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Summer Research in Mathematics

This is a list of CMS summer research projects suggested by supervisors. To remove a listing, please contact Jacob Rasmussen

Statistical physics of positive and negative chemotaxis in biofilm bacteria

Project Title Statistical physics of positive and negative chemotaxis in biofilm bacteria
Contact Name Nuno Miguel Oliveira
Contact Email nmdso2@cam.ac.uk
Period of the Project No restrictions
Brief Description of the Project

Bacteria often form surface-attached communities called biofilms that are critical for their impacts on us, including bioremediation, chronic infections and antibiotic resistance. Biofilm cells are typically pictured as sluggish or even sessile when compared to their free-swimming counterparts, but we have recently shown that biofilm bacteria can effectively climb gradients of nutrients and other canonical chemoattractants (chemotaxis) with unprecedented precision.

While positive chemotaxis was already reported in biofilm bacteria, the statistical physics of this novel phenomenon has not been explored in detail. Moreover, we still do not know if and how biofilm cells avoid chemicals (i.e. negative chemotaxis). Our recent data show that biofilm bacteria actively move away from their own secreted compounds, and that the motility strategy cells employ is different the one used for positive chemotaxis. This summer project aims to compare and contrast positive and negative chemotaxis in biofilm bacteria using the tools of statistical physics. The ideal candidate should have experience with image analysis (particularly particle-tracking software) and statistical physics, or willingness to learn these skills. Students do not need to have any “wet lab” experience and can focus on quantitative analysis and mathematical modelling if they prefer. However, students wishing to obtain their own experimental data and further explore the system are welcome as well. References:

- Cremer J, et al. (2019) Chemotaxis as a navigation strategy to boost range expansion. Nature (575):658-663.

- Hall-Stoodley L, Costerton JW, Stoodley P (2004) Bacterial biofilms: From the natural environment to infectious diseases. Nat Rev Microbiol (2):95-108.

- Oliveira NM, Foster KR, Durham WM (2016) Single-cell twitching chemotaxis in developing biofilms. Proc. Natl. Acad. Sci. USA 23(113):6532-6537.

- Wadhams GH, Armitage JP (2004) Making sense of it all: bacterial chemotaxis. Nat Rev Mol Cell Biol (5):1024-1037.

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Microbial interactions, antibiotic gradients and the evolution of resistance

Project Title Microbial interactions, antibiotic gradients and the evolution of resistance
Contact Name Nuno Miguel Oliveira
Contact Email nmdso2@cam.ac.uk
Period of the Project No restrictions
Brief Description of the Project

The evolution and spread of antimicrobial resistance is currently a major public concern as it threats global health, food security and socio-economic development. While there is a large literature on how bacteria respond to antimicrobials, the vast majority of these works ignores two key aspects of environmental and clinical settings. Firstly, current literature typically ignores the fact that concentration gradients of antimicrobial drugs always emerge in natural conditions, and we know little about how these gradients affect the evolution of resistance. Secondly, bacteria often live in complex microbial communities where they meet and interact with a plethora of bacterial species, including antibiotic-producing species. Indeed, we often forget that most antibiotics we use in the clinic and animal husbandry are natural compounds produced by microbes such as bacteria and fungi to inhibit their neighbouring competitors. We have recently shown that bacteria actively control their motility in antibiotic gradients. Unexpectedly, bacteria actively move towards increasing concentrations of antibiotics–both microfluidic gradients and natural gradients produced by neighbouring bacteria–and are able to tolerate concentrations that would be lethal in homogenous environments.

This summer project aims to develop mathematical models that describe quantitatively bacterial adaptation in antibiotic gradients, namely those that emerge when bacteria grow side-by-side with antibiotic-producing species. It is suggested that students start by building upon the so-called “staircase model” (stochastic lattice model), which can provide adaptation rates of moving cells in concentration gradients of drugs. However, students are encouraged to consider other mathematical frameworks, namely systems of partial differential equations, to model explicitly the evolution of antibiotic resistance via microbial interactions.

References:

- Abruden M, et al. (2015) Socially mediated induction and suppression of antibiotics during bacterial coexistence. Proc. Natl. Acad. Sci. USA 112:11054-11059.

- Be’er A, et al. (2010) Lethal protein produced in response to competition between sibling bacterial colonies. Proc. Natl. Acad. Sci. USA 107(14):6258-6263.

- Greulich P, Waclaw B, Allen RJ (2012) Mutational pathway determines whether drug gradients accelerate evolution of drug-resistant cells. Phy Rev Lett 109:088101.

- Hermsen R, Deris JB, Hwa T (2012) On the rapidity of antibiotic resistance evolution facilitated by a concentration gradient. Proc. Natl. Acad. Sci. USA 109(27):10775-10780.

- Kelsic E, et al. (2015) Counteraction of antibiotic production and degradation stabilizes microbial communities. Nature 521:516-519.

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Biased motility of ameboid cells in shear stress gradients

Project Title Biased motility of ameboid cells in shear stress gradients
Contact Name Nuno Miguel Oliveira
Contact Email nmdso2@cam.ac.uk
Period of the Project No restrictions
Brief Description of the Project

Many types of cells, from bacterial to mammalian, can actively move and bias their motion as a response to physicochemical stimuli, and it is now well appreciated that such directed cell migration is fundamental to several pathophysiological processes, including wound healing, immune cell trafficking, and cancer metastasis. Accordingly, understanding how physical and chemical cues guide cells, is a major research topic in cell biology and related fields.

This project aims to explore how ameboid cells control their motion in well-defined and stable gradients of shear stress. Ameboid cells such as immune and cancer cells are often submerged and experience changes in fluid shear, but we still do not know how exactly they respond to temporal and spatial gradients of shear stress. This project will address this lack of understanding. The summer student will build on our preliminary data, which shows that ameboid cells actively move away from low shear locations, and that this response is conditional on previous shear conditions that cells have experienced before, suggesting an unexpected form of shear stress memory. The ideal candidate should have experience with image analysis (particularly particle-tracking software) and fluid physics, or willingness to learn these skills. Students do not need to have any “wet lab” experience and can focus on quantitative analysis and mathematical modelling if they prefer. However, students wishing to obtain their own experimental data and further explore the system are welcome as well.

References:

- Artemenko Y, et al. (2016) Chemical and mechanical stimuli act on common signal transduction and cytoskeletal networks. Proc. Natl. Acad. Sci. USA 113(47):E7500-E7509.

- Décave E, et al. (2003) Shear flow-induced motility of Dictyostelium discoideum cells on solid substrate. J Cell Sci 116:4331-4343.

- Fache S, et al. (2005) Calcium mobilization stimulates Dictyostelium discoideum shear-flow-induced cell motility. J Cell Sci 118:3445-3457.

- Lima WC, et al. (2014) Role of PKD2 in Rheotaxis in Dictyostelium. Plos One 9:e91457.

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Generalising homomorphisms

Project Title Generalising homomorphisms
Contact Name Marj Batchelor
Contact Email mb139@cam.ac.uk
Period of the Project 8 weeks over the summer by arrangement.
Brief Description of the Project Universal measuring coalgebras (see Moss E. Sweedler' book "Coalgebras") provide a generalisation of homomorphisms of algebras. The concept has been around for over 50 years, but until a CMP project four years ago, even simple examples had not been calculated. The calculation involved the introduction of an algebra with interesting functorial properties. The following four years have been spent extracting the categorical structures behind that calculation. The project this summer would involve assisting with the preparation of the paper introducing this construction, and exploring possible applications.
Prerequisite Skills The paper is essentially algebraic; neither coalgebra theory or category theory is required, but students working on the project will need to learn enough of both to be able to use those ideas. Beyond that, cheerful patience is a must!
Other Skills Used in the Project The construction has potential applications in homological algebra, number theory and might serve as a framework for understanding internal symmetry groups in physics, so students enthusiastic about any of those three areas might enjoy the project.

 

Effective dynamics on bounded domains

Project Title Effective dynamics on bounded domains
Contact Name Simon Becker
Contact Email slb214@cam.ac.uk
Period of the Project Some time between End of June-End of September
Brief Description of the Project Many PDEs are defined on unbounded domains like Rn. We want to study under what conditions it suffices to analyze them with numerical methods on suitably chosen bounded domains to describe the dynamics locally with full error control.
Prerequisite Skills Apart from its "numerical motivation", this project is mainly analytical. A strong background in analysis (linear analysis and measure theory) is mandatory, the numerical analysis skills can be learned on the job.
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Weakly nonlinear analysis of Holmboe waves

Project Title Weakly nonlinear analysis of Holmboe waves
Contact Name Dr Adrien Lefauve
Contact Email aspl2@cam.ac.uk
Period of the Project e.g. 29 June - 21 August (but flexible)
Brief Description of the Project

This project is theoretical/numerical but is motivated by experiments on stratified shear flows made in the GKB Laboratory in DAMTP. The Stratified Inclined Duct experiment [1] sets up a two-layer exchange flow between two reservoirs of fluids at different densities (saltwater/freshwater). For a given parameter range, we observe Holmboe waves, which are finite-amplitude travelling waves on the density interface [2]. As some flow parameters are varied (especially the Reynolds number), the flow becomes intermittently turbulent, and then fully-turbulent. The motivation behind this project is to understand the dynamical link between the Holmboe wave regime and the transition to turbulence. The first step in that direction is to understand the nonlinear saturation of Holmboe waves from their linear instability, i.e. how an infinitesimally small, but exponentially growing wave saturates to a finite-amplitude. In this project, you will be carrying out numerical calculations called 'weakly nonlinear stability analysis', which takes into account the first order nonlinearities in an asymptotic expansion to derive the saturated amplitude, as a function of flow parameters (e.g. the Reynolds number).

[1] Meyer & Linden, J. Fluid. Mech. 753:242-253 (2014)

[2] Lefauve et al., J. Fluid Mech. 848:508-544 (2018)

Prerequisite Skills Interest in research, relative independence. Preferred Part II student, ideally with good knowledge of Part II Fluid Dynamics. Good knowledge and independent coding abilities on Matlab.
Other Skills Used in the Project By decreasing order of importance: Part II Dynamical systems, Asymptotic methods, Waves.

 

Flow of glacial ice sheets over deformable material

Project Title Flow of glacial ice sheets over deformable material
Contact Name Katarzyna Kowal
Contact Email k.kowal@damtp.cam.ac.uk
Period of the Project 8 weeks in the summer
Brief Description of the Project The ice sheet of West Antarctica alone has the potential to raise sea levels by about 7 metres should it become unstable and flow rapidly into the ocean in response to global warming, altering coast lines and metropolitan areas as we know them today. Such sea level rise can result from the dynamics of ice sheets, alone, and depends greatly on conditions underneath the ice, causing rapid acceleration in ice discharge and melt rates. In particular, glacial ice sheets flow over unconsolidated, water-saturated subglacial sediment that serves to lubricate the base of the ice and accelerate the flow of ice towards the ocean. The project seeks to explore the dependence of the flow of viscous fluids, such as glacial ice sheets on the large scale, on what lubricates it from below and on the accumulation of the underlying material. There will be the opportunity for designing and conducting small- scale fluid-dynamical laboratory experiments involving viscous fluids, such as syrup as well as the opportunity to model the flow mathematically from first principles.
Prerequisite Skills Knowledge of fluid mechanics (such as from an introductory course on fluid mechanics) is useful but not essential as the project is self-contained.
Other Skills Used in the Project  

 

Dynamics of lubricated ice-sheet grounding zones

Project Title Dynamics of lubricated ice-sheet grounding zones
Contact Name Katarzyna Kowal
Contact Email k.kowal@damtp.cam.ac.uk
Period of the Project 8 weeks in the summer
Brief Description of the Project The stability of marine ice sheets, such as those of West Antarctica, depends heavily on the dynamics of ice-sheet grounding zones - regions separating grounded ice sheets, in contact with the bedrock, from freely floating ice shelves. Curiously, it has been found that subglacial till (a mixture of water, clay and subglacial sediment) that lubricates the base of ice streams accumulates into sedimentary wedges, or till-deltas, in these grounding zones. Such sedimentation can have immediate consequences in stabilizing grounding zones against retreat in response to rising sea levels. The aim of the project is to explore the formation of grounding zone wedges and their effect on the large-scale dynamics of ice sheets using principles of viscous fluid mechanics. There will be the opportunity for designing and conducting small-scale fluid-dynamical laboratory experiments involving viscous fluids, such as syrup, as well as the opportunity to model the flow mathematically from first principles.
Prerequisite Skills Knowledge of fluid mechanics (such as from an introductory course on fluid mechanics) is useful but not essential as the project is self-contained.
Other Skills Used in the Project  

Predicting glioblastoma molecular subtypes using magnetic resonace and histology images

Project Title Predicting glioblastoma molecular subtypes using magnetic resonace and histology images
Contact Name Chao Li
Contact Email cl647@cam.ac.uk
Period of the Project 29/06/2020 - 30/09/2020
Brief Description of the Project In the proposed project, we aim to predict the molecular subtypes of glioblastoma by integrating magnetic resonance and histology images using deep learning approach. Our hypothesis is that the image-derived phenotypes of glioblastoma can identify the molecular phenotypes for precision treatment of patients. This collaboration project will be jointly hosted by Professor Carola-Bibiane Schönlieb, Department of Applied Mathematics and Theoretical Physics and Mr Stephen Price, Consultant neurosurgeon, Division of Neurosurgery, Department of Clinical Neurosciences. Two researchers, Dr Chao Li and Dr Veronica Corona will be supervising the project.
Prerequisite Skills image processing, programming, statistics
Other Skills Used in the Project data visualisation

 

Hölder continuity of shape-constrained projections, with applications to estimation under model misspecification

Project Title Hölder continuity of shape-constrained projections, with applications to estimation under model misspecification
Contact Name Richard Samworth, Oliver Feng
Contact Email rjs57@cam.ac.uk
Period of the Project 8 weeks in the summer
Brief Description of the Project

A statistical model is a family of distributions representing plausible data generating mechanisms for a data set. Many statistical estimation procedures can be thought of as a projection of a distribution onto a (finite- or infinite-dimensional) model. One can then seek to understand the statistical properties of an estimator by studying the analytic properties (e.g. continuity, differentiability) of this projection. See [1] for a recent illustration in the context of log-concave density estimation. This project would seek to extend these ideas to other shape-constrained estimation problems, such as decreasing density estimation and convex regression. 

[1] Barber, R. F. and Samworth, R. J. (2020) Local continuity of log-concave projection, with applications to estimation under model misspecification. http://arxiv.org/abs/2002.06117

Prerequisite Skills The project would be well suited to a finishing third-year student with strong analysis skills in addition to an interest in Statistics.
Other Skills Used in the Project  

Random Forest: Change-point regression meets machine learning

Project Title Random Forest: Change-point regression meets machine learning
Contact Name Florian Pein
Contact Email fp366@cam.ac.uk
Period of the Project Any time in summer
Brief Description of the Project

Random forest [1] is a machine learning approach for classification or regression that constructs multiple decision trees at training. In short, growing a decision tree consists of randomly selecting one of the covariates and splitting the data according to a certain rule, for instance at the point that maximises a given statistic. In other words, each split can be understood as finding a single change-point in a univariate data sequence. Hence, growing a single tree has a strong connection with binary segmentation [2, 3], which is a universal approach for multiple change-point regression. Here one tests firstly for a single change-point in the data. If a change-point is found, the data set is split at this point and the procedure is recursively applied to both subsets until no further change-point can be found.

While this approach is simple and easy to implement, it often lacks statistical power and finds suboptimal split points. In this project we will explore more advanced change-point procedures and see how they can be used to grow a decision tree. One idea could be to use wild binary segmentation [3] instead. Another idea is to still look recursively for single change-points, but instead of assuming that the left and right part are homogeneous to fit them by a fused Lasso [4].

A good outcome of the project would be to develop a new methodology using one of those ideas, implement it in an R package, and demonstrate its performance through computer simulations. There might be also a possibility of submitting a write-up of the work to a statistical journal for publication.

References

[1] Breiman, L. (2001). Random forests. Machine learning, 45(1), 5-32.

[2] Vostrikova, L. (1981). Detecting 'disorder' in multidimensional random processes. SovietMath. Dokl. 2455–5

[3] Fryzlewicz, P. (2014). Wild binary segmentation for multiple change-point detection. The Annals of Statistics, 42(6), 2243-2281.

[4] Tibshirani, R., Saunders, M., Rosset, S., Zhu, J., & Knight, K. (2005). Sparsity and smoothness via the fused lasso. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(1), 91-108.

Prerequisite Skills

Mathematics of Machine Learning or comparable knowledge

Good programming skills, preferable in R and C++

Other Skills Used in the Project Solid statistical knowledge, e.g. IB Statistics, II Statistical Modelling and/or II Principles of Statistics

 

Enhancing the Image quality of cone-beam CT to predict the treatment response in nasopharyngeal carcinoma patients

Project Title Enhancing the Image quality of cone-beam CT to predict the treatment response in nasopharyngeal carcinoma patients
Contact Name Dr Chao Li
Contact Email cl647@cam.ac.uk
Period of the Project 01/06/2020 - 30/09/2020
Brief Description of the Project Nasopharyngeal carcinoma (NPC) is a malignant head and neck cancer, for which radiotherapy (RT) is the main treatment modality. Although induction chemotherapy plus concurrent chemoradiotherapy has significantly improved patient survival, approximately 30% of these patients suffer from treatment failure. Cone-beam CT (CBCT) is the main modality for RT planning, their image quality however is affected by noise and artifacts, In this project, we aim to improve the image quality of CBCT using the super resolution technique. Further, the enhanced images will be trained using machine learning models for treatment adaptation and treatment response prediction. Please contact me for more details.
Prerequisite Skills imaging processing, programming, statistics
Other Skills Used in the Project data visualisation, clinical knowledge