (up-step). Ford (2006), Dong, Goldschmidt and Martin (2006), Marchal (2008), Chen et al. (2009) and Pitman and Winkel (2009) generalised Rémy's tree-growth process to larger classes of tree-growth processes with some recursive structure. These references include studies of the asymptotics in n, establishing limiting continuum random trees in the sense of Aldous (1991). Key ingredients are more elementary combinatorial structures such as partition- and composition-valued Chinese Restaurant Processes (see e.g. Pitman (2006) and Pitman and Winkel (2009)). Forman et al. (2016) and Rogers and Winkel (work in progress) are studying limiting evolutions of composition-valued down-up Markov chains. This is part of a wider collaborative project to understand the limiting evolution of Aldous's tree-valued down-up chain, ultimately aiming to establish the existence of a diffusive scaling limit as conjectured by David Aldous. The purpose of this project is to explore generalisations of Aldous's tree-valued down-up Markov chain to the context of the one- and two-parameter classes of tree-growth processes referred to above. Avenues to explore include basic structural properties of the Markov chain including stationary distributions and asymptotic properties such as mixing times and relaxation times as studied by Aldous and Schweinsberg in their special case. Further directions include asymptotic studies alongside the wider collaborative project mentioned above or wider/related classes of Markov chains models on combinatorial structures such as partitions, compositions, trees and more general connected graphs such as sparse connected graphs with surplus edges etc. This project is situated at the Probability side of the general research theme of "Statistics and Applied Probability", with some interaction with the Combinatorics side of "Logic and Combinatorics" and with "Mathematical Analysis". References: - Aldous (1991) The Continuum Random Tree I. Ann. Probab., 19(1):1-28 - Aldous (2000) Mixing time for a Markov chain on Cladograms. Combin. Probab. Comput., 9(3):191-204 - Chen, Ford and Winkel (2009) A new family of Markov branching trees: the alpha-gamma model. Elec. J. Prob. 14, 400-430 - Dong, Goldschmidt and Martin (2006) Coagulation-fragmentation duality, Poisson- Dirichlet distributions and random recursive trees. Ann. Appl. Probab. 16(4): 1733-1750 - Ford (2006) Probabilities on cladograms: introduction to the alpha model. Ph.D. thesis, Stanford University, 241p. - Forman, Pal, Rizzolo and Winkel (2016) Diffusions on a space of interval partitions with Poisson-Dirichlet stationary distributions. ArXiv:1609.06706 - Forman, Pal, Rizzolo and Winkel (work in progress) Aldous diffusion. - Marchal (2008) A note on the fragmentation of a stable tree. In Fifth Colloquium on Math. and Comp. Sci., Discr. Math. Theor. Comput. Sci. Proc., AI, pages 489-499 - Pitman (2006) Combinatorial Stochastic Processes, volume 1875 of LNM. Springer Berlin - Pitman and Winkel (2009) Regenerative tree growth: binary self-similar continuum random trees and Poisson-Dirichlet compositions. Ann. Probab. 37(5), 1999-2041 - Rémy (1985) Un procédé itératif de dénombrement d'arbres binaires et son application à leur génération aléatoire. RAIRO Inform. Théor., 19(2):179-195 - Rogers and Winkel (work in progress) Chinese Restaurants from a Lévy process - Schweinsberg (2002) An O(n2) bound for the relaxation time of a Markov chain on cladograms. Random Structures Algorithms, 20(1):59-70

My anticipated (general) area of research focus: Literature around the economic impact of British interference in India at a national level is abundant, however inter-regional variations, especially covering the western Tribal Areas and Balochistan have been relatively sparsely covered. On the one hand, the East India Company established the port cities of Madras, Bombay and Calcutta with the commercial opportunities here attracting the migration of many Indians (especially merchants and bankers). Eventually, institutions such as colleges and universities were established there making English language the lingua franca of the region, producing highly skilled doctors, civil servants and lawyers. The British Raj then expanded the access of these Indian merchants to overseas markets and the volume of trade through these three ports increased fivefold between 1863-1913. Other regions such as the United Provinces, Gujrat and Punjab experienced vast agricultural development with cultivated acreage across these regions increasing by a factor of 2.11 between 1600 and 1910, according to research by Professor Stephen Broadberry. In contrast, the Tribal Areas of western British India mainly experienced British military aggression and by 1947, there were no schools, hospitals, railways or other infrastructure within these Tribal Areas, except for those used exclusively by the British. In fact, the Waziristan region is reported to have had more troops stationed there in 1936 than anywhere else in the entire subcontinent. Other regions such as Bihar witnessed significant economic deindustrialisation in the 19th century. Balochistan and the Tribal Areas were given significant autonomy, although without any plans for economic development or institution building. This view is supported by Professors Acemoglu, Johnson and Robinson who argue that colonisers less likely to settle in a specific region (possibly because of facing higher mortality like the British faced in the Tribal Areas and Balochistan), were more likely to establish extractive states there. These extractive states then determine the quality of institutions and the subsequent economic performance of the region. I would like to use regional British and Indian military mortality figures as a proxy for institutions, investigating heterogeneity across British Indian regions and then comparing it to the economic performances of these regions today to evaluate Acemoglu et al.'s hypothesis. Moreover, the Tribal Areas and Balochistan mainly consist of the Pashtun tribes of Mohmand, Shinwari, Achekzai and Kakar amongst many others, who were involuntarily separated from their fellow tribespeople between Afghanistan and British India in the late 19th century. I would like to compare the modern economic performances of these tribes settled in Balochistan and the Tribal Areas who have experienced the legacy of British colonial rule with their fellow identical tribespeople in Afghanistan who lived under the Durrani and Barakzai empires. This comparison can then hopefully be used as a robustness check of the overall long-term economic impact of the British colonial rule. By focusing on the economic impact of inter-regional variations of colonial strategies and covering the Tribal Areas and Balochistan amongst other British Indian regions, I will hopefully be filling up a gap within the academic discourse. Being natively fluent in Pashto and understanding Dari and Urdu, I would like to use primary and secondary sources in these languages to contribute to my research and to the wider academic discourse, from a non-European perspective.

Feedback control is a fundamental engineering principle that has been used successfully for decades to improve the robustness and performance of engineered and natural systems, and to drive their output to desirable set-points. Examples abound, ranging from Watt's fly-ball governor to medical life-support systems to fly-by-wire planes. With advances in computation and instrumentation, as well as new theoretical frameworks, the field is now moving towards distributed, data-driven, real-time optimal feedback control solutions. The ultimate aim is to address the challenge of controlling systems with significant uncertainty - both in their modeling, as well as in the environment they need to operate. Indeed, systems of this type are becoming more and more prevalent, and autonomous systems solutions are being sought across technology and biomedicine. The development of new theory, tools, and instrumentation for these challenges creates new opportunities, and new horizons, with significant impact in areas where feedback control has not had widespread application yet. One such area is biological evolution. Defined as the change in inherited characteristics over successive generations of a species, it is a slow optimisation process that over time (sometimes millions of years) naturally improves traits and species fitness. Directed evolution - a cyclic process of gene diversification, screening, and selection - has had significant impact on biotechnology in the past few years, but this is largely a passive, uncontrolled process. The benefits of dynamically steering evolution so as to improve single-cell designs in a predictable fashion autonomously would be hugely significant - with applications across biotechnology and beyond. However this has not been attempted in closed loop before, not least because the experimental paradigm and feedback control frameworks required to realise this vision are currently missing. In this new horizons project we will combine advanced instrumentation with feedback control theory to develop a first-of-its-kind platform for closed-loop directed evolution. In particular, we will create a robotic microscope platform to monitor the behaviour of ~1 million bacterial cells as they grow continuously, confined in a microfluidic chip for long periods of time. Continuous imaging of the population will be used to quantify each cell's performance, with this information driving a feedback controller that will in turn decide which cells to propagate and which not, in order to achieve a target distribution of behaviours over time. Feedback control at this scale has not been attempted before, and this application area will push the development of new theory and algorithms. Indeed, our controlled evolution platform will advance the boundaries of what is possible right now both from the standpoint of instrumentation and control, as well as in terms of understanding and manipulating evolution. The development of this robotic control approach is motivated by the need for real-time, accurate actuation for directed evolution and is facilitated by nascent microscopy and image processing technologies. Moreover, the development of the theory is challenged by the size of the system to be controlled (millions of states) as well as uncertainties involved in modeling and observing the stochastic evolutionary process. On the whole, this high-risk high-reward project will unlock a number of biotechnological applications, as well as opening new research opportunities in biological evolution and feedback control.

This student is enrolled on the AIMS CDT funded through EPSRC CDT grant. The research project and department will be chosen at the end of the first year of study and this record will be updated shortly after.