We propose a new phase of the successful Mathematics for Real-World Systems (MathSys) Centre for Doctoral Training that will address the call priority area "Mathematical and Computational Modelling". Advanced quantitative skills and applied mathematical modelling are critical to address the contemporary challenges arising from biomedicine and health sectors, modern industry and the digital economy. The UK Commission for Employment and Skills as well as Tech City UK have identified that a skills shortage in this domain is one of the key challenges facing the UK technology sector: there is a severe lack of trained researchers with the technical skills and, importantly, the ability to translate these skills into effective solutions in collaboration with end-users. Our proposal addresses this need with a cross-disciplinary, cohort-based training programme that will equip the next generation of researchers with cutting-edge methodological toolkits and the experience of external end-user engagement to address a broad variety of real-world problems in fields ranging from mathematical biology to the high-tech sector. Our MSc training (and continued PhD development) will deliver a core of mathematical techniques relevant to all applied modelling, but will also focus on two cross-cutting methodological themes which we consider key to complex multi-scale systems prediction: modelling across spatial and temporal scales; and hybrid modelling integrating complex data and mechanistic models. These themes pervade many areas of active research and will shape mathematical and computational modelling for the coming decades. A core element of the CDT will be productive and impactful engagement with end-users throughout the teaching and research phases. This has been a distinguishing feature of the MathSys CDT and is further expanded in our new proposal. MSc Research Study Groups provide an ideal opportunity for MSc students to experience working in a collaborative environment and for our end-users to become actively involved. All PhD projects are expected to be co-supervised by an external partner, bringing knowledge, data and experience to the modelling of real-world problems; students will normally be expected to spend 2-4 weeks (or longer) with these end-users to better understand the case-specific challenges and motivate their research. The potential renewal of the MathSys CDT has provided us with the opportunity to expand our portfolio of external partners focusing on research challenges in four application areas: Quantitative biomedical research, (A2) Mathematical epidemiology, (A3) Socio-technical systems and (A4) Advanced modelling and optimization of industrial processes. We will retain the one-year MSc followed by three-year PhD format that has been successfully refined through staff experience and student feedback over more than a decade of previous Warwick doctoral training centres. However, both the training and research components of the programme will be thoroughly updated to reflect the evolving technical landscape of applied research and the changing priorities of end-users. At the same time, we have retained the flexibility that allows co-creation of activities with our end-users and allows us to respond to changes in the national and international research environments on an ongoing yearly basis. Students will share a dedicated space, with a lecture theatre and common area based in one of the UK's leading mathematical departments. The space is physically connected to the new Mathematical Sciences building, at the interface of Mathematics, Statistics and Computer Science, and provides a unique location for our interdisciplinary activities.

A landscape model consists of a parameterised family of potential functions together with a Riemannian metric. The dynamical system associated with this is given by the corresponding gradient vectorfield. Any Morse-Smale dynamical system with only rest point attractors and any system that admits a filtration admits such a representation except in a small neighbourhood of attractors and repellers. Such landscape models are of great interest in Developmental Biology because they correspond to Waddington's famous epigenetic landscapes but can also be rigorously associated with network models of the relevant genetic systems. When used to model the dynamics of a cell the parameters of the landscape correspond to signals being received by the cell. These can be due to morphogens in the cell's environment or signals coming from other cells. When these signal are altered, the landscape changes and this can cause bifurcations which destroy the attractor governing a cell's state and this can lead to a change in the cell's state. This is cellular differentiation, the way by which cell can change their cell type and specification. For example, stem cells differentiate in this way eventually to provide cells for all the tissue types in the body. The formation of the vertebrate trunk provides an important example of how cell fate decisions in developing tissues are made by signal controlled gene regulatory networks. Our biological collaborators have been studying part of this, namely the time course of differentiation of mouse embryonic stem cells to anterior neural or neural-mesodermal progenitors using such multidimensional single cell data. These experiments and the associated mathematical analysis has suggested that underlying this system is a highly non-trivial landscape of a complexity significantly greater than any published. This will be a key exploratory system that we will use to develop our ideas and we will work closely with the Briscoe and Warmflash labs to do this. However, it is important to stress that the purpose of this proposal is to focus strongly on developing mathematical ideas and tools and not just to be embedded in a particular biological project. On the other hand, access to state-of-the art data is very important. It ensures biological relevance and work with real data, rather than simulated data, raises real mathematical challenges. More and more powerful biological tools are becoming available to study such processes but the increasing amount and complexity of the data produced and the fact that the processes are carried out by complex systems means that new mathematical tools are need to help understand what is going on. In particular, biologists can now measure the numbers of multiple molecules in each of tens of thousands of cells in a single experiment. The key aim of this project is to increase our understanding of landscape models and combine this with state-of-the-art statistical techniques to provide new tools to analyse such data and to use it to probe the mechanisms of cellular differentiation and cellular decision-making in some important biological systems. The project involves deep collaboration with biological labs both in terms of data and biological ideas. It will be an excellent example of data science since it involves informatics (bioinformatics), statistics, mathematics (analysis, geometry & probability), hp computing and science (biology). It provides a new method of date dimension reduction a key theme in data science.

Complex system often exhibit a dynamics that can be regarded as superpositionof several dynamics on different time scales.A simple example is a Brownian partice that moves in an inhomogeneousenvironment which exhibits temperature fluctuations in space and time on a relatively large scale. There is a superposition of two relevant stochastic processes,a fast one given by the velocity of the particle and a much slower onedescribing changes in the environment. It has become common to call thesetypes of systems 'superstatistical' since they consist of a superposition of twostatistics, a fast one as described by ordinary statistical mechanicsand a much slower one describing changes of the environment. The superstatistics is very general and has been recently applied to a variety of complex systems, including hydrodynamicturbulence, pattern forming nonequilibrium systems, solar flares, cosmic rays,wind velocity fluctuations, hydro-climatic fluctuations, share price evolution,random networks and random matrix theory.The aim of the research proposal is twofold.On the theoretical side, the aim is to develop a generalisedstatistical mechanics formalism that describes a large variety of complexsystems of the above type in an effective way. Rather thantaking into account every detail of the complex system, one seeksfor an effective description with few relevant variables. For thisthe methods of thermodynamics are generalised:One starts with more general entropy functionsthat take into account changes of the environment(or, in general, large-scale fluctuations of a relevant system parameter) as well. An extended theory also takes into account how fast the local system relaxes to equilibrium,thus describing finite time scale separation effects.On the applied side, the aim is to apply the above theory to a large variety of time series generated by different complexsystems (pattern forming granular gases, brain activityduring epileptic seizures, earthquake activity in Japan and California, evolutionof share price indices, velocity differences in turbulent flows).It will be investigated which superstatistical phenomena are universal(i.e. independent of details of the complex system studied) and whichare specific to a particular system. Possible universality classeswill be extracted directly from the data. Application-specific modelswill be developed to explain the observed probability distributionsof the slowly varying system parameters.

Research context: Populations of the same species in locations hundreds of kilometers apart often fluctuate in unison or partly in unison, a phenomenon called synchrony. For instance, British aphid species, of economic importance because they are a major agricultural pest, outbreak 80% in synchrony over short distances and 50% in synchrony over distances of 200km, a huge distance for most aphid species. In fact, synchrony is widespread, and has been detected in birds, lemmings, fish such as cod, human pathogens such as measles, amphibians, and numerous other species. Many species exhibiting synchrony are of major conservation, economic, or health importance. Population synchrony has practical importance for several reasons. For instance, synchronized pest or disease populations require a coordinated response. An endangered species whose populations are synchronized is in accentuated danger of final extinction because populations are simultaneously low and might all go extinct by chance at once. An exploited synchronized species is periodically unavailable or less available across a wide area in many markets. Synchrony has been measured with methods that characterize the degree of synchrony between two populations only by a single number from 1 (perfect synchrony) down to -1 (perfect asynchrony). This approach is useful but limited: our results show synchrony is too complex to be captured with one number. Synchrony between two populations can occur mainly on short time scales, with little to no synchrony on long time scales; or on long time scales, with little or no synchrony on short time scales; or on any range of time scales. Synchrony between environmental variables in different locations has the same complexity. For instance, temperatures in London and Glasgow rise and fall largely together on annual time scales (seasonal variation) and multi-annual time scales (the North Atlantic Oscillation), but short-time-scale (day-to-day) temperature variation in London may resemble that in Glasgow much less. Different time scales of synchrony have different ecological and extinction-risk implications, and may have different implications for optimal control strategies for pests. In addition, new and important preliminary results show that the time-scale-specific structure of environmental synchrony is changing as part of climate change, and likely affects population synchrony, and thereby extinction risk. Research aims: We will use large spatio-temporal databases, new theory, and new lab experiments to obtain a broad time-scale-specific description of environmental and population synchrony, and to assess the implications of observed patterns for climate change, extinction risks, and inference of what mechanisms cause synchrony in the field. Applications: We will provide information about a newly observed and previously unrecognized aspect of climate change and a global assessment of its overarching importance for conservation and pest management applications and for ecological understanding.

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