Doctoral Training Partnerships: a range of postgraduate training is funded by the Research Councils. For information on current funding routes, see the common terminology at https://www.ukri.org/apply-for-funding/how-we-fund-studentships/. Training grants may be to one organisation or to a consortia of research organisations. This portal will show the lead organisation only.

Active Flow Control (AFC) mechanisms have tremendous potential in improving flow characteristics in a wide variety of sectors. For effective AFC design, it is essential to determine the sensitivity of each of the control parameters to the flow property of interest. Adjoint methods provide the sensitivity of the objective function to any number of input parameters at a reasonable additional cost. They are based on simple RANS turbulence modelling, which captures poorly the physics of flow especially in the presence of complex flow features such as flow separation, which can be effectively eliminated by flow control. Recently developed Reynolds Stress Models (RSM) display significant improvement in the flow prediction capacity as compared to conventional RANS models, yet adjoints with RSM are not yet available. An alternative for accurate flow prediction is Large Eddy Simulation (LES). Unfortunately resolving the chaotic turbulent motion results in exponential growth of the gradients. Our first objective is to develop an effective discrete adjoint method with an accurate and stable RSM, and compute sensitivities in complex flows involving active control mechanism applied to realistic wing geometries. Although RSM outperforms conventional turbulence models in a host of applications, there is a need for further physics-based calibration for specific flows. Our second objective is to use the adjoint method to drive the model coefficients to their optimum value such that the model results match with high-fidelity simulation data yielding a better turbulence model, which can be applied for effective flow control design in bluff body with severe rear flow separation. Our third objective will be to develop adjoint approaches for chaotic LES flows using the hosting groups' innovative gappy checkpointing approach that retains the accuracy of the LES but regularises the chaotic motion for the reverse adjoint pass, hence avoiding the exponential blowup of the sensitivities.

This work will have a direct impact on our understanding of chronic inflammatory disease and may foster new therapeutic approaches to the treatment of such diseases, therefore this proposal will advance our understanding of biomedical sciences and promote the translation and application of biomedical research to the clinic. In addition, as a New Investigator Award, this proposal will support a vibrant research environment by enabling young promising researchers, such as myself, to establish independent research careers here in the UK. The results of this research will be communicated in peer reviewed publications. In addition, results will be presented at scientific meetings to communicate progress with the scientific community. Close interaction with the communications department of Imperial College and the MRC will ensure that new findings and porgress are dissseminated to the public in an appropriate and timely fashion.

This research proposal utilises Dynamic Difficulty Adjustment (DDA) and Procedural Content Generation (PCG) [11] [12], and equips it to virtual reality (VR) games in order to help players enhance their skills. VR technology has improved commercially, using eye, facial and hand tracking technology [1] [10]; this can be utilised to capture players' psychological signals [2]. The project addresses difficulty spikes [3] in rhythm-based VR games like Beat Saber, improving skills and motivation. By combining DDA, PCG, and VR tracking, personalised learning experiences are created. The research explores effectiveness, player engagement, skill progression, and hypotheses on skill development and motivation. Core steps include detecting psychological signals, implementing DDA for dynamic adjustment, developing PCG algorithms for personalised maps, and creating an AI Coach system. The project aims to expand findings to other rhythm games and contexts. Beneficiaries include players enjoying tailored experiences, game developers enhancing player retention, and the health and fitness industry benefiting from accessible VR workouts.

The idea of a group is ubiquitous in mathematics, and has a variety of applications in the sciences. As a starting point, we take a physical object, and consider all its symmetries: transformations (such as rotations and reflections) that leave that object fixed. The symmetry group of the object is the list of all these symmetries together with the operation of combining those symmetries by first applying one then the other. This combining operation satisfies certain simple rules - for example, for every symmetry there is an inverse symmetry which gets you back where you started. This leads to the idea of an abstract group, which is a set of symbols and a way of combining these symbols which satisfies the same rules. Group theory asks what different abstract groups exist and what general properties they have. The study of groups is a major branch of mathematics, and its most impressive achievement is the classification of finite groups which are simple, which means they can't be broken down into smaller groups. Representation theory reverses the above process: given an abstract group, it asks how we can realise that group as the symmetry group of a physical object. This question is asked in an algebraic way through matrix representations: each symmetry of n-dimensional space is encoded as an n by n matrix of numbers, and combining symmetries corresponds to multiplying matrices together. As with classifying groups, we want to classify the simple (or "irreducible") representations of a given group, i.e. those which can't be broken down into smaller representations. This project looks at what happens when we change the number system by taking a prime number p and replacing each matrix entry with its remainder modulo p. Now a representation which was irreducible can become reducible, and one of the most important tasks is to work out what its irreducible constituents are. This project looks at a special case of this question, by asking for a classification of which irreducible representations remain irreducible when reduced modulo p. This is interesting because it means that part of the task of classifying and constructing the irreducible representations modulo p (which in general is very hard) is done automatically. In this project we will look at this question for the simple groups (and some closely related groups called quasi-simple and almost-simple groups). For one important family of almost simple groups the problem is solved: these are the symmetric groups S(n), where n is a positive integer and S(n) is the group of all permutations of the numbers 1,...,n. From this starting point, we intend to solve our main problem for the double cover of S(n): this group is twice as large as S(n), consisting of permutations accompanied by a + or - sign, and has important applications in physics. Having tackled these groups, we intend to look at a very large family, namely the groups of Lie type. These are groups which are already by definition groups of matrices, over various number systems and satisfying various algebraic conditions. The family of finite groups of Lie type is very complex and varied, so we intend to start with the easiest members of the family to begin with. These are the general linear groups, which are the groups of all n by n matrices over a given number system, with no additional condition other than having inverses. The representation theory of these groups is quite well understood, and has a remarkably close relationship to the representation theory of the symmetric groups. This means that it is an excellent initial case of our main question to solve, and acts as a test case for studying the groups of Lie type in general. The outcomes of the project will be journal articles and conference presentations, together with a repository of information on the internet making data and the outcomes of preliminary investigations more widely available.