## 2023 Summer Research in Maths (SRIM) projects

Below is a list of the SRIM project proposals advertised for summer 2023.

## Specific Projects

**Lloyd Fung (DAMTP / Goldstein Lab) - Spherical Harmonics-based solver for the Smoluchowski equation that governs dilute suspensions of active Brownian particles**

**Keywords:***Mathematical Biology, Emergent Behavior, Smoluwchowski equation, Bio-active Fluids, Numerical Analysis***Alison Ming (Atmos-Ocean dynamics) - Dynamical processes controlling tropical lower stratospheric temperature variability**

**Keywords:***Atmosphere, Stratosphere dynamics, Numerical computing***Laura Cimoli (DAMTP) - Mapping of oceanic heat and carbon uptake using machine learning**

**Keywords:***Physical oceanography, tracer sequestration, machine learning***Laura Cimoli (DAMTP) - Inferring ocean turbulent mixing from tracer observations**

**Keywords:***Physical oceanography, ocean mixing, tracer observations***Stephen Eglen (DAMTP / Computational Biology) - Developing a toolkit for spike train analysis in the Julia programming language**

**Keywords:***electrophysiology, spike train analysis, visualisation, functional connectivity.***Anna Breger (DAMTP, COVID-19 AIX-COVNET) - Impact of clinical data preprocessing**

**Keywords:***Covid, Clinical Data, Data preprocessing, Machine learning, Classification*

## General Project Outlines / Possibilities

**Adrien Lefauve (DAMTP, Environmental and Industrial Fluid Dynamics) - Possible experimental or theoretical projects looking at decompression sickness****Adrien Lefauve (DAMTP, Environmental and Industrial Fluid Dynamics) - Possible project working on datasets from the 'Stratified Inclined Duct' experiment****Nuno Miguel Oliveira (DAMTP Goldestein/Oliveira) - The evolution of microbial cooperation in space****Nuno Miguel Oliveira (DAMTP Goldestein/Oliveira) - The evolution of bacterial chemotaxis towards antibiotics****Nuno Miguel Oliveira (DAMTP Goldestein/Oliveira) - The flash dynamics of bioluminescent biofilms under flow****Timothy Gowers (DPMMS) - A project on automatic theorem proving****Sumit Kumar Birwa (DAMTP/Goldstein lab) - Bioluminescence in waves****Maria Tatulea-Codrean (DAMTP, Biological Fluid Mechanics) - Synchronization of Biological Oscillators****Rajesh Kumar Bhagat (DAMTP) - New understanding of fluid's energy equation: a theory for surface viscosity****Rajesh Kumar Bhagat (DAMTP) - New understanding of fluid's energy equation: a theory for non-Newtonian interfacial flows****Can Cui (DAMTP, Astrophysical Fluid Dynamics) - Global MHD Simulations of Gas in Debris Disks****Richard Samworth (DPMMS, Stats Lab) - What is the future of AI?****Richard Samworth (DPMMS, Stats Lab) - Missing data****Arieh Iserles (DAMTP) - Convergence of orthogonal series****Rajen Shah (DPMMS, Statistical Laboratory) - Nonparametric estimation of the incubation period distribution of Covid-19****Rajen Shah (DPMMS, Statistical Laboratory) - Inference in high-dimensional regression models****Camille Scalliet (Soft Matter - DAMTP) - Machine learning chemical reactions****Matthew Colbrook (DAMTP) - Computing Spectral Properties of Operators on Surfaces****James Davies (DPMMS, Combinatorics) - Euclidean Ramsey Theory****Amelia Drew (DAMTP, General Relativity group) - Cosmic Strings with Adaptive Mesh Refinement****Sergii Strelchuk (DAMTP, CQIF) - Classical simulation of Topological Quantum Computation**

## Specific Projects

## Spherical Harmonics-based solver for the Smoluchowski equation that governs dilute suspensions of active Brownian particles

Project Title | Spherical Harmonics-based solver for the Smoluchowski equation that governs dilute suspensions of active Brownian particles |

Keywords | Mathematical Biology, Emergent Behavior, Smoluwchowski equation, Bio-active Fluids, Numerical Analysis |

Project listed | 5 January 2023 |

Project status | Closed |

Contact Name | Lloyd Fung |

Contact Email | lsf27@cam.ac.uk |

Department/Lab/Group | DAMTP / Goldstein Lab |

Period of the Project | 8 weeks |

Project Open to | Part IB students, Part II students |

Background Information | Motile or actively sinking microorganism underpins many vital environmental processes, such as the carbon cycle and marine ecology. An accurate model of their transport can also help engineers improve industrial processes such as wastewater treatment and biofuel production. "Active Brownian particles" is a popular model for this type of suspended particles, but there is a lack of numerical solvers for the Smoluchowski equation that governs their transport. |

Brief Description of the Project | The summer student is expected to develop a new numerical scheme using spherical harmonics and implement the scheme using MATLAB. The development of the scheme is effectively an extension of Methods IB, where the student is expected to apply Sturm-Liouville theory in spherical coordinates. The student will also learn about numerical stability analysis as they develop the time-stepping method. Implementing the numerical scheme will involve a lot of coding in MATLAB, so good programming skills are highly desirable. |

Work Environment | The student will be working with Dr Lloyd Fung mostly, but they are also encouraged to join Eric Lauga's group meeting. There will be a meeting about once a week, but remote work is also acceptable. |

References | "A Physical Introduction to Suspension Dynamics" by Guazzelli & Morris |

Prerequisite Skills | Fluids, PDEs |

Other Skills Used in the Project | Numerical Analysis, Simulation |

Programming Languages | MATLAB |

## Dynamical processes controlling tropical lower stratospheric temperature variability

Project Title | Dynamical processes controlling tropical lower stratospheric temperature variability |

Keywords | Atmosphere, Stratosphere dynamics, Numerical computing |

Project listed | 5 January 2023 |

Project status | Closed |

Contact Name | Alison Ming |

Contact Email | adk33@cam.ac.uk |

Department/Lab/Group | Atmos-Ocean dynamics |

Period of the Project | 8 weeks |

Project Open to | Part IB students, Part II students |

Background Information | The tropical tropopause layer is the main inflow region for air entering the stratosphere and exerts a strong control on the whole climate system. The dryness of the stratosphere is modulated by temperature variations in this region which cause the freeze-drying out of water vapour out of the upwelling air. The amount of water vapour is the main source of hydrogen oxide molecules that destroys ozone. This in turn affects stratospheric temperatures and also contributes to the formation of polar stratospheric clouds in the polar vortex. Hence, the state of the stratosphere is determined by a complex interplay between dynamical processes (including transport), radiative heating and chemistry. These processes and their feedbacks are important to variability of the tropical lower stratosphere and have far reaching impacts on other parts of the climate system. Understanding the determinants of temperature and compositional variability on subseasonal timescales remains an important and open question. This proposal seeks to use data driven techniques to understand the main control on the temperature of the tropical tropopause layer. The aim is to build a linear inverse model which is able to predict variation on the temperature and decompose the climate variability into distinct types of dynamical climate modes. We will also investigate the sensitivity of the results to the choice of variables in the state vector. |

Brief Description of the Project | We will use existing large gridded climate datasets (ERA5 reanalysis) for the input variables. Machine learning will be used to train a linear inverse model on part of the dataset and the model will then be applied to the remainder of the dataset to assess the skill in predicting tropical lower stratospheric temperatures on a seasonal timescale. The student will start by reproducing the recent work of Albers and Newman (2021) which predicts North America winter variability. The student will then apply the same techniques to the problem of tropical lower stratospheric temperatures. This project is expected to identify the main dynamical modes that contribute to the temperature variability. They will gain a knowledge of atmospheric dynamics and better understand the complex interactions in the atmosphere as well as how data driven tools can help with disentangling these interactions. |

Work Environment | I will be the main supervisor but other people (different universities) will also be available to advise. I would expect them to be in the office during working hours and can work remotely for part of it if they wish. |

References | Fueglistaler, S., Dessler, A. E., Dunkerton, T. J., Folkins, I., Fu, Q., and Mote, P. W. (2009), Tropical tropopause layer, Rev. Geophys., 47, RG1004, https://doi:10.1029/2008RG000267 John R Albers and Matthew Newman 2021 Environ. Res. Lett. 16 044024 https://doi.org/10.1088/1748-9326/abe781 P. Haynes, P. Hitchcock, M. Hitchman, S. Yoden et al. (2021). The influence of the stratosphere on the tropical troposphere. J. Meteor. Soc. Japan 99, 803â€“845. https://doi:10.2151/jmsj.2021-040 |

Prerequisite Skills | Mathematical physics, PDEs, Data Visualization, Strong programming skills |

Other Skills Used in the Project | |

Programming Languages | Python, C++, No Preference |

## Mapping of oceanic heat and carbon uptake using machine learning

Project Title | Mapping of oceanic heat and carbon uptake using machine learning |

Keywords | Physical oceanography, tracer sequestration, machine learning |

Project listed | 5 January 2023 |

Project status | Closed |

Contact Name | Laura Cimoli |

Contact Email | lc929@cam.ac.uk |

Department/Lab/Group | DAMTP |

Period of the Project | 8 weeks |

Project Open to | Part IB students, Part II students |

Background Information | The ocean circulation has played a crucial role in climate change mitigation by removing anthropogenic carbon and heat from the atmosphere. However, the temporal and spacial variability of the rates at which they are sequestered in the ocean interior are still largely unknown. In the last few decades, anthropogenic inert tracers, such as chlorofluorocarbons (CFCs) and sulfur hexafluoride (SF6), have been used to reveal the ocean circulation patterns. CFCs and SF6 are also taken up from the atmosphere, and so "mimic" the sequestration of anthropogenic heat and carbon. Unfortunately, CFCs and SF6 observations are sparse in time and space, making it challenging to adequately inform us about the ocean circulation and its variability. This project will help better understanding the oceanic anthropogenic carbon and heat via the analysis and mapping of available observations. |

Brief Description of the Project | The goal of the project is to produce global, gridded maps of CFCs and SF6. The starting point is a new observation-based annual reconstruction of anthropogenic transient tracers (See DOI: 10.1002/essoar.10512537.1). This product returns new estimates of tracer sequestration rates and ocean steady-state circulation. However, it still has large spatial sparsity, which can be addressed by machine learning algorithms trained with other water properties observations. The global gridded result will be useful (i) to investigate hidden pathways of the ocean circulation, and (ii) to better understand the sequestration rates of other tracers, such as heat and carbon. The student will learn about the large-scale ocean circulation and water properties, inverse methods based on available observations, and machine learning algorithms (involving coding in python or MatLab). |

Work Environment | This is an individual project, where the student will be expected to be in the office regularly to promote interactions with the supervisor(s). Remote work can be discussed as needed. Potential collaborators: professors John Taylor and Colm Caulfield (DAMTP) and Ali Mashayek (Earth Sciences). |

References | DOI: 10.1002/essoar.10512537.1 |

Prerequisite Skills | Statistics, Fluids |

Other Skills Used in the Project | |

Programming Languages | Python, MATLAB |

## Inferring ocean turbulent mixing from tracer observations

Project Title | Inferring ocean turbulent mixing from tracer observations |

Keywords | Physical oceanography, ocean mixing, tracer observations |

Project listed | 5 January 2023 |

Project status | Closed |

Contact Name | Laura Cimoli |

Contact Email | lc929@cam.ac.uk |

Department/Lab/Group | DAMTP |

Period of the Project | 8 weeks |

Project Open to | Part IB students, Part II students |

Background Information | The ocean interior is a stably density stratified environment where internal waves propagate, break and contribute to the mixing of water masses with different properties. In the deep ocean, mixing contributes to regulating (i) the strength of the deep branch of the global overturning circulation and (ii) the timescale and spatial distribution of tracers taken up from the atmosphere (ex. anthropogenic heat and carbon). Measuring cross-density mixing is challenging due to its high spatial and temporal intermittency and the small scales on which it occurs. Therefore, we often resort to estimating it from other variables, for example from variations in the gradients of tracer concentrations or velocity shear. However, these estimates can have large differences, and sometimes even be 1-2 orders of magnitude apart. This project will focus on (I) inferring bulk measures of mixing (i.e. averaged over time and space) from observations of several tracers (such as cholofluorocarbons), (II) comparing the outcome with localized velocity-based estimates of mixing (the most common direct observational method), and (III) use statistics and possibly machine learning to converge the two approaches and offer a unifying framework for global ocean mixing. |

Brief Description of the Project | The student will use tracer and in-situ observations to infer mixing in different deep ocean regions. The first region of interest might be the Samoan Passage, a crucial pathway for the northward spreading on abyssal waters formed around Antarctica (see https://www.jstor.org/stable/26845652). As the first approach, the project will focus on inferring cross-density mixing from a new observation-based annual reconstruction of anthropogenic transient tracers (such as cholorofluorocarbons and sulfur hexafluoride, which are the perfect tool because they are inert in seawater and their atmospheric histories are well known; See: DOI: 10.1002/essoar.10512537.1). The second approach will involve estimating mixing from from velocity and turbulence dissipation observations. The student will learn about the large-scale ocean circulation, the role of mixing in the circulation, density stratified turbulence, statistics, and potentially some machine learning. |

Work Environment | This is an individual project, where the student will be expected to be in the office regularly to promote interactions with the supervisor(s). Remote work can be discussed as needed. Potential collaborator: Professor Colm Caulfield (DAMTP). |

References | Carter et al., 2019: "A spatial geography of abyssal turbulent mixing in the Samoan Passage". Oceanography. Cimoli et al., 2022. "Annually-resolved propagation of CFCs and SF6 in the global ocean over eight decades". Final review in JGR Oceans. |

Prerequisite Skills | Statistics, Fluids |

Other Skills Used in the Project | |

Programming Languages | Python, MATLAB |

## Developing a toolkit for spike train analysis in the Julia programming language

Project Title | Developing a toolkit for spike train analysis in the Julia programming language |

Keywords | electrophysiology, spike train analysis, visualisation, functional connectivity. |

Project listed | 19 January 2023 |

Project status | Closed |

Contact Name | Stephen Eglen |

Contact Email | sje30@cam.ac.uk |

Company/Lab/Department | DAMTP / Computational Biology |

Period of the Project | 8 weeks |

Project Open to | Part IB students; Part II students |

Background Information | Neurons mostly communicate with each other by generating action potentials, also known as 'spikes'. We now have the technology to record the electrical activity of many neurons simultaneously, which should allow us to begin to understand the function of neural circuits. However, the data collected can often be very large, and need computational methods to analyse them reliably. The aim of this project is to develop some of those computational methods. |

Brief Description of the Project |
The aim of this project is to build computational tools in the Julia programming language for spike train analysis. The starting point will be to implement some core methods for reading in electrophysiological data sets, and performing "burst analysis" whereby groups of spikes occur in quick succession to generate a burst. After this, I would like the student to implement pairwise methods for detecting correlated activity in pairs of spike trains. Time permitting, the student would be able to develop new methods for detecting 'functional connectivity' between all neurons in a recording of 60--4000 neurons recorded simultaneously. The student will also need to develop good data visualisation methods for this project, as they are critical for the efficient communication of large volumes of data. Finally, I would like reproducible test data sets to be created, so that others can evaluate our methods, and compare with their own. I have chosen the Julia programming language for this project. Most of our group's work to date has been in the R programming language. Part of the aim of the project would be to evaluate the utility of Julia. Based on earlier work from two summer students, I believe that Julia has a lot of promise https://sje30.github.io/catam-julia/ particularly in making code that is easy to write and fast. Efficiency (in both speed and memory usage) is an issue here as we have large volumes of data. In our current R implementations, we often have to rewrite key portions in the C programming language, which makes it harder to develop, debug and maintain. A feasible output from this program would be to make the preliminary version of the toolkit freely available on the internet, visible through the Julia package manager. This would hopefully then seed further work in this area in collaboration with partners. |

Work Environment | The student will work directly with me on the project. Remote supervision is likely. |

References |
Some key papers from our earlier work: Gelfman S, Wang Q, Lu Y-F, Hall D, Bostick CD, Dhindsa R, Halvorsen M, McSweeney KM, Cotterill E, Edinburgh T, Beaumont MA, Frankel WN, Petrovski S, Allen AS, Boland MJ, Goldstein DB, Eglen SJ (2018) meaRtools: An R package for the analysis of neuronal networks recorded on microelectrode arrays. PLoS Comput Biol 14:e1006506 Available at: http://dx.doi.org/10.1371/journal.pcbi.1006506. Cotterill E, Charlesworth P, Thomas CW, Paulsen O, Eglen SJ (2016) A comparison of computational methods for detecting bursts in neuronal spike trains and their application to human stem cell-derived neuronal networks. J Neurophysiol 116:306-321 Available at: http://dx.doi.org/10.1152/jn.00093.2016. |

Prerequisite Skills | Statistics |

Other Skills Used in the Project | Simulation; Data Visualization |

Programming Languages | Julia |

## Impact of clinical data preprocessing

Project Title | Impact of clinical data preprocessing |

Keywords | Covid, Clinical Data, Data preprocessing, Machine learning, Classification |

Project listed | 22 March 2023 |

Project status | Closed |

Contact Name | Anna Breger |

Contact Email | ab2864@cam.ac.uk |

Company/Lab/Department | DAMTP, COVID-19 AIX-COVNET |

Period of the Project | 4 weeks |

Project Open to | Part IB students; Part II students |

Background Information | The National COVID-19 Chest Imaging Database (NCCID) is a centralised UK database of thoracic imaging and corresponding clinical data. It is made available by the National Health Service Artificial Intelligence (NHS AI) Lab to support the development of machine learning tools focussed on Coronavirus Disease 2019 (COVID-19). A bespoke cleaning pipeline for NCCID, developed by the NHSx, was introduced in 2021. We published in 2023 an extension to the original cleaning pipeline for the clinical data of the database. It has been adjusted to correct additional systematic inconsistencies in the raw data such as patient sex, oxygen levels and date values. The pipeline is publicly available on GitHub and will allow global users to work with more consistent data for the development of machine learning tools without being an expert. The actual impact of the cleaning evaluated in downstream tasks will give important insights for future data handling. |

Brief Description of the Project | The student should evaluate with simple machine learning algorithms how the cleaning of the NCCID clinical data impacts the outcome, e.g. in classification tasks such as death vs alive (annotations provided in the data set). It will be important to start with simple models that do not ask for many parameters, such as linear regression, in order to see the direct impact of the cleaned versus not cleaned input data. An extension to more complicated methods is possible but not necessary for that project, the focus will be on the analysis parts. A successful outcome would be any understanding of how our data cleaning impacts the results of downstream tasks. This would be of great interest for our research group as no such experiments have been conducted so far with that data. |

Work Environment | The student will be embedded in the international COVID-19 AIX-COVNET collaboration and supervised by me remotely, clinical contact person will be Dr Ian Selby (Department of Radiology). Remote work is encouraged, office space possible upon request. |

References | https://nhsx.github.io/covid-chest-imaging-database/ https://gitlab.developers.cam.ac.uk/maths/cia/covid-19-projects/nccidxclean |

Prerequisite Skills | Statistics; Predictive Modelling ;Data Visualization; Basic programming and data handling skills in Matlab or Python as well as basic knowledge about machine learning |

Other Skills Used in the Project | Image processing; Database Queries |

Programming Languages | Python; MATLAB |

## General Project Outlines / Possibilities

## Possible experimental or theoretical projects looking at decompression sickness

Project Title | Possible experimental or theoretical projects looking at decompression sickness |

Project listed | 5 January 2023 |

Project status | Closed |

Contact Name | Adrien Lefauve |

Contact Email | aspl2@cam.ac.uk |

Department/Lab/Group | DAMTP, Environmental and Industrial Fluid Dynamics |

Brief Description of the Project | Decompression sickness (a.k.a. the bends) is a serious medical condition limiting the practice of scuba diving. It occurs when dissolved breathing gases in body tissues are released in gaseous form into tissue when ascending from depth. Most previous research assumed that this gas release takes the form of small spherical bubbles, not only in bodily fluids but also in soft tissues, which are then assumed to deform elastically and reversibly. However, we have shown recently that decompressed polyacrylamide hydrogels beads (an experimental model for soft tissues) can also be subject to large gas-filled penny-shaped cracks, which are thin, grow linearly in time, and irreversibly fracture the material. A fascinating follow-up problem is to understand the transition from small, spherical, non- fracturing bubbles (which are 'safe') to large, thin, fracturing cracks (which are 'unsafe'). Possible projects include furthering the experimental or theoretical investigation of this problem. This would suit someone with multidisciplinary interests and skills. |

## Possible project working on datasets from the 'Stratified Inclined Duct' experiment

Project Title | Possible project working on datasets from the 'Stratified Inclined Duct' experiment |

Project listed | 5 January 2023 |

Project status | Closed |

Contact Name | Adrien Lefauve |

Contact Email | aspl2@cam.ac.uk |

Department/Lab/Group | DAMTP, Environmental and Industrial Fluid Dynamics |

Brief Description of the Project | One of the flagship experiments in the G. K. Batchelor Laboratory in DAMTP is the 'Stratified Inclined Duct' experiment, which consists of two 400-litre reservoirs of salt solutions at different densities connected by a long tilted channel. This setup sustains a two-layer exchange flow across the channel with varying levels of turbulence and mixing across the interface between the two fluids. This flow is an excellent model of some important oceanic flows, where different water masses flowing past one another and exchange heat or salt, e.g. in deep ocean basins, in straits, or in low-lying estuaries. We collected a dataset of > 100 high-resolution shadowgraph movies (totalling approximately 1 TB), which show the two-dimensional temporal evolution of turbulent density interfaces under different flow conditions. A possible project is to reduce the complexity of this dataset with machine learning techniques, in particular auto-encoders (unsupervised learning). After reducing complex movies to lower-dimensional "encoded" data, we would want to classify the data into a small number of dynamically relevant regimes to learn insightful flow physics. |

## The evolution of microbial cooperation in space

Project Title | The evolution of microbial cooperation in space |

Project listed | 5 January 2023 |

Project status | Closed |

Contact Name | Nuno Miguel Oliveira |

Contact Email | nmdso2@cam.ac.uk |

Department/Lab/Group | DAMTP Goldestein/Oliveira |

Brief Description of the Project | It has been argued that bacteria rely on other strains and species to grow as they often lose important genes that code for leaky traits they can acquire from other genotypes, and that this adaptive gene loss could be a route for the evolution of microbial cooperation (1). However, current evolutionary theory predicts that competition dominates microbial interactions and that mutual dependencies between microbial genotypes are rare, particularly in spatially heterogenous environments where it is harder for microbial genotypes to find the partners they need for their growth (2). While this theory has found experimental support in the literature, it ignores a key piece of bacterial biology: cell motility and chemotaxis. In natural and clinical settings, bacteria not only display different forms of active motility, ranging from swimming and twitching to swarming, but they can also bias their motion in chemical gradients (chemotaxis). This project seeks to understand if and how bacterial motility and chemotaxis promote or jeopardize the evolution of microbial cross-feeding. Methodologically, it is expected that the student builds upon our current lattice model for the evolution of within-genotype cooperation and extends the model for interactions between different genotypes. Our current model is a modified version of Reichenbach et al's lattice model, which the authors use to study the effect of motility on biodiversity (4), and we have used it to study how different types of motility affect the evolution of within-genotype cooperation in microbial communities. Once the dynamics of the lattice model for between-genotype cooperation is well-understood, it is expected that the student develops a deterministic version to confirm and extend the results obtained with the stochastic model (5). |

References | (1) Morris JJ, Lenski RE, Zinser ER. (2012) The Black Queen Hypothesis: Evolution of dependencies through adaptive gene loss. MBio 3: e00036-12. (2) Oliveira NM, Niehus R, Foster KR. (2014) Evolutionary limits to cooperation in microbial communities. Proc Natl Acad Sci USA 111(50):17941-17946. (3) Oliveira NM, Martinez-Garcia E, Xavier J, Durham WM, Kolter R, Kim W, Foster KR. (2015) Biofilm formation as a response to ecological competition. PloS Biol 13(7). (4) Reichenbach T, Mobilia M, Frey E. (2007) Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games. Nature 448:1046-1049. (5) Wakano JY, et al. (2009) Spatial dynamics of ecological public goods. Proc Natl Acad Sci USA 106(19):7910-4. |

## The evolution of bacterial chemotaxis towards antibiotics

Project Title | The evolution of bacterial chemotaxis towards antibiotics |

Project listed | 5 January 2023 |

Project status | Closed |

Contact Name | Nuno Miguel Oliveira |

Contact Email | nmdso2@cam.ac.uk |

Department/Lab/Group | DAMTP Goldestein/Oliveira |

Brief Description of the Project | In natural and clinical environments, bacteria are often exposed to gradients of antibiotics and other antimicrobials, but we still know little about how bacterial cells respond to such gradients. Until recently, we did not even know how bacteria control their motility in gradients of antibiotics, and cell motility is a major feature for bacterial survival and reproduction. We recently discovered that, unexpectedly, bacteria actively move towards lethal concentrations of antibiotics from different classes and modes of action, which suggested a general response (1). Moreover, we further found that the motile response we uncovered was consistent with positive chemotaxis towards the antibiotics themselves (as opposed to other emergent chemicals), and that a subpopulation of migrating bacteria produces their own antibiotics as they move along the gradient (1). These observations made us argue that bacterial chemotaxis towards antibiotics evolved as a response to antibiotic-producing species, where bacteria counterattack with their own toxins when they sense an incoming threat, akin to what we find in many animal societies. However, we still do not know how such perplexing behaviour contributes to the reproductive success of bacteria. Moreover, we have recently found by means of computer simulations and analytical theory that positive chemotaxis towards antibiotics is expected to reduce the adaptation rate of bacteria to antibiotics (2), which makes it even harder to understand how bacterial chemotaxis towards antibiotics evolved. To address this issue, we need to quantify the benefits and costs of such behaviour in direct competition with antibiotic-producing species, and this research project aims to provide such understanding. The student is expected to build upon our current mathematical model of bacterial evolution in antibiotic gradients (2), and upon source-sink theory more generally (3,4). More precisely, the student will study if, and how, moving up an antibiotic landscape can be advantageous when this landscape is dynamic and results from antibiotic-producing competitors that grow in the neighbourhood of the motile genotype. Evolutionary game theory will be used to understand what the optimal motile strategy is to outcompete antibiotic-producing species. |

References | (1) Oliveira NM, Wheeler JHR, Deroy C, Booth S, Walsh E, Durham WM, Foster KR. Suicidal chemotaxis in bacteria. Nature Communications. Accepted. Preprint in bioRxiv at: doi: https://doi.org/10.1101/2021.12.21.473623 (2) Piskovsky V, Oliveira NM. Bacterial motility governs the evolution of antibiotic resistance in spatially heterogenous environments. In review. Preprint in bioRxiv at: doi: https://doi.org/10.1101/2022.10.21.513270 (3) Hermsen R, Hwa T. (2010) Sources and sinks: a stochastic model of evolution in heterogenous environments. Phys Rev Lett 105:248104. (4) Hermsen R, Deris J, Hwa T. (2012) On the rapidity of antibiotic resistance evolution facilitative by a concentration gradient. Proc Natl Acad Sci USA 109:10775-89. |

## The flash dynamics of bioluminescent biofilms under flow

Project Title | The flash dynamics of bioluminescent biofilms under flow |

Project listed | 5 January 2023 |

Project status | Closed |

Contact Name | Nuno Miguel Oliveira |

Contact Email | nmdso2@cam.ac.uk |

Department/Lab/Group | DAMTP Goldestein/Oliveira |

Brief Description of the Project | Bacterial bioluminescence has intrigued natural philosophers for centuries and the biochemistry of the process is now well established after major technical and conceptual development in the 20th century. Like in most bioluminescent organisms, light emission in bacteria is the result of a chemical reaction where an enzyme (luciferase) catalyses the oxidation of a light-emitting species (luciferin) in the presence of oxygen. An interesting and unique aspect of the luminescence reaction in bacteria when compared to what happens in other luminous organisms is that it competes for oxygen and reductive power with the reactions of bacterial respiration. However, this competition between luminescence and respiration for reagents has been poorly explored, and the ecological conditions where it may play a role are still unknown. Another interesting aspect about the literature on bacterial bioluminescence is that the vast majority of studies has focused on planktonic suspensions of free-living bacteria, while in natural conditions luminous bacteria most often grow on inert or living surfaces as biofilms, and we still do not know the dynamics of light emission in such surface-attached communities. We have recently found that biofilms of luminous bacteria growing under flow conditions flash in a minute-time-scale when the fluid flow where they grow is suddenly blocked. Importantly, not all cells of the community flash at the same time. Instead, there is a striking spatial pattern of light emission in the collective where light emission is triggered first at the periphery of the community, from where a travelling wave of light forms and moves toward the core of the biofilm. This research project aims to model and understand quantitatively the spatiotemporal dynamics of this flow-induced flash. To understand the temporal dynamics of the flash, the student is expected to develop a quantitative model of enzyme kinetics of bacterial bioluminescence with a system of ordinary differential equations that considers the competition for reagents between the luminescence and respiration reactions, with which one can understand how the stimulation or inhibition of respiration may affect light emission. In particular, the goal is to understand how the inhibition of bacterial respiration induced by blocking fluid flow stimulates light emission. Once the temporal dynamics of the flash is understood, the goal is to understand its spatial dynamics, namely by extending the former model and consider instead a system of partial differential equations to characterize how the stimulation of light percolates through the community from the periphery to its centre. Given that bacterial bioluminescence is essentially a marine phenomenon where changes in fluid flow are commonplace, understanding how bacterial bioluminescence is affected by flow will help us to understand why it may have evolved in nature. Despite centuries of research, we still know little about why bacteria produce light. |

References | (1) Hastings JW (1952) Oxygen concentration and bioluminescence intensity. J Cell and Comp Physiol 39(1):1-30. (2) Nealson KH, Hastings JW (1979) Bacterial bioluminescence: its control and ecological significance. Microbiol Rev 43(4):496-518. (3) Schoepfle GM (1940) Kinetics of luminescent flashes in the bacterium Achromobacter fisheri at different temperatures. J Cell and Comp Physiol 16:341-360. (4) Watanabe H, et al. (1977) Luminescence and respiratory activities of Photobacterium phosphoreum: II. Control by monovalent cations. J Biochem 82(6):1707-1714. (5) Widder EA (2010) Bioluminescence in the Ocean: Origins of biological, chemical, and ecological diversity. Science 328(5979):704-709. |

## A project on automatic theorem proving

Project Title | A project on automatic theorem proving |

Project listed | 5 January 2023 |

Project status | Closed |

Contact Name | Timothy Gowers |

Contact Email | wtg10@cam.ac.uk |

Department/Lab/Group | DPMMS |

Brief Description of the Project | I run a group that is working on automatic theorem proving -- that is, on trying to program computers to find proofs of theorems. This summer I expect to be able to fund one or two people to participate in the project. This would involve attending meetings twice a week for most of July and September, and working on some of the many aspects of the project, probably in collaboration with existing group members, in between meetings. It is not yet possible to propose precise projects because we do not know what we will have already done by next summer. (Just by way of illustration though, one possibility might be to focus on an area of mathematics that the program is not yet good at and to try to improve its performance in that area.) Strong programming ability would be a big advantage, but the most important quality is a deep appreciation of the difference between the notion of a proof and the notion of a proof discovery process (e.g., an obvious sign that they are different is that there can be quite simple proofs that took a long time to discover), and the ability to focus on the discovery part. For much more detail about the project, you can explore our website: https://wtgowers.github.io/human-style-atp/ |

## Bioluminescence in waves

Project Title | Bioluminescence in waves |

Project listed | 5 January 2023 |

Project status | Closed |

Contact Name | Sumit Kumar Birwa |

Contact Email | skb61@cam.ac.uk |

Department/Lab/Group | DAMTP/Goldstein lab |

Brief Description of the Project | Some marine microorganisms exhibit bioluminescence, a unique ability through which they can convert chemical energy into light. These microorganisms when in abundance can discolour the water resulting in coloured tides or milky seas. Marine organisms use bioluminescence for several purposes like startling an enemy, as a burglar alarm, etc. Large eukaryotic organisms such as dinoflagellates exhibit a brief burst of light when mechanically stimulated [1], while bacteria can produce light continuously for long periods of time. While it is known that dinoflagellates can bioluminesce in response to fluid shear generated by waves breaking in oceans, it is believed that bacteria do not show any such response. In this project, we will create a tabletop experimental set-up that generates non-linear standing waves to mimic the conditions experienced by microorganisms in the ocean. These experiments are expected to give us a deep insight into the bioluminescent response that microorganisms exhibit due to mechanical stimulus generated in the ocean. The standing waves will also lead to some interesting phenomena like the clustering of microorganisms and pattern formation which may open up many new problems and challenges. Hand-in-hand with experiments, we will also analytically derive the stress profile and flow generated in these surface waves using simple fluid mechanics. For this project, students will be required to do both experiments and theory. Some basic understanding of fluid mechanics, non-linear dynamics and numerical simulations will be helpful. |

References | [1] Jalaal, M. et al. Stress-induced dinoflagellate bioluminescence at the single cell level. Phys. Rev. Lett. 125, 028102 (2020). |

## Synchronization of Biological Oscillators

Project Title | Synchronization of Biological Oscillators |

Project listed | 5 January 2023 |

Project status | Closed |

Contact Name | Maria Tatulea-Codrean |

Contact Email | mt599@cam.ac.uk |

Department/Lab/Group | DAMTP, Biological Fluid Mechanics |

Brief Description of the Project | The spontaneous coordination of independently driven oscillators is a ubiquitous natural phenomenon, from singing crickets and cardiac rhythms, to firing neurons and undulating flagella. The most famous example is the synchronization of pendulum clocks observed by Christiaan Huygens in 1665. Mathematically, these systems are described by coupled nonlinear ODEs, making their analysis accessible to students finishing Part II of the Mathematical Tripos. My personal expertise lies in the modelling of synchronization between rotating bacterial flagella (MTC & Eric Lauga, 2022, Physical Review Letters, 128: 208101), a topic which remains poorly understood and can be explored further in many different directions corresponding to different physical mechanisms. The 8-week project will start with a short literature review followed by an analytical or computational investigation, or a mixture of both. The student will have an opportunity to interact with members of the Biological Fluid Mechanics group and attend informal seminars. An accessible article about the beauty of synchronization and the history of its mathematical modelling was written by Steven Strogatz and Ian Stewart for The Scientific American, under the title "Coupled Oscillators and Biological Synchronization", but there is no pre-requisite technical reading before the start of the project. The following courses are desirable: Part IA Differential Equations, Part II Mathematical Biology, Part II Dynamical Systems. The project can be adapted to suit the interests and mathematical strengths of the student. |

## New understanding of fluid's energy equation: a theory for surface viscosity

Project Title | New understanding of fluid's energy equation: a theory for surface viscosity |

Project listed | 5 January 2023 |

Project status | Closed |

Contact Name | Rajesh Kumar Bhagat |

Contact Email | rkb29@cam.ac.uk |

Department/Lab/Group | DAMTP |

Brief Description of the Project | Interfacial flows are common in both industrial processes and the natural world. One particularly relevant example is the transmission of airborne diseases through droplets and aerosols containing pathogens. The formation and evolution of these droplets and aerosols are described by the Navier-Stokes equation and the energy equation. However, the conventional energy equation, which is based on the premise that the internal energy of the fluid is defined by state variables or on a per-unit-mass basis, ignores the surface energy at interfaces and therefore cannot explain interfacial flows. The Navier-Stokes equation, on the other hand, is a statement of Newton's laws in differential form and can explain these phenomena when applied correctly. The surface energy is attached to a two-dimensional slice at the interface, and the pointwise energy equation cannot include it as a boundary condition. I am offering two projects to test the implications of this new understanding. In the first project, we will aim to gain a basic understanding of the elusive and phenomenological concept of surface viscosity and its relation to surface energy. Theoretically, we will develop equations for flatlanders, an imaginary two dimensional world with surface energy, and connect it to the 3D world. The students will have the opportunity to conduct lab experiments and develop new theories. The ultimate practical goal of this project is to understand droplet formation in dynamically similar environments to the human vocal tract. For further information, please contact Dr. Rajesh Bhagat @ (rkb29@cam.ac.uk) or feel free to drop by at H1.03 |

References | 1. Bhagat, R.K., Jha, N.K., Linden, P.F. and Wilson, D.I., 2018. On the origin of the circular hydraulic jump in a thin liquid film. Journal of Fluid Mechanics, 851. 2. Bhagat, R.K. and Linden, P.F., 2020. The circular capillary jump. Journal of Fluid Mechanics, 896. 3. Is the influence of surface tension fully contained in Laplace pressure? (Available on request; soon available online) |

## New understanding of fluid's energy equation: a theory for non-Newtonian interfacial flows

Project Title | New understanding of fluid's energy equation: a theory for non-Newtonian interfacial flows |

Project listed | 5 January 2023 |

Project status | Closed |

Contact Name | Rajesh Kumar Bhagat |

Contact Email | rkb29@cam.ac.uk |

Department/Lab/Group | DAMTP |

Brief Description of the Project | Interfacial flows are common in both industrial processes and the natural world. One particularly relevant example is the transmission of airborne diseases through droplets and aerosols containing pathogens. The formation and evolution of these droplets and aerosols are described by the Navier-Stokes equation and the energy equation. However, the conventional energy equation, which is based on the premise that the internal energy of the fluid is defined by state variables or on a per-unit-mass basis, ignores the surface energy at interfaces and therefore cannot explain interfacial flows. The Navier-Stokes equation, on the other hand, is a statement of Newton's laws in differential form and can explain these phenomena when applied correctly. The surface energy is attached to a two-dimensional slice at the interface, and the pointwise energy equation cannot include it as a boundary condition. I am offering two projects to test the implications of this new understanding. In the second project, we will develop a model to describe the energy equation for non-Newtonian fluids. These fluids pose a unique challenge, and we will use existing models to describe the interfacial phenomena for them. The students will have the opportunity to conduct lab experiments and develop new theories. The ultimate practical goal of this project is to understand droplet formation in dynamically similar environments to the human vocal tract. For further information, please contact Dr. Rajesh Bhagat @ (rkb29@cam.ac.uk) or feel free to drop by at H1.03 |

References | 1. Bhagat, R.K., Jha, N.K., Linden, P.F. and Wilson, D.I., 2018. On the origin of the circular hydraulic jump in a thin liquid film. Journal of Fluid Mechanics, 851. 2. Bhagat, R.K. and Linden, P.F., 2020. The circular capillary jump. Journal of Fluid Mechanics, 896. 3. Is the influence of surface tension fully contained in Laplace pressure? (Available on request; soon available online) |

## Global MHD Simulations of Gas in Debris Disks

Project Title | Global MHD Simulations of Gas in Debris Disks |

Project listed | 9 January 2023 |

Project status | Closed |

Contact Name | Can Cui |

Contact Email | cc795@cam.ac.uk |

Department/Lab/Group | DAMTP, Astrophysical Fluid Dynamics |

Brief Description of the Project | Newborn (pre-main sequence) stars are surrounded by gas-rich protoplanetary disks. Most of these disks dissipate over a timescale of a few million years, leaving a tenuous debris disk composed of planetesimals and the dust derived from them, as well as gas and planets. Debris disks were sometimes defined as being gas-poor. However, mounting observational evidence in the past few years has indicated that circumstellar gas is present during the debris disk phase. The origin of the gas and its subsequent evolution has become the most active area of research. The theoretical understanding of gas dynamics lags behind the observational progress. If these disks are encompassed by magnetic fields, analogous to our Solar System, MHD processes can be important. In this project, we will self-consistently study the gas dynamics in debris disks, by conducting the very first global 3D MHD simulations with Athena++ code applicable to debris disk systems. 1.1 Ideal MHD and the magneto-rotational instability: severely depleted in hydrogen, debris disk systems are overabundant in atomic carbon and oxygen. Carbon atoms will be ionized in very short timescales, and hence its ionization fraction is high. For sufficiently ionized gas, the magneto-rotational instability can operate, driving vigorous turbulence and transporting angular momentum. We will use the techniques developed in my previous work on the magneto-rotational instability and apply it to conduct global 3D ideal MHD simulations with Athena++ in the context of debris disks. We aim to study the gas kinematics induced by the magneto-rotational instability. The results can be compared to gas observations (ALMA and Herschel) of β Pictoris, a debris disk archetype considered with high ionization. |

References | https://ui.adsabs.harvard.edu/abs/2021MNRAS.507.1106C/abstract |

## What is the future of AI?

Project Title | What is the future of AI? |

Project listed | 9 January 2023 |

Project status | Closed |

Contact Name | Richard Samworth |

Contact Email | rjs57@cam.ac.uk |

Department/Lab/Group | DPMMS, Stats Lab |

Brief Description of the Project | We are at the early stage of the Artificial Intelligence (AI) revolution. Scientific fields such as genetics, medical imaging, particle physics, climate science, astrostatistics, social science and many others stand on the verge of being able answer questions that would have seemed unimaginable only a decade ago. Meanwhile, AI-driven technologies such as ChatGPT and driverless cars promise to have greater and greater impacts on our lives. At the core of these advances are algorithms, and our increasing dependence on their automated responses makes it vital that these methodologies are robust, fair and efficient. What areas exploit AI effectively? What areas are ripe for further innovation? How can maths and statistics contribute? |

References | Jordan, M. I. (2019) Arti cial Intelligence—The Revolution Hasn’t Happened Yet. Harvard Data Science Review, 1.1 |

## Missing data

Project Title | Missing data |

Project listed | 9 January 2023 |

Project status | Closed |

Contact Name | Richard Samworth |

Contact Email | rjs57@cam.ac.uk |

Department/Lab/Group |
DPMMS, Stats Lab |

Brief Description of the Project | One of the ironies of working with Big Data is that missing data play an ever more significant role, and often present serious difficulties for analysis. For instance, a common approach to handling missing data is to perform a so-called `complete-case analysis', where we restrict attention to cases with no missing attributes. If we have an n by d data matrix in which each entry is missing independently with probability 0.01, then when d = 5, a complete-case analysis would result in around 95% of the individuals (rows) being retained, but even when we reach d=300, only around 5\% of rows will have no missing entries, and the approach becomes infeasible. A student might like to consider the methodological and theoretical effects of missing data in a modern statistical problem of their choice. |

References |
Zhu, Z., Wang, T. and Samworth, R. J. (2022) High-dimensional principal component analysis with heterogeneous missingness. J. Roy. Statist. Soc., Ser. B, 84, 2000--2031. Follain, B., Wang, T. and Samworth, R. J. (2022) High-dimensional changepoint estimation with heterogeneous missingness. J. Roy. Statist. Soc., Ser. B, 84, 1023--1055. Loh, P.-L. and Wainwright, M. J. (2012) High-dimensional regression with noisy and missing data: provable guarantees with nonconvexity. Ann. Statist., 40, 1637--1664. |

## Convergence of orthogonal series

Project Title | Convergence of orthogonal series |

Project listed | 19 January 2023 |

Project status | Closed |

Contact Name | Arieh Iserles |

Contact Email | ai10@cam.ac.uk |

Company/Lab/Department | DAMTP |

Brief Description of the Project |
While a great deal is known about the speed of convergence of classical L_2 orthogonal series to analytic functions in compact intervals, there is definite absence of theory once we remove any phrase ("classical", "L_2", "analytic" or "compact") from this sentence. Indeed, all we have are highly incomplete computational and asymptotic results that indicate that something strange is going on. The project will have three components: |

## Nonparametric estimation of the incubation period distribution of Covid-19

Project Title | Nonparametric estimation of the incubation period distribution of Covid-19 |

Project listed | 19 January 2023 |

Project status | Closed |

Contact Name | Rajen Shah |

Contact Email | rds37@cam.ac.uk |

Company/Lab/Department | DPMMS, Statistical Laboratory |

Brief Description of the Project |
From a regional outbreak in Wuhan, Covid-19 became a global pandemic. In the early stages of the pandemic, travellers from Wuhan who subsequently tested positive for Covid-19 were likely to have contracted the disease during their stay in Wuhan. Using data (available from the paper referenced) on the dates travellers entered and left Wuhan, and the dates at which they reported symptoms, the goal of the project would be to estimate the distribution of the incubation period, that is the time from transmission of the virus to the onset of symptoms. The referenced paper makes some parametric assumptions about the form of this distribution, but the aim here would be to construct a method that avoids making such assumptions, and develop theoretical guarantees on its performance. This project would likely be suitable for a 3rd year undergraduate with some prior knowledge of theoretical statistics as for example taught in the Cambridge Part II course Principles of Statistics. |

References | https://arxiv.org/pdf/2004.07743.pdf |

## Inference in high-dimensional regression models

Project Title | Inference in high-dimensional regression models |

Project listed | 19 January 2023 |

Project status | Closed |

Contact Name | Rajen Shah |

Contact Email | rds37@cam.ac.uk |

Company/Lab/Department | DPMMS, Statistical Laboratory |

Brief Description of the Project |
High-dimensional data, where the number of variables is large compared to the number of observations, are becoming increasingly common across a range of scientific disciplines and industry. Classical statistical methods often perform poorly for such data, or do not work at all. There is therefore a need to develop new methods that are able to cope well with the high-dimensionality of the data. Great strides have been made in this direction over the last couple of decades, and we now have well-established methods for providing point estimates of high-dimensional parameters in a variety of common statistical models. Whilst this represents great progress, much remains to be done. One of the key goals of statistical methodology is uncertainty quantification, and there is a need to develop high-dimensional analogues of classical statistical tools for forming confidence intervals and performing hypothesis tests, for example. One approach that has gained great traction in recent years is that of the debiased Lasso (see references). While in theory this can provide confidence intervals for parameters even in challenging high-dimensional settings, there remain substantial challenges in using it in practice in terms of both its statistical and computational performance. This project could look at ways of addressing these issues, with the aim of developing a software package for the statistical computing language R implementing the methodology. This project would be particularly suitable for those with prior computing experience in R and / or C++. |

References | http://www.statslab.cam.ac.uk/~rds37/teaching/modern_stat_methods/notes2.pdf chapters 2 and 4 https://arxiv.org/abs/1303.0518 https://arxiv.org/abs/1110.2563 https://arxiv.org/abs/1909.10828 |

## Machine learning chemical reactions

Project Title | Machine learning chemical reactions |

Project listed | 20 January 2023 |

Project status | Closed |

Contact Name | Camille Scalliet |

Contact Email | cs2057@cam.ac.uk |

Company/Lab/Department | Soft Matter - DAMTP |

Brief Description of the Project | The goal of this original research project is to use recently developed machine learning tools (based on neural networks) to investigate the dynamics of large chemical reaction networks. This project will complement my current efforts in collaboration with Eric de Giuli to derive a general theoretical framework for the dynamics of chemical reaction networks. We are particularly interested in the case where many chemical reactions compete and standard simulation tools have shortcomings. The student should have experience using python and a strong interest in computer simulations, in connection with theoretical developments. |

References | My webpage: https://www.damtp.cam.ac.uk/user/cs2057/ Reference on neural network for reaction networks: https://arxiv.org/abs/2210.01169 Theory for large chemical reaction networks: https://iopscience.iop.org/article/10.1088/1751-8121/aca3df |

## Computing Spectral Properties of Operators on Surfaces

Project Title | Computing Spectral Properties of Operators on Surfaces |

Project listed | 24 January 2023 |

Project status | Closed |

Contact Name | Matthew Colbrook |

Contact Email | mjc249@cam.ac.uk |

Company/Lab/Department | DAMTP |

Brief Description of the Project |
The goal of this project is to develop provably convergent and efficient algorithms for computing spectral properties of operators (e.g., PDEs, integral operators etc.) on surfaces. Surface-bound phenomena arise in a wide variety of applications, including electromagnetics, plasma physics, biological pattern formation, bulk–surface diffusion processes, biomechanics, and fluid dynamics. However, computing spectral properties in the ensuing infinite-dimensional Hilbert spaces is challenging! This project will combine several recent breakthroughs in this area. To deal with eigenvalue problems (discrete spectra), the student will combine methods based on complex contour integration and residuals to compute error bounds. These methods will make use of high-order methods that solve PDEs on surfaces. As well as combining and generalising previous ideas, there is considerable scope here for novelty. For example, by combining with eigenvalue asymptotics, this may lead to the first method that can accurately compute high frequency eigenmodes on generic surfaces. Another avenue could be solving time-dependent PDEs via contour methods. A second part of the project will focus on so-called continuous spectra. These provide a generalisation of diagonalisation of linear operators in infinite dimensions. Again the tool here is solving shifted linear systems through surface PDE software. Finally, if time, the project will look at the foundations of computation for surface spectral problems – i.e., proving rigorous theorems on what can and cannot be done. This is a potentially high impact project that touches upon a range of ideas in analysis and computing. A successful outcome will be transforming the results of this project into a paper for a peer-reviewed journal. There is room to complete this final outcome after the completion date, and such an outcome would also provide a competitive edge for the student if they apply to further studies (i.e., PhD). |

Work Environment | |

References | Colbrook, Matthew J., Bogdan Roman, and Anders C. Hansen. "How to compute spectra with error control." Physical Review Letters 122.25 (2019): 250201. Colbrook, Matthew, Andrew Horning, and Alex Townsend. "Computing spectral measures of self-adjoint operators." SIAM Review 63.3 (2021): 489-524. Colbrook, Matthew J. "Computing semigroups with error control." SIAM Journal on Numerical Analysis 60.1 (2022): 396-422. Ben-Artzi, Jonathan, et al. "Computing Spectra--On the Solvability Complexity Index Hierarchy and Towers of Algorithms." arXiv preprint arXiv:1508.03280 (2020). Horning, Andrew, and Alex Townsend. "FEAST for differential eigenvalue problems." SIAM Journal on Numerical Analysis 58.2 (2020): 1239-1262. Fortunato, Daniel. "A high-order fast direct solver for surface PDEs." arXiv preprint arXiv:2210.00022 (2022). |

## Euclidean Ramsey Theory

Project Title | Euclidean Ramsey Theory |

Project listed | 24 January 2023 |

Project status | Closed |

Contact Name | James Davies |

Contact Email | jgd37@cam.ac.uk |

Company/Lab/Department | DPMMS, Combinatorics |

Brief Description of the Project | Euclidean Ramsey Theory focuses on the following general question: Is it true that no matter how we colour the points of the d-dimensional Euclidean space by k colours, we can always find a monochromatic subconfiguration of a certain type? This project would investigate such questions. Proofs in this area can combine a variety of techniques in combinatorics, analysis, and geometry, and a background in some of these areas is desirable. Good programming ability may also be helpful. |

## Cosmic Strings with Adaptive Mesh Refinement

Project Title | Cosmic Strings with Adaptive Mesh Refinement |

Project listed | 30 January 2023 |

Project status | Closed |

Contact Name | Amelia Drew |

Contact Email | ad652@cam.ac.uk |

Company/Lab/Department | DAMTP, General Relativity group |

Brief Description of the Project |
This project will involve evolving configurations of cosmic strings using an adaptive mesh refinement code. Cosmic strings are topological defects that may have formed due to a phase transition in the early Universe, and are a potential source of dark matter axions and gravitational waves. There is scope for discussion about the type of project that would interest the student. Some possible ideas could be: |

References |
My publications in this area: https://arxiv.org/abs/2211.10184, https://arxiv.org/abs/2201.03458, https://arxiv.org/abs/2112.10567, https://arxiv.org/abs/1910.01718 Code website: https://www.grchombo.org/ Background reading: the papers referenced in my first author papers above, and Vilenkin and Shellard, Cosmic Strings and Other Topological Defects |

## Classical simulation of Topological Quantum Computation

Project Title | Classical simulation of Topological Quantum Computation |

Project listed | 21 February 2023 |

Project status | Closed |

Contact Name | Sergii Strelchuk |

Contact Email | ss870@cam.ac.uk |

Company/Lab/Department | DAMTP, CQIF |

Brief Description of the Project |
Topological quantum computation (TQC) remains one of the most promising roads towards practical quantum computers. In a topological quantum computer, computations are performed via the braiding of anyons, exotic quasi-particles that arise in 2 dimensional models exhibiting topological order. The topological nature of the braiding process makes it particularly robust against local perturbations, and this resistance to noise is what makes TQC such a prominent contender for a practical implementations in which noise is inevitable. Furthermore, it can easily be shown that even simple models, such as those with Fibonacci anyons, are universal for quantum computation. Despite the promise of TQC, the exotic nature of these anyonic excitations makes it challenging to engineer systems that exhibit them in the lab. A different model, known as permutational quantum computing (PQC), is based on the same principles but instead works with particles that have a symmetric braiding. This means that the topological nature of these braiding operations is irrelevant, and the swapping of two particles is just a permutation. The advantage is that this model can be realized using e.g. ordinary spin-1/2 particles, which are much less exotic than anyons. In this setting, it connects to an intriguing model of quantum computation whose origins can be traced back to Roger Penrose: Permutational Quantum Computing (PQC). Inspired by these developments, this project aims to reinvestigate the classical simulation of TQC. Recent work on generalized symmetries recasts exotic anyonic excitations in a more conventional language, and brings these anyons much closer to the types of excitations used in PQC. It is expected that many of the tools introduced to classically simulate PQC can be generalized to TQC. Given that TQC is universal however, the classical algorithms proposed for PQC must break down at some point, and one of the goals of this project is to pinpoint exactly where and why this happens. |

References | Suggested (not mandatory) reading: https://arxiv.org/abs/1801.04795 https://arxiv.org/pdf/1705.04103.pdf https://arxiv.org/abs/2207.07250 |