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This is a list of projects hosted by the Cambridge Mathematics Placements site. Projects should be aimed at students working in the Faculty of Mathematics. To remove a listing, please contact CMP. If a project title has a line through it, that project has now been filled.


Certified Quantum Algorithms

Contact Name Prof. Lawrence Paulson and Dr. Anthony Bordg
Contact Email apdb3@cam.ac.uk
Company Name Computer Laboratory
Address University of Cambridge 15 JJ Thomson Avenue Cambridge CB3 0FD
Period of the Project Summer 2019 (flexible)
Project Open to Undergraduates, Part III students, PhD students
Deadline to Register Interest February 22 2019
Brief Description of the Project Interactive proof assistants such as Isabelle and Coq have attracted prominence recently because of their use in formalising significant results in mathematics such as the Kepler conjecture and the odd order theorem. In the former case, the original mathematical proof was distrusted because of its reliance on a lengthy computation, as in the more familiar four colour theorem. The latter result was formalised to demonstrate the power of the technology. With more and more intricate proofs in mathematics and computer science, proof assistants become appealing. In this project we propose to formalise some quantum algorithms of the candidate's choice, extending an existing basic library. Other topics suitable for formalisation during this project include the mathematical background needed for general relativity (tensor analysis, differential geometry ...) and financial mathematics, depending on the candidate's taste and skills. The candidate will receive training in the use of the Isabelle proof assistant.
Skills Required Knowledge of the requisite mathematics (for instance linear algebra in the case of quantum computing) and basic computer literacy. No knowledge of Isabelle or quantum computing required.
Skills Desired Prior familiarity with any proof assistant would be valuable.

 

Exploiting dynamic information in 3d/4d/5d fluorescence microscopy

Contact Name Leila Muresan
Contact Email lam94@cam.ac.uk
Company Name Dept. of Physiology, Development and Neuroscience /CAIC
Address Anatomy Building, Downing site, CB2 3DY Cambridge
Period of the Project Start: second half of June
Project Open to Part III students, PhD students
Deadline to Register Interest  
Brief Description of the Project With the advent of the light sheet microscopy robust image analysis becomes crucial in unlocking the potential of this technique. This project will focus on two aspects of microscopy image analysis: improving tracking cells by better detection of cell division events and analysis of the obtained cell trajectories. The newly designed approaches will be tested on images of growing embryos (a collaboration between CAIC and Steventon lab, Dept. of Genetics). We are open for both explorative or well-defined project format (opting for a machine learning based approach for the latter). Reference: Amat et al., Efficient processing and analysis of large-scale light-sheet microscopy data, Nature Protocols, volume 10, pages 1679–1696 (2015)
Skills Required programming skills
Skills Desired statistical analysis

 

Inverse scattering problem for microswimmers in an obstacle lattice

Contact Name Dr Otti Croze
Contact Email oac24@cam.ac.uk
Company Name Cavendish Laboratory (Physics)
Address oac24@cam.ac.uk
Period of the Project 8 weeks, to be agreed with the student.
Project Open to Undergraduates, Part III students, PhD students
Deadline to Register Interest February 22
Brief Description of the Project Microswimmers obey peculiar rules for scattering off surfaces and obstacles. For example, swimming bacteria slide along the surface of a circular obstacle before leaving its surface, in contrast with the specular reflection of a classical Lorentz gas particle. These different scattering rules translate into marked differences in the predicted diffusive transport in a lattice of obstacles, as we showed in a recent paper [https://arxiv.org/abs/1807.04117]. We did this by analyzing a direct scattering problem: particle trajectories were inferred based on a known scattering rule (sliding around an obstacle by a given central angle). In this project, we will tackle the inverse scattering problem of deriving scattering rules from particle trajectories, without information on scattering (black box collisions). The ultimate goal of this research is to infer scattering rules from experimental trajectories, without the need to analyse microscopic scattering events in detail. The project will: 1) develop mathematical and computational tools (including machine learning) to solve the inverse scattering problem; 2) test the method to trajectories using simulations with known scattering; 3) if possible, apply the method to real experimental data of bacteria swimming in microfluidic lattices of obstacles. The project is open-ended.
Skills Required Strong analytical modelling and programming skills.
Skills Desired Knowledge or strong interest inverse problems, stochastic dynamics, active matter

 

Modeling Drug Toxicology

Contact Name Rahuman Sheriff
Contact Email sheriff@ebi.ac.uk
Company Name European Bioinformatics Institute (EMBL-EBI)
Address Wellcome Trust Genome Campus, Hinxton, Cambridge CB10 1SD
Period of the Project minimun 8 weeks
Project Open to Part III students, PhD students
Deadline to Register Interest Until filled
Brief Description of the Project We have an opening for an internship position in the BioModels Team (www.ebi.ac.uk/biomodels) , which is a part of Molecular Systems Cluster at European BioInformatics Institute, European Molecular Biology Laboratory (EMBL-EBI), UK. This project is being carried out within the EU IMI Translational Quantitative Systems Toxicology (TransQST) consortium (ww.transqst.org) in collaboration with University of Vienna, Austria. The TransQST consortium is an academic - industrial partnership that aims to improve human safety during clinical trials by developing novel systems toxicology models. The goal of this interdisciplinary translational research collaboration is to develop systems toxicology models to predict toxicity of drugs in liver/kidney/heart and gut in humans based on non-clinical datasets. The outcome of the internship project will be initially shared with the consortium members which includes leading pharmaceutical companies and eventually published in a scientific journal as a part of project. We have developed a computational approach (unpublished) that employs network propagation to predict the biological pathways that are affected by a known drug or compound. The result of our approach provides a drug effect on pathway (DEP) score that describes how likely a drug or compound affects a given biochemical pathway in human. We are investigating and establishing the relationship between the DEP and adverse / toxic effect of drug and validating our approach by comparing it with existing datasets. We would like to classify all potential drugs and compounds by their effect on biochemical pathways and identify DEP signatures that can predict the adverse effect and toxicity of the drugs in order to facilitate drug discovery and non-clinical to clinical translation. In the first part of the internship project we would like to cluster all the clinical candidate compounds and approved drugs available in the market based on their DEP score and identify patterns. In the second part, we would like investigate the statistical association of these clusters of the drugs with clinically relevant parameters such drug class, disease prescribed for, toxicity and adverse effects, etc  by retrieving these information from public repositories including chEMBL, FDA, etc.
Skills Required The candidate is expected to have sufficient background in statistical analysis of the data including supervised and unsupervised clustering to explore the multidimensional dataset and visualise emerging patterns. Good programming skill in R, MatLab, Python, or any similar language is expected.
Skills Desired Some basic knowledge on computationally retrieving big data from databases using APIs is desirable.

 

Targeted Curation of Systems Biology Models in Immmuno-Oncology

Contact Name Rahuman Sheriff
Contact Email sheriff@ebi.ac.uk
Company Name European Bioinformatics Institute (EMBL-EBI)
Address Wellcome Trust Genome Campus, Cambridge, CB10 1SD PS: Free transport facility to campus available
Period of the Project minimun 8 weeks
Project Open to Undergraduates, Part III students, PhD students
Deadline to Register Interest Until filled
Brief Description of the Project Recently, mathematical modeling approaches have become increasingly common in cell biology research. Such a quantitative approach supports cell signalling and cancer research by helping to elucidate mechanisms and provide predictions that can be validated (Altrock et al. 2015 Nat Rev Cancer). Current trend suggests that many experimental researchers use mathematical models and simulations to emphasize the validity of their hypothesis and often present them as the last figure of their publications. BioModels (https://www.ebi.ac.uk/biomodels/) is one of the world leading reposition of mathematical models of biological processes hosted at EMBL-EBI (Chelliah et al. 2015, Nuc Acid Res, Stanford et al. 2015, Mol Sys Bio). In the proposed curation internship, you will work with the curation team at BioModels and will learn to curate and annotate mathematical models from published scientific literature and deposit them to BioModels. You will further learn to use simulation software such as COPASI to reproduce the simulation results from the reference publication, and learn to encode models in standard modelling formats such as Systems Biology Markup Language (SBML). Specifically, you will work on targeted curation of literature-based models in immuno-oncology. Tumor immune interaction has become a key area of research due to promising success in cancer immunotherapy. Immunosuppression is the key mechanism through which cancer escapes immune targeting and involves a complex interplay between cancer and immune cells. Cancer immunotherapy involves targeting or use of immune system components (including antibodies, cancer vaccines, T cells, etc.,) to kill tumour cells and supress cancer progression. The complex interaction between immune cells and cancer cells are modelled to study systemic behavior and develop new immune therapy strategies for cancer. There are several models proposed in recent times and new models are coming frequently (Arulraj and Barik, PLOSone 2018 ; Ruby Kim, SPORA 2018 ; Kaitao Li, JBC 2017) which makes curation much needed and interesting. You will gain strong exposure to cancer immunology and cancer biology (in general) by participating in the targeted curation of these models. In the past, BioModels targeted curation of literature-based models on diabetes and neurodegenerative diseases has resulted in scientific publications; Lloret-Villas et al. 2017 and Ajmera et al. 2013, respectively. The cancer immunotherapy models you will curate under the supervision of the curation team will be published in the BioModels repository, disseminated to a broader scientific community and furthermore can be potentially summarized into a scientific publication. Thus, in this internship, in addition to professional growth, you will also contribute to the growth of the open access BioModels repository, which has around 20,000 distinct users per month.
Skills Required knowledge in differential equations and basic cell biology
Skills Desired Coding in any programming language.

 

How stable are virtual sounds when listeners move their heads?

Contact Name Alan Archer-Boyd
Contact Email alan.archer-boyd@mrc-cbu.cam.ac.uk
Company Name Medical Research Council Cognition and Brain Sciences Unit
Address 15 Chaucer Rd Cambridge CB2 7EF United Kingdom
Period of the Project 8 weeks
Project Open to Undergraduates, Part III students
Deadline to Register Interest February 22
Brief Description of the Project The increased availability of motion-tracking systems, real-time audio signal processing, and the development of augmented reality (AR) devices has led to renewed interest in how head movement affects the way listeners perceive sound sources. One question has been how best to maximize the perceived stability of virtual sound sources during listener head movements, and what factors may contribute to that stability. The aim of the proposed research project is to evaluate the effect of interaural cross-correlation (i.e. how spatially narrow or wide a sound is perceived by a listener) on the stability of virtual stationary sound sources during listener head movements. The project will build directly upon previous recently published work (Freeman et al., 2017). The results could be useful for future AR devices, hearables, and situation-aware hearing aids. The project will involve some experimental design and calibration, data collection, and analysis, and will be supervised by a post-doc and the group leader. The student will use a Nintendo Wiimote-based motion-tracking system, combined with a MATLAB/C-based real-time audio system. This is an exciting opportunity for a keen student to apply their theoretical knowledge to a new area of auditory research, and to experience being part of a research group at the MRC CBU. Freeman, T.C.A., Culling, J.F., Akeroyd, M.A., Brimijoin, W.O. (2017). “Auditory compensation for head rotation is incomplete. Journal of experimental psychology: human perception and performance 43 (2), p371
Skills Required Basic MATLAB and/or Python programming skills Interest in psychoacoustics, experimental methods, and/or motion tracking
Skills Desired Real-time signal processing Statistical modelling in R or Bayesian statistics in JASP or similar Knowledge of psychoacoustics, experimental methods, and/or motion tracking

 

Learning wavelet representations

Contact Name Austen Lamacraft
Contact Email al200@cam.ac.uk
Company Name Cavendish Lab / Department of Physics
Address Cavendish
Period of the Project 8 weeks
Project Open to Part III students
Deadline to Register Interest  
Brief Description of the Project Wavelets offer a multiscale representation of data (audio, images, etc.) that combines the merits of the real space and Fourier representations. Many naturally occurring signals are sparse in the wavelet basis, which therefore provides a natural starting point for a host of tasks including compression and classification. The goal of this project is to use recently developed deep learning architectures to learn wavelet representations that are adapted to particular datasets, and in this way improve upon our current handcrafted bases.
Skills Required Python programming
Skills Desired Familiarity with a deep learning framework (TensorFlow, PyTorch) would be a bonus.

 

Music and Mathematics: Formal Methods in Style Detection

Contact Name Francis Knights
Contact Email fk240@cam.ac.uk
Company Name Faculty of Music
Address Fitzwilliam College
Period of the Project 8 weeks
Project Open to Undergraduates, Master's (Part III) students
Deadline to Register Interest February 22
Brief Description of the Project

The project focuses on developing mathematical methods to classify and establish authorship of musical material. From a practical perspective we have developed and applied several tools from statistics, graph theory, information theory and machine learning and implemented them computationally. We can then classify musical works depending on their stylistic similarity, and we have to date addressed many specific musicological questions in the FMM project (see https://formal-methods-in-musicology.webnode.com). Additionally, the results have been of interest in algorithmic composition and generative music as well as in a more theoretical approach to the evolution of style and musical analysis. Currently, we are exploring three new projects and are looking for interested students for each one:

1) Stylistic differences in plainchant and other single-line music. The aim of the project is to apply the classification methods developed so far in order to understand different traditions in this repertoire. Specifically, the importance of ornaments and differences in notation.

2) Analysis and stylistic discrimination in the music of the Beatles. The idea is to explore stylistic classification and authorship emerging in the music by the Beatles. In particular, whether stylistic signatures can be detected in order to understand the different contributions from Lennon and McCartney in the many co-authored songs they wrote.

3) Stylistic trends in improvised music. This project deals with stylistic differences in improvised music, as opposed to composed music. In principle the analysis will involve studying Indian music as an important representative of such an improvisational tradition, but other traditions such as jazz can also be approached. The students are expected to work on the theoretical (developing new methods and ideas) as well as on the computational aspects of the project. Some previous musical training would be a real advantage.

Skills Required  
Skills Desired  

Moment-based inference for stochastic models

Contact Name Olivier Restif
Contact Email or226@cam.ac.uk
Company Name University of Cambridge
Address Department of Veterinary Medicine Madingley Road
Period of the Project Flexible
Project Open to Undergraduates, Part III students
Deadline to Register Interest February 22
Brief Description of the Project We recently developed a likelihood-free and simulation-free algorithm to estimate the parameters of linear, continuous-time Markovian processes, using experimental data on bacterial infection dynamics [1]. The goal of this project is to generalise the algorithm to a broader class of models. Specifically, we consider birth-death-migration processes in a network of compartments representing the organs of an infected patient. [1] https://doi.org/10.1371/journal.pcbi.1005841
Skills Required Stochastic processes, scientific programming.
Skills Desired Statistics, R programming

 

Image processing methods for light microscopy

Contact Name Jerome Boulanger
Contact Email jeromeb@mrc-lmb.cam.ac.uk
Company Name MRC-LMB
Address MRC Laboratory of Molecular Biology Francis Crick Avenue, Cambridge Biomedical Campus Cambridge CB2 0QH, UK
Period of the Project June to September (flexible)
Project Open to Part III students, PhD students
Deadline to Register Interest February 22
Brief Description of the Project Computational and mathematical methods offer new opportunities to enhance and screen optical microscopy images unlocking numerous applications in life science. Resolution, sensitivity and speed are the key parameters in this context. We are interested in image reconstruction approaches in low light conditions resulting in extreme signal-to-noise ratio. The aim is to improve the sensitivity and imaging speed of microscopes by designing efficient reconstruction methods using new regularization paradigms. In addition to improving the acquired images, the extraction of quantitative information is crucial for the understanding of the observed biological processes. Advanced methods for image segmentation and classification using convolutional network and/or their sparse modeling counterpart have shown a dramatic increase of performance in the recent years. We are interested in reformulating several image analysis problems up to the acquisition protocols to further extend our capacity to extract quantitative information. Finally, this project lies in the realm of image processing methods for light microscopy and is rather open.
Skills Required  
Skills Desired image processing, optimisation

 

Modeling DNA Evolution

Contact Name Nick Goldman
Contact Email goldman@ebi.ac.uk
Company Name EMBL-European Bioinformatics Institute
Address Welcome Trust Genome Center Hinxton, Cambridge CB10 1SD
Period of the Project 8 weeks
Project Open to Undergraduates, Part III students
Deadline to Register Interest  
Brief Description of the Project

Group: at the EMBL-EBI Goldman group, one of the main interests is the study and modelling of DNA evolution. In particular, we aim to provide more accurate and efficient methods to reconstruct the evolutionary history of animals, plants, bacteria, viruses, and tumors, from DNA sequence data.

Project: The student will simulate DNA evolution under one of several scenarios (protein coding sequence evolution in the presence of complex mutations, pathogen evolution within an outbreak, geographic migration of animals and plants, or bacterial recombination); the student will then test the accuracy and efficiency of existing methods in reconstructing the evolutionary history from the simulated DNA data.

The student will acquire familiarity with models used in molecular evolution and with basic concepts in genetics, will learn how to write basic programs, and how to use scientific software for statistical inference and simulation in phylogenetics, sequence alignment, epidemiology and/or phylogeography. **Ambitious students might aim at developing new simulation algorithms or inference methods.

Skills Required Basic ability to code in a programming laguage (preferentially C, Python or Java) is required.
Skills Desired Basic knowledge of probability, statistics, linear algebra, Markov models and differential equations are a plus. Familiarity with either population genetic models, epidemiological models, or phylogenetic models is also useful.

 

Finsler optimisation on matrix manifolds

Contact Name Dr Cyrus Mostajeran & Professor Rodolphe Sepulchre
Contact Email csm54@cam.ac.uk
Company Name Engineering Department
Address Engineering Dept, Trumpington St, Cambridge CB2 1PZ
Period of the Project 8-10 weeks from late June to end of September
Project Open to Part III students, PhD students
Deadline to Register Interest  
Brief Description of the Project An important special class of constraints in optimisation theory are geometric constraints, which require that the solution of the optimisation problem lie on a nonlinear manifold. Such problems can be recast as optimisation on manifolds, whereby the algorithms are made to be intrinsic in the sense that they do not rely on a characterisation of the constrained search space by an embedding in a larger linear space. Optimisation on matrix manifolds finds applications to many areas of engineering including high-dimensional imaging, computer vision, machine learning, and statistical analysis. The project will focus on one of several possible topics in optimisation on manifolds utilizing a Finslerian structure instead of the more common Riemannian structure. The specific project will be tailored to the applicant's particular strengths and interests. The shift from Riemannian to Finslerian geometry is achieved when the smoothly varying inner product associated to the manifold is replaced by a smoothly varying norm. This generalisation can be quite powerful as norms that do not correspond to an inner product are often more suited to particular tasks. Finsler optimisation on manifolds is a relatively young and exciting area of research which offers the promise of considerable improvements in performance and convergence properties of existing Riemannian optimisation algorithms. The aim of the project will be to arrive at algorithms with a favourable trade-off between scalability, accuracy, robustness, and interpretability. References: [1] Absil, P-A., Robert Mahony, and Rodolphe Sepulchre. Optimization algorithms on matrix manifolds. Princeton University Press, 2009 and [2] https://www.manopt.org.
Skills Required Good programming skills (in Matlab, Python, or C++), strong interest in numerical analysis and optimisation
Skills Desired Differential geometry

 

Integrating hormonal and mechanical signals to regulate plant cellular growth and morphogenesis

Contact Name Dr. Alexander Jones
Contact Email Alexander.Jones@slcu.cam.ac.uk
Company Name Sainsbury Laboratory, University of Cambridge
Address 47 Bateman Street, Sainsbury Laboratory, Cambridge CB2 1LR
Period of the Project Eight weeks (Longer internships possible)
Project Open to Part III
Deadline to Register Interest  
Brief Description of the Project Differential growth is a key feature that governs plant growth and behaviour throughout their life-cycle. The ability of plant organs to germinate, curl, bend, twist, sense gravity, and respond to temperature and light fluctuations all involves the differential growth. Plant hormone auxin and plant microtubule (MT) cytoskeleton plays a fundamental role in determining differential growth rates and growth directions. For example, the plant hormone auxin and tensile stress patterns (i.e. wall tension within cell walls) can reorient MTs and influence growth patterns during plant development. A significant gap in our understanding is how a cell integrates differential effects of auxin and the direction of wall tension on MT reorientation and growth patterns. To address this, we are utilizing the differential cell growth associated with apical hook opening (part of how a plant seedling responds to light), and combining live cell imaging with computational models to generate and test emergent hypotheses. Live cell imaging is performed that captures dynamic cellular growth, auxin response output, and MT reorientation patterns during differential growth, and these data have indicated that MT reorientation patterns tightly correlate with spatiotemporal patterns of auxin response output and cellular growth. The goal of the project will be to develop and analyse mathematical models and finite element simulations representing alternative hypotheses that predicts differential effect of auxin on local MT reorientation patterns. To establish causal relationships, genetic mutants defective in MT reorientation and auxin signalling/metabolism are being analysed, and this data will also become available to the student to incorporate into their models and simulations. The student will work closely with a postdoc (Ankit Walia) who has generated all the cell biological data and will experimentally test the key predictions of the models. This iterative combination of experiments and modelling will help to gain mechanistic insights into how cells integrate hormonal and mechanical signals to regulate growth and morphogenesis. These mechanisms are of broad interest, both in plant and human biology. Note: Simulation analyzing code (in C++ and custom made python scripts) utilized by our current collaborators and another laboratory may become available as a starting resource so that the student can build their own simulation code and resources for the project.
Skills Required Programming in C++ code and Python for data analysis.
Skills Desired  

Constrained optimisation Schur-Convex functions and applications

Contact Name Paul Kattuman
Contact Email p.kattuman@jbs.cam.ac.uk
Company Name Judge Business School
Address Judge Business School, Cambridge CB2 1AG
Period of the Project 8 weeks from late June
Project Open to Undergraduates, Part III
Deadline to Register Interest Mid-April
Brief Description of the Project This is a well defined project related to measurement of inequality, concentration, risk and similar aspects of dispersion. Many popular indices are in common use for the measurement of such features. All share the feature that they are Schur-convex functions. The differ in the weights they place on different parts of the distribution and hence conclusions on the extent of inequality etc. and whether it has increased or decreased can readily appear to be dramatically contradictory. This project is focussed on applying sharp results that have been obtained on maximising/minimising any schur-convex function, subject to constraints defined in terms of the value of any other schur-convex function, to define upper and lower bounds on inequality measured by any index, conditioned on the value of any other index. A number of applications follow.
Skills Required Statistical theory, computing (R)
Skills Desired