# Summer Research in Mathematics

This is a list of Research in the CMS projects for 2021. To remove a listing, please contact Arti Sheth Thorne.

Students are encouraged to find their own Research in the CMS projects by directly approaching potential project supervisors. Projects do not need to be listed here in order for you to apply for a bursary.

Projects that have been filled or withdrawn are indicated with strikethrough text. Please do not contact the host in question if a project has been filled or withdrawn.

## Mathematics of bilayer graphene

 Project Title Mathematics of bilayer graphene Contact Name Simon Becker Contact Email slb214@cam.ac.uk Project Dates 8 weeks during summer 2021 Suggested Deadline to contact project supervisor 26 February 2021 - the earlier you get in touch, the better Brief Description of the Project Graphene is a material that consists of carbon atoms situated at the vertices of a honeycomb lattice. It has received a lot of attention for its peculiar properties. We shall analyze the properties of two sheets of graphene stacked on top of each other in an external magnetic field. We shall then derive the de Haas-van Alphen and quantum Hall effect. For a single sheet of graphene, this has has been done here https://arxiv.org/pdf/1801.01931.pdf and you should expect that the result for multiple layers of graphene will involve similar mathematics. Keywords Schrödinger equation, Functional Analysis, Spectral Theory, Microlocal Analysis, Mathematical Physics References https://link.springer.com/article/10.1007/s00220-019-03409-4 Magnetic oscillations in a model of graphene S Becker, M Zworski Communications in Mathematical Physics 367 (3), 941-989 http://www.numdam.org/item/AIHPA_1990__52_4_303_0/ On diamagnetism and de Haas-van Alphen effect Helffer, B. ; Sjöstrand, J. Annales de l'I.H.P. Physique théorique, Tome 52 (1990) no. 4, pp. 303-375. Helffer B., Sjőstrand J. (1989) Equation de Schrödinger avec champ magnétique et équation de Harper. In: Holden H., Jensen A. (eds) Schrödinger Operators. Lecture Notes in Physics, vol 345. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51783-9_19 Prerequisite Skills Mathematical physics, PDEs, Mathematical Analysis Other Skills Used in the Project A basic background knowledge in quantum mechanics is desirable. Knowledge or the will to learn about non-trivial results in functional analysis, harmonic analysis, and spectral theory are needed. To consult some references knowledge of the french language may be useful.

## Semiparametric Generalised Linear Models

 Project Title Semiparametric Generalised Linear Models Contact Name Oliver Y. Feng and Richard J. Samworth Contact Email r.samworth@statslab.cam.ac.uk Project Dates Any 8 week period over the long vacation Suggested Deadline to contact project supervisor 8 March 2021 Brief Description of the Project Generalised Linear Models (GLMs), as covered in the Part II Statistical Modelling course, are the most popular way of seeking to understand the way that a response variable depends on covariates. Logistic regression (for binary responses), Poisson regression and, of course, standard linear models are the most common examples. GLMs require the statistician to postulate a link function that relates the mean response to the linear predictor, and this is normally pre-specified for convenience. The aim of this project is to develop methodology that allows simultaneous estimation of the (inverse) link function and the parameter vector of interest, thereby greatly enhancing the flexibility and scope of GLMs. It will build on ongoing work by the supervisors and others on the nonparametric estimation of S-shaped functions. The project will involve a combination of algorithmic development and implementation, with the potential also for theoretical analysis. It would be best suited to a finishing Part II student, but may also be accessible to a finishing Part IB student. Keywords Generalised linear models, S-shaped function estimation, semiparametric inference References Prerequisite Skills Interest in Statistics, some programming experience (ideally in R, but other languages also fine) Other Skills Used in the Project Interest in analysis and/or optimisation

## The mechanics of biofilm morphogenesis

 Project Title The mechanics of biofilm morphogenesis Contact Name Nuno Miguel Oliveira Contact Email nmdso2@cam.ac.uk Project Dates 8 weeks during summer 2021 Suggested Deadline to contact project supervisor As soon as possible but no deadline. Brief Description of the Project Bacteria most often grow as surface-attached communities called biofilms that are critical for their impacts on us, ranging from chronic infections and antibiotic resistance to biofouling, bioremediation and nutrient cycling. This realization fostered the study of biofilm growth, the expectation being that, if we know how biofilms develop, we are better equipped to manipulate these collectives, promoting the growth of those that are beneficial, while limiting those that are deleterious. Biofilm growth can be described as a developmental process akin to what we find in multicellular organisms where cells experience a range of surface-specific mechanics to which they respond to throughout biofilm morphogenesis. However, we still have very limited understanding about the mechanics of biofilm growth, partially because it is hard to measure the forces that bacteria experience during community development. This problem has recently been circumvented in our lab by the use of bioluminescent bacteria that literally light up when they are mechanically stimulated. In short, we can now use bacterial bioluminescence as a probe to understand the mechanics of biofilm morphogenesis. The summer student is expected to analyse experimental data where bacteria were grown on different stiffness substrates, which generate different biofilm morphologies and light patterns. From these patterns, and with a suitable calibration curve of force versus light intensity, the student will be able to reconstruct the forces generated during biofilm morphogenesis with high precision for the first time. In addition to this characterization, it is expected that the student develops a mathematical model that can capture and explain the key patterns observed experimentally and, hopefully, make predictions that can be tested experimentally next. The suggested bibliography provides relevant information about the available mathematical models that students can use to describe the system, namely those rooted on mechanical instability theory. The ideal candidate should be familiar with image analysis, programming, fluid mechanics and pattern formation. Keywords morphogenesis, biofilms, mechanics References - Fei C, et al (2020) Nonuniform growth and surface friction determine bacterial biofilm morphology on soft substrates. PNAS 117(14):7622-7632. - Matoz-Fernandez DA, et al (2020) Wrinkle patterns in active viscoelastic thin sheets. PRR 2:013165. - Persat A, et al (2015) The mechanical world of bacteria. Cell 161(5):988-997. - Widder EA (2010) Bioluminescence in the Ocean: Origins of biological, chemical, and ecological diversity. Science 328(5979):704-709. - Yan J, et al (2019) Mechanical instability and interfacial energy drive biofilm morphogenesis. eLife 8:e43920. Prerequisite Skills See project description Other Skills Used in the Project See project description

## The Governing Dynamics of Milky Seas

 Project Title The Governing Dynamics of Milky Seas Contact Name Nuno Miguel Oliveira Contact Email nmdso2@cam.ac.uk Project Dates 8 weeks during summer 2021 Suggested Deadline to contact project supervisor As soon as possible but no deadline. Brief Description of the Project A long-standing mystery of maritime lore is the formation of the so-called "milky seas", a large-scale luminous phenomenon at the surface of the ocean. Milky seas are rather common as they have been reported numerous times according to the logs of merchant ships since at least the 17th century and can now be detected by satellites orbiting the Earth. However, we know surprisingly little about what drives and sustains this dazzling phenomenon. We do know milky seas have biological origin, and that they result from bioluminescent bacteria. Many marine organisms, ranging from bacteria and fungi to animals such as fish, have the ability to produce light by converting chemical energy, and this phenomenon is known as bioluminescence. Bacteria are unique among these in that they produce light continuously while all other organisms produce short bursts of light when stimulated - often mechanically - which is consistent with what we know about milky seas, sustained light production at the surface of the ocean that can last several days. Accordingly, bioluminescent bacteria have been isolated from milky seas. However, explaining the evolution of light production in bacteria remains a major puzzle because light production takes up to 20% of the energy of cells and it is not always clear what is the benefit it provides. In particular, we do not know how bacterial bioluminescence can evolve and be maintained by natural selection in milky seas as simple evolutionary theory predicts that light-producing cells should be outcompeted by dark mutants that do not pay the costs of light production, and these often emerge in nature and labs. This research project will address this question. The summer student is expected to develop the first eco-evolutionary model that can capture the dynamics of milky seas by combining evolutionary game theory and the mathematics of dynamical systems (namely ODEs, PDEs and spatially explicit individual-based models). In particular, the student will consider the population dynamics of light producers, dark mutants, and an overlooked player of milky seas: algae. In milky seas, bioluminescent bacteria grow on the surface of algal blooms and these algae can be beneficial or detrimental to bacteria, but their effect on the evolution of light production has not been studied. The student will test two independent models for the evolution of bioluminescence: (1) Symbiosis-based model: Given that algae use light for producing oxygen during photosynthesis, and light production in living organisms is an oxygen-dependent process, the student will investigate if bacterial bioluminescence can evolve and be maintained through symbiosis between algae and light producing bacteria; (2) Detoxification-based model: Given that algae are major producers of reactive oxygen species that cause oxidative stress in cells, and the biochemical reactions associated with light production can work as potent antioxidants, the student will investigate if bacterial bioluminescence can evolve and be maintained as a response to algae-induced oxidative stress. The ideal candidate should be familiar with systems of ordinary and partial differential equations, Markov processes and should have an interest for understanding evolutionary dynamics. Keywords Game theory, Dynamical Systems, Bioluminescence Evolution References - Bergstrom CT, Lachmann M (2003) The Red King effect: When the slowest runner wins the coevolutionary race. Proc. Natl. Acad. Sci. USA 100(2):593-598. - Doekes HM, et al. (2019) Toxin production spontaneously become regulated by local cell density in evolving bacterial populations. Plos Comput Biol 15(8):e1007333. - Miller SD, et al. (2005) Detection of a bioluminescent milky sea from space. Proc. Natl. Acad. Sci. USA 102(40):14181-14184. - Nealson KH, Hastings JW (2006) Quorum Sensing on a Global Scale: Massive Numbers of Bioluminescent Bacteria Make Milky Seas. App Env Microbiol 72(4):2295-2297. - Oliveira NM, Niehus R, Foster KR (2014) Evolutionary limits to cooperation in microbial communities. Proc. Natl. Acad. Sci. USA 111(50):17941-17946. Prerequisite Skills See project description Other Skills Used in the Project See project description

## Mendelian randomisation using genome-wide association studies

 Project Title Mendelian randomisation using genome-wide association studies Contact Name Qingyuan Zhao Contact Email qz280@cam.ac.uk Project Dates 8-10 weeks, starting around 1st July Suggested Deadline to contact project supervisor 15 March 2021 Brief Description of the Project Mendelian randomisation (MR) is a popular research design in epidemiology and genetics to examine the causal effect of a risk exposure on diseases. From a statistical perspective, MR is an instance of the instrumental variables method with genetic variants acting as an instrument for the exposure of interest. Most of the current MR studies utilize summary statistics obtained in large-scale genome-wide association studies (GWAS). This internship will involve three steps: 1. Read the literature on Mendelian randomisation [1-3] and get familiar with the current practice. (2 weeks) 2. Try to improve the spike-and-slab empirical Bayes shrinkage in [3] using Tweedie's formula or the more flexible $g$-modelling [4]. (2-3 weeks) 3. Develop an R package to streamline several analyses developed by the supervisor's team [5]. Integrate it with the MR-Base platform and write a markdown tutorial. (4-5 weeks) See also this page: http://www.statslab.cam.ac.uk/~qz280/post/summer-intern-2021/ Keywords Statistics, Causal Inference, Instrumental Variables, Empirical Bayes, R Software Development References [1] George Davey Smith, Shah Ebrahim, 'Mendelian randomization': can genetic epidemiology contribute to understanding environmental determinants of disease?, International Journal of Epidemiology, 32(1):1â€“22. 2003. doi:10.1093/ije/dyg070 [2] Didelez V, Sheehan N. Mendelian randomization as an instrumental variable approach to causal inference. Statistical Methods in Medical Research. 16(4):309-330. 2007. doi:10.1177/0962280206077743 [3] Qingyuan Zhao, Yang Chen, Jingshu Wang, Dylan S Small, Powerful three-sample genome-wide design and robust statistical inference in summary-data Mendelian randomization, International Journal of Epidemiology, 48(5):1478-1492. 2019. doi:10.1093/ije/dyz142 [4] Efron, Bradley. Two modeling strategies for empirical Bayes estimation. Statistical Science 29(2):285-301. 2014. doi:10.1214/13-STS455. [5] https://github.com/qingyuanzhao/mr.raps; https://github.com/jingshuw/GRAPPLE; https://github.com/danieliong/MRPATH. Prerequisite Skills Part II Statistical Modelling or equivalent Other Skills Used in the Project Experience or interest in statistical computing

## Gas Dynamics by the Vertical Shear Instability in Protoplanetary Disks

 Project Title Gas Dynamics by the Vertical Shear Instability in Protoplanetary Disks Contact Name Can Cui Contact Email cc795@cam.ac.uk Project Dates 8 weeks during summer 2021 Suggested Deadline to contact project supervisor Brief Description of the Project The mechanisms of angular momentum transport and the level of turbulence in protoplanetary disks (PPDs) are crucial for understanding many aspects of planet formation. In recent years, it has been realized that MHD turbulence tends to be suppressed in PPDs, and hydrodynamic instabilities are likely generate vigorous turbulence and drive disk accretion. In this project, we will look at the gas dynamics induced by the vertical shear instability, which is one of the most promising hydrodynamic instability in PPDs. The methods can be either numerical simulations with the state-of-the-art astrophysical code Athena++, or analytical such as linear analysis, based on student interests. It can run into publications depending on studentâ€™s contribution. Keywords MHD, Turbulence, Astrophysical Disks, Linear analysis, Simulations References [1] Nelson R. P., Gressel O., Umurhan O. M., 2013, MNRAS, 435, 2610; Linear and non-linear evolution of the vertical shear instability in accretion discs  https://doi.org/10.1093/mnras/stt1475 [2] Lin M.-K., Youdin A. N., 2015, ApJ, 811, 17; Cooling Requirements for the Vertical Shear Instability in Protoplanetary Disks https://doi.org/10.1088/0004-637X/811/1/17 Prerequisite Skills Other Skills Used in the Project Python; Basic Knowledge in Fluid Dynamics

## Evidence synthesis in conservation science

 Project Title Evidence synthesis in conservation science Contact Name Alec Christie (Department of Zoology) and Qingyuan Zhao (Statistical Laboratory) Contact Email apc58@cam.ac.uk and qz280@cam.ac.uk Project Dates 8-10 weeks, starting around 1st July Suggested Deadline to contact project supervisor 15 March 2021 Brief Description of the Project With the drastic loss of biodiversity, we urgently need scientific evidence to inform the best ways to conserve wildlife. The scientific evidence base in conservation science is known to be messy, patchy, and made up of many poor quality study designs [1, 2]. This makes combining the results of different research studies to make reliable evidence-based recommendations a major methodological problem. Covid-19 has shown that these issues extend to other disciplines too, such as evidence-based medicine and healthcare, where uptake of mask-wearing by Western countries was arguably delayed by reluctance of scientists to consider observational studies and rely on limited evidence from randomised experiments [3]. Therefore, there is an urgent need to find robust methods to combine evidence from these different types of studies. Meta-analysis is a conventional method to combining study results that typically involves weighting individual studies by their sample size to obtain a pooled estimate of some effect of interest. More sophisticated meta-analyses may fit a linear mixed-effect model to account for heterogeneity of the studies. However, conventional meta-analysis rarely takes into account the design of studies and the quality of the evidence. This is a major problem in conservation science because there is a lot of variation in the quality of study designs used to test conservation interventions to conserve species [1,2]. To account for the quality of evidence, people often rely on a subjective risk-of-bias assessment (or 'critical appraisal') to exclude poorly designed studies and stratify the rest [4-6]. This manual approach is highly subjective and cumbersome, lacks scalability for future large-scale synthesis projects, and may have unintended consequences by excluding studies from evidence synthesis. Fortunately, it may be possible to "meta-analyse meta-analyses" to quantitatively evaluate the reliability of different study designs. In a recent paper [7], we set out a prototype hierarchical Bayesian model for this purpose. In principle, this model can be used to synthesise studies in the future. We want a student to help us refine and integrate this method with other types of 'bias-adjusted meta-analyses' to solve the issue of how to robustly combine the results of different experimental and observational studies. We want to trial and apply new methods on the Metadataset platform [8] that allows decision-makers to dynamically interrogate meta-analyses online. If successful, the outputs of this project proposal could be widely used to help decision-makers gain reliable evidence-based recommendations from scientific research and help to better conserve wildlife species in the future. Keywords Meta-analysis, evidence synthesis, experimental, observational, bias References Variation in quality of scientific evidence in conservation science: [1] Christie, A.P., Amano, T., Martin, P.A., Petrovan, S.O., Shackelford, G.E., Simmons, B.I., Smith, R.K., Williams, D.R., Wordley, C.F.R., Sutherland, W.J., 2020. The challenge of biased evidence in conservation. Conserv. Biol. cobi.13577. https://doi.org/10.1111/cobi.13577 [2] Christie, A.P., Amano, T., Martin, P.A., Petrovan, S.O., Shackelford, G.E., Simmons, B.I., Smith, R.K., Williams, D.R., Wordley, C.F.R., Sutherland, W.J., 2020. Poor availability of context-specific evidence hampers decision-making in conservation. Biol. Conserv. 248, 108666. https://doi.org/10.1016/j.biocon.2020.108666 Covid-19 issues in varying quality of evidence for face mask-wearing: [3] Royal Society 2020. Face masks and coverings for the general public: Behavioural knowledge, effectiveness of cloth coverings and public messaging. https://royalsociety.org/-/media/policy/projects/set-c/set-c-facemasks.p... Bias-adjusted meta-analysis: [4] Turner, R.M., Spiegelhalter, D.J., Smith, G.C.S., Thompson, S.G., 2009. Bias modelling in evidence synthesis. J. R. Stat. Soc. Ser. A (Statistics Soc. 172, 21-47. https://doi.org/10.1111/j.1467-985X.2008.00547.x [5] Stone, J.C., Glass, K., Munn, Z., Tugwell, P., Doi, S.A.R., 2020. Comparison of bias adjustment methods in meta-analysis suggests that quality effects modeling may have less limitations than other approaches. J. Clin. Epidemiol. 117, 36-45. https://doi.org/10.1016/j.jclinepi.2019.09.010 [6] Rhodes, K.M., Savovic, J., Elbers, R., Jones, H.E., Higgins, J.P.T., Sterne, J.A.C., Welton, N.J., Turner, R.M., 2020. Adjusting trial results for biases in meta-analysis: combining data-based evidence on bias with detailed trial assessment. J. R. Stat. Soc. Ser. A (Statistics Soc. 183, 193-209. https://doi.org/10.1111/rssa.12485 Prototype hierarchical Bayesian model for combining results of different study designs: [7] Christie, A.P., Abecasis, D., Adjeroud, M. et al. Quantifying and addressing the prevalence and bias of study designs in the environmental and social sciences. Nat Commun 11, 6377 (2020). https://doi.org/10.1038/s41467-020-20142-y [8] Metadataset platform: www.metadataset.com. Prerequisite Skills Familiarity with statistical modelling Other Skills Used in the Project Familiarity with statistical software (R or other languages). Interest in making a difference and working on the cutting edge of evidence synthesis.

## Droplets breakup: the capillary retraction of liquid filaments

 Project Title Droplets breakup: the capillary retraction of liquid filaments Contact Name Francesco Paolo Conto Contact Email fpc25@cam.ac.uk Project Dates 8 weeks during summer 2021 Suggested Deadline to contact project supervisor Brief Description of the Project Droplets formed from the breakup of liquid jets and capillary retracting filaments are not only fundamental phenomena in nature (the breakup of ocean spume is an example), but they are also relevant to a wide range of industrial applications: ink-jet printing, atomisation, spraying, microfluidics and particle technology are only some of the fields that rely on these dynamics. For this reason, the capillary retraction of liquid filaments (elongated drops) in a passive ambient fluid (air) is a classical problem that has fascinated such scientists as Lord Rayleigh, Taylor and Culick. It has been widely studied especially during the last two centuries through theoretical, numerical and experimental methods. At this time, the topic is particularly relevant to the formation of infectious droplets responsible for the spread of COVID-19. Saliva droplets are formed from the breakup of ligands both in the mouth through due to the movements of the tongue and lips while speaking, but also in the lungs with the collapsing and inflation of the smallest airways and alveoli. We have recently detected a new regime and a new breakup mode in Newtonian liquid filaments. Our work shows the importance of capillary waves interaction on the liquid-gas interface: our simple but effective model can serve as a new mechanism for producing and controlling microdroplets potentially improving the current design strategies in technologies such as ink-jet printing. This project is devoted to the study of the breakup of capillary retracting liquids. In particular, the pinch-off usually occurs through a dynamic capillary instability. The student will look at the thinning neck formed prior droplets break-off. The main aim of the project is to investigate and characterise surface tension and rheology effects (liquids of interest such as biofluids and inks are usually complex and/or non-Newtonian) on the dynamics. The student will work on analytical/numerical models which could extend the capabilities of our earlier work. Keywords Hydrodynamics, Droplets, Breakup, Capillary waves References https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/artic... https://www.nature.com/articles/s41598-019-51824-3 Prerequisite Skills Differential equations, Asymptotic methods, MATLAB/Maple Other Skills Used in the Project Fluid Dynamics, Image analysis

## Partial Differential Equations: Finite Elements vs. Deep Learning

 Project Title Partial Differential Equations: Finite Elements vs. Deep Learning Contact Name Tamara Grossmann, Jonas Latz and Carola-Bibiane Schönlieb Contact Email tg410@cam.ac.uk Project Dates 8 weeks during summer 2021 Suggested Deadline to contact project supervisor 8 March 2021 Brief Description of the Project Mathematical models are used throughout science and engineering to represent the behaviour of systems of interest. Such systems are, e.g., turbulent flow, heat conduction, or biological growth. Moreover, models are sometimes defined implicitly through a partial differential equation (PDE). This PDE usually then needs to be solved numerically to obtain a computer-based simulation of the system of interest. Over the past decades, finite element methods (FEM) have been popularly used to approach such simulations. Here, the PDE solution's underlying function space is being replaced by a finite-dimensional vector space on which the PDE is solved numerically. Representing complicated functions, however, is also the main objective of many modern machine learning models, such as deep neural networks. Thus, many scientists have recently replaced classical methods such as FEM by deep learning techniques for the approximation of PDE-based models. Those 'physics-informed neural networks' have produced very promising results throughout various applications. The aim of the project is therefore to conduct a comparative study concerning FEM and physics-informed neural networks. The summer student will get practical insights into numerical analysis of partial differential equations, as well as modern machine learning approaches. Keywords PDEs, physics-informed neural networks, deep learning, machine learning, finite element method, finite difference method, numerical analysis References - Lu et al. (2020): 'DeepXDE: a deep learning library for solving differential equations' https://arxiv.org/abs/1907.04502 - Higham et al. (2019): 'Deep Learning: An Introduction for Applied Mathematicians', SIAM Rev., 61(4), 860-891 https://epubs.siam.org/doi/pdf/10.1137/18M1165748 - Quarteroni and Valli (1994): 'Numerical Approximation of Partial Differential Equation', Springer, Berlin, Heidelberg - Braess (2007): 'Finite Elements', 3rd edition, Cambridge University Press. Prerequisite Skills Numerical Analysis, programming experience (ideally in Python or C/C++) Other Skills Used in the Project Prior knowledge in finite element analysis and deep learning is useful but not required.

## Data-driven image priors for inverse problems

 Project Title Data-driven image priors for inverse problems Contact Name Subhadip Mukherjee (jointly with Prof. Carola-Bibiane Schönlieb) Contact Email sm2467@cam.ac.uk Project Dates 8-10 weeks during summer 2021 Suggested Deadline to contact project supervisor 28 February 2021 Brief Description of the Project Inverse problems deal with reconstructing an unknown model parameter from indirect and noisy measurements and arise in virtually all imaging applications. In recent years, surge in deep learning research has led to considerable attention to data-driven techniques for solving inverse problems, primarily because of their superior performance as compared to classical model-based approaches. The goal of the project is to develop and analyze data-driven image-priors for image reconstruction problems, especially in the context of medical imaging. Specifically, we would like to leverage ideas from latent-variable-based generative modeling (such as generative adversarial networks (GANs), variational auto-encoders (VAEs), etc.) to learn the underlying image distribution and use it subsequently to regularize the variational reconstruction framework. The project aims to better understand the computational and theoretical properties of data-driven methods for inverse problems; and develop novel, efficient, and flexible learning protocols and reconstruction algorithms. Keywords Inverse problems, generative modeling References 1. Adversarial regularization: https://arxiv.org/abs/1805.11572 2. Regularization by denoising: https://arxiv.org/abs/1806.02296 3. GAN-prior for inverse problems: https://arxiv.org/abs/1802.08406 4. Network Tikhonov (NETT): https://arxiv.org/pdf/1803.00092.pdf 5. Sampling from the posterior distribution using a GAN: https://arxiv.org/abs/1811.05910 Prerequisite Skills Strong background in programming (preferably in Python), familiarity with at least one deep learning framework (preferably PyTorch) Other Skills Used in the Project Elementary knowledge of linear algebra, matrix computations, probability theory, optimization, and functional analysis

## Optimal portfolio construction with low frequency data

 Project Title Optimal portfolio construction with low frequency data Contact Name Parley Ruogu Yang Students who are interested in applying shall email Parley their CV and a sample of their recent python script. The script is preferred to be, though not necessarily, related to quantitative finance. Python codes related to other relevant fields, e.g. statistical modelling, data analysis are also welcomed. Contact Email ry266@cam.ac.uk Project Dates 8 weeks in the summer, depending on student's interests and coding abilities Suggested Deadline to contact project supervisor 5 March 2021 Brief Description of the Project We shall look at optimal combination of portfolio --- this could be, for example, hedged long on US equities, systematic options trading, macroeconomic predictions and forecasts and etc. We focus on low-frequency data --- data that are available mostly in public and are updated on a daily basis. This means that we assume the portfolio manager to trade at most once a day, and by restricting so, we hope that an optimal strategy would be rather wise and robust. Click here to watch a video presentation about the project. Keywords Finance, portfolio optimisation, financial time series, low-frequency data, asset management, stock markets, strategic investment, algorithmic trading References Prerequisite Skills Key requirements for the students: 1. Passionate about strategic investment and portfolio construction. Knowledge in finance is preferred but not required. 2. Ability to handle mathematical and statistical tasks. A Maths Part-IB level statistical understanding or Economics Part-IIA level econometric understanding or equivalent is preferred, or if not satisfied, a passion and strive to enrich the understanding to the mentioned level is required. 3. Ability to code in Python at an intermediate level. Previous experience in Python related projects is preferred but not required. A passion to learn further on Python is required. Ability to engage with standard data science packages in Python is required. Other Skills Used in the Project See above

## Hyperbolic equations on black hole spacetimes

 Project Title Hyperbolic equations on black hole spacetimes Contact Name Zoe Wyatt, Claude Warnick Contact Email zoe.wyatt@maths.cam.ac.uk Project Dates 8 weeks Suggested Deadline to contact project supervisor Brief Description of the Project Many interesting properties about Einstein's theory of gravity have been discovered by studying solutions to geometric wave equations on curved spacetime backgrounds. In this project we will look at the Klein-Gordon equation on a rotating black hole spacetime. In the sub-extremal case, where the angular momentum (per unit mass) is strictly less than the black hole mass, it is known that there exist smooth, finite energy, and exponentially growing solutions to the Klein-Gordon equation (https://arxiv.org/abs/1302.3448). This project will seek to find an equivalent result for the extremal case, where the black hole angular momentum is equal to its mass. The project is best suited to a finishing Part II student, but may also be accessible to a finishing Part IB student. Funding for this project is secured, a bursary funding application is not required. Keywords PDEs, Mathematical Analysis, General Relativity References https://arxiv.org/abs/1302.3448 https://arxiv.org/abs/1910.02854 Prerequisite Skills Essential courses: Methods, Analysis and Topology, Complex Analysis and/or Complex Methods. Other Skills Used in the Project General Relativity and quantum mechanics are both useful but not essential.

## Digital quantum simulations of quantum theories

 Project Title Digital quantum simulations of quantum theories Contact Name Bipasha Chakraborty Contact Email bc335@cam.ac.uk Project Dates Two months over summer (online) Suggested Deadline to contact project supervisor ASAP, no strict deadline but preferably by 14th March to meet funding application deadline of 19th March Brief Description of the Project This project proposes to use existing novel algorithms and develop new ones for digital quantum simulations of quantum theories such as Schwinger Model (quantum electrodynamics) or Gross-Neveu model. Both are prototype models for testing futuristic algorithms for next generation computing for more complicated gauge theories like Quantum Chromodynamics with is tremendously important to understand the Standard Model of particle physics. Keywords Quantum simulation, Schwinger Model, Gross-Neveu model References https://arxiv.org/pdf/2001.00485.pdf Prerequisite Skills Advanced Quantum Mechanics, Python Programming Other Skills Used in the Project Quantum Computation or Quantum Information Science, Quantum Field Theory (although none of these is necessary as a pre-requisite)

## Quantum Machine Learning & deep learning for particle physics

 Project Title Quantum Machine Learning & deep learning for particle physics Contact Name Bipasha Chakraborty Contact Email bc335@cam.ac.uk Project Dates Two months over the summer (online) Suggested Deadline to contact project supervisor ASAP, no strict deadline but preferably by 14th March to meet funding application deadline of 19th March Brief Description of the Project This project would test and apply deep learning based on artificial neural networks for classical and quantum computation of quantum theories. Lattice QCD is a successful framework for handing Quantum Chromodynamics where we use Monte-Carlo methods for classical computation. The use of machine learning to handle huge amount of data or reduce the computational challenge is inspiring. Similarly, I propose to test neural networks in the quantum computation of some prototype models for Quantum Chromodynamics for developing futuristic algorithms. Keywords Machine learning, Neural Networks, Deep Learning, Quantum Computation References https://arxiv.org/abs/1807.05971 https://arxiv.org/abs/1801.05784 https://www.nature.com/articles/s41467-020-14454-2 Prerequisite Skills Quantum Mechanics, Python programming Other Skills Used in the Project Machine Learning

## Developing a tool for automated quality triaging of COVID-19 chest X-rays

 Project Title Developing a tool for automated quality triaging of COVID-19 chest X-rays Contact Name Mike Roberts Contact Email mr808@cam.ac.uk Project Dates 8 weeks in the summer Suggested Deadline to contact project supervisor Brief Description of the Project "Garbage in, garbage out" is a fundamental philosophy for machine learning. In our COVID-19 collaboration we have tens of thousands of chest X-ray images which require checking manually by a radiologist to ensure they are of sufficient quality to use in our machine learning algorithms. The aim of this project will be to develop a tool for automated quality triaging of the images, to highlight unacceptable images early and remove them from our dataset. This will be very useful to our project and to the community. The goal of the project is to take several algorithms developed by the collaboration for detecting particular issues in the chest x-ray images (inversion, rotation, contrast issues, artefacts etc) and consolidate them into one simple software package. This will also require developing other new tools to detect other issues with the images. Success in this project would be (a) training the algorithm using a large radiologist annotated dataset, (b) validating the algorithm on another large radiologist annotated dataset and (c) prospectively using the tool on several large unannotated datasets. The tool will be released open source and a publication describing the tool development is also very likely. Keywords Chest X-ray, COVID-19, machine learning, imaging, quality control References Prerequisite Skills Image processing, basic machine learning, Python/C++ Other Skills Used in the Project