M.Sc. in Mathematical Modelling and Scientific Computing

M.Sc. in Mathematical Modelling and Scientiﬁc Computing Dissertation Projects

December 2019

Contents

1 Numerical Analysis Projects 3 1.1 Operator Preconditioning and Accuracy of Linear System Solutions . . . . 3 1.2 Computing Common Zeros of Trivariate Functions ...... 4 1.3 Univariate Optimisation for Smooth Functions ...... 4 1.4 Algorithms for Least-Squares and Underdetermined Linear Systems Based on CholeskyQR ...... 5 1.5 Tensor and Matrix Eigenvalue Perturbation Theory ...... 6 1.6 Chebfun Dissertation Topics ...... 7 1.7 Rational Functions Dissertation Topics ...... 8 1.8 Low-Rank Plus Sparse Matrix Model and its Application ...... 9 1.9 Eﬃcient Exemplar Selection for Representation Learning ...... 10 1.10 Numerical Solution of a Problem in Electrochemistry ...... 11 1.11 Nonconvex Optimisation ...... 12

2 Biological and Medical Application Projects 13 2.1 Localisation of Turing Patterning in Discrete Heterogeneous Media . . . . 13 2.2 Distributed Delay in Reaction-Diﬀusion Systems ...... 14 2.3 Aspects of Flagellate Swimmer Dynamics and Mechanics ...... 16 2.4 Learning Time Dependence with Minimal Time Information ...... 17 2.5 Surrogate Modelling for Particle Inﬁltration into Tumour Spheroids . . . . 18 2.6 Mathematical Modelling of polyelectrolyte hydrogels for Application in Regenerative Medicine ...... 19

3 Physical Application Projects 21 3.1 Evaporation-Driven Instabilities in Complex Fluids ...... 21

1 3.2 Phase Separation in Swollen Hydrogels ...... 22 3.3 Pattern Formation in Polymers ...... 23

4 Data Science 25 4.1 Computational Topology for Analysing Neuronal Branching Models and Data ...... 25

5 Research Interests of Academic Staﬀ 26

2 1 Numerical Analysis Projects

1.1 Operator Preconditioning and Accuracy of Linear System Solu- tions

Supervisors: Prof Yuji Nakatsukasa and Dr Carolina Urzua Torres Contact: [email protected] and [email protected] Preconditioning is widely recognised as a crucial technique for improving the convergence of iterative (usually Krylov) methods for linear systems Ax = b. A major body of research in numerical linear algebra is devoted to devising preconditioners for certain classes of matrices. One aspect that is seldom discussed is that if A is ill-conditioned, then even if a good preconditioner M is used, usually the final (relative) accuracy achievable is limited by the condition number of the original linear system O(uκ2(A)), where u is the unit roundoff. This is because applying A or M involves errors of that order. Another, more recent, line of work is operator preconditioning (e.g. [1, 3]) in the context of solving differential equations (which usually lead to a large linear system). Here the idea is to precondition the problem at the operator level to reduce the problem to that of a well-conditioned differential operator, which is then discretised to give Ax˜ = ˜b. This procedure hints that the effect of numerical errors could be significantly reduced, and hence the accuracy would no longer be limited by uκ2(A) but by uκ2(A˜), which can be much better, potentially as small as O(u); a similar effect is described in [2]. The goal of this work is to explore operator preconditioning in terms of the solution accuracy. The project could involve a subset of the following: (i) analyzing the properties of the linear systems and solutions (forward and backward errors), (ii) roundoff error analysis, (iii) examining various equations and preconditioners, and (iv) implementing the preconditioners. The study might also lead to the design of good preconditioners at the matrix level. Exposure to basic numerical linear algebra and differential equations would be helpful for the project.

References

[1] R. Hiptmair. Operator preconditioning. Computers and Mathematics with Applica- tions, 52(5):699–706, 2006. [2] S. Olver and A. Townsend. A fast and well-conditioned spectral method. SIAM Rev., 55(3):462–489, 2013. [3] C. Urzua-Torres, R. Hiptmair, and C. Jerez-Hanckes. Optimal operator preconditioning for Galerkin boundary element methods on 3D screens. SIAM J. Numer. Anal., 2019.

3 1.2 Computing Common Zeros of Trivariate Functions

Supervisor: Prof Yuji Nakatsukasa Contact: [email protected] Chebfun can deal with functions in 1D, 2D, and 3D. Its functionalities in 1D are mostly complete and highly sophisticated. In higher dimensions, and in particular 3D, the situation is sometimes different. Among the key missing operations in Chebfun3 is roots(f,g,h)—finding common zeros of three trivariate functions, which generically have zero-dimensional solutions consisting of disjoint points. The 1D version roots(f) is re- markably efficient and reliable, based on domain subdivision and colleague eigenvalue problems. The 2D version was developed in [1], which also uses these two ideas, together with resultant methods to eliminate variables. The goal of this project is to develop and implement a Chebfun3 roots(f,g,h) algorithm and command. It is envisaged that a successful algorithm will involve a mixture of mathematical analysis and numerical/computational techniques. In particular, conditioning and numerical stability will need to be investigated carefully, in light of the analysis in [2]. Since there is little software publicly available for trivariate rootfinding, a successful completion would be an important contribution to multivariate polynomial computing. Background in approximation theory would be very helpful. The Part C course “Ap- proximation of Functions” is highly recommended.

References

[1] Y. Nakatsukasa, V. Noferini, and A. Townsend. Computing the common zeros of two bivariate functions via B´ezoutresultants. Numer. Math., 129:181–209, 2015. [2] V. Noferini and A. Townsend. Numerical instability of resultant methods for multidimensional rootﬁnding. SIAM J. Numer. Anal., 54(2):719–743, 2016.

1.3 Univariate Optimisation for Smooth Functions

Supervisor: Prof Yuji Nakatsukasa Contact: [email protected]

Finding the (global) minimum of a univariate function minx f(x) is a classical problem in continuous optimisation. It arises naturally when a search direction is found, and one looks for an appropriate step size, e.g. in line search algorithms. For example in the currently very popular stochastic gradient descent method, usually the step size is fixed. However, an optimal step size can significantly outperform fixed and suboptimal choices. When the function is convex (or unimodal), bisection or golden section are established techniques for finding the global minimum. Another powerful approach would be to approximate the function with a polynomial and minimize the polynomial. This technique is used in Chebfun (together with interval subdivision as necessary), but appears

4 not to be widely used elsewhere. We expect this approach to have a number of ad- vantages, including (i) it can easily deal with nonconvex functions, (ii) it can ﬁnd local minima/maxima, and (iii) the convergence should be fast (fewer samples are needed) if f is smooth, e.g. analytic. The goals of this project include exploring these aspects, analyzing convergence, and developing eﬃcient implementations. Extensions can be considered, such as using rational approximation or minimising bivariate functions. The student should have some prior exposure to basic optimisation [1], and ideally approximation theory [2]. The Part C courses “Continuous Optimisation” and “Ap- proximation of Functions” are recommended.

References

[1] J. Nocedal and S. J. Wright. Numerical Optimization. Springer New York, second edition, 1999. [2] L. N. Trefethen. Approximation Theory and Approximation Practice. SIAM, Philadelphia, 2013.

1.4 Algorithms for Least-Squares and Underdetermined Linear Sys- tems Based on CholeskyQR

Supervisor: Prof Yuji Nakatsukasa Contact: [email protected] CholeskyQR—based on RT R = AT A (Cholesky factorization) and Q = AR−1—is an algorithm for computing the QR factorization of matrices, which is very attractive in speed especially in parallel computing architectures. It is unfortunately numerically unstable, and hence has not been widely used. However, the instability can be improved significantly by techniques such as repeating and shifting [1, 2]. The QR factorization is used in various problems in numerical linear algebra, including least-squares problems and underdetermined linear systems. This project aims to explore the use of CholeskyQR in these contexts: (1) We observe that when CholeskyQR is used for least-squares problems (together with a Krylov solver; a neat mixture of direct and iterative linear algebra methods), we arrive at a robust and efficient solver, achieving up to ×5 speedup. Establishing backward stability, however, remains an important open problem, which we hope to address here. Other possible directions include exploring HPC implementations and exploring its use in applications. (2) Similarly, CholeskyQR can be used for finding the minimal solution for an underdetermined linear system. This line of work is at a young stage and many directions are possible: proving stability, refining the algorithm, implementation, and applications (e.g. compressed sensing). A good background in linear algebra is desirable (e.g. the part C course Numerical Linear Algebra).

5 References

[1] T. Fukaya, R. Kannan, Y. Nakatsukasa, Y. Yamamoto, and Y. Yanagisawa. Shifted CholeskyQR for computing the QR factorization of ill-conditioned matrices, arXiv:1809.11085, 2018. [2] Y. Yamamoto, Y. Nakatsukasa, Y. Yanagisawa, and T. Fukaya. Roundoﬀ error analysis of the CholeskyQR2 algorihm. Electron. Trans. Numer. Anal., 44:306–326, 2015.

1.5 Tensor and Matrix Eigenvalue Perturbation Theory

Supervisor: Prof Yuji Nakatsukasa Contact: [email protected] I would be very happy to supervise any topic in numerical linear algebra, especially related to eigenvalue problems. Below is a sample project; I also have a handful of other potential projects, and would be happy for anyone to propose their own directions. n×n An important result in eigenvalue perturbation theory [1] is: Let A, E ∈ R be symmetric matrices partitioned as

A 0 0 ET A = 1 ,E = 1 . (1) 0 A2 E1 0

In words, the perturbation in the eigenvalues scales quadratically in the off-diagonal perturbation E, rather than linearly (which is the case without the structure in (1) imposed). Put another way, E lies in the tangent space of the manifold of matrices similar to A. Analogous results hold for nonsymmetric block diagonal matrices. This fact has important ramifications in a number of topics in numerical analysis: for example the cubic convergence of the symmetric QR algorithm, cubic convergence of the (nonsymmetric) Rayleigh quotient iteration, analysis and convergence of the Rayleigh- Ritz method, and the surprising efficacy of randomized SVD algorithms. The main goal of this project is to explore analogues of the above result for tensor eigenvalue problems [2]. Tensors are multidimensional arrays that generalize matrices. Roughly, the goal is to characterize the class and structure of peturbation such that the tensor eigenvalue(s) change only quadratically with respect to the perturbation in the tensor. The importance of tensors is increasing rapidly in the era of data science, and

6 such results can have significant implications in many areas of computational mathematics. Specific directions may include the following: (i) prove quadratic perturbation bounds for (symmetric) tensor eigenvalues, (ii) exploring analogous results for nonsymmetric tensors, (iii) deriving analogous results for tensor singular values, (iv) investigating ramifications of the quadratic perturbation bounds, and (v) Related problems in matrix eigenvalue perturbation theory. A good understanding of linear algebra is highly desirable. The Part C course “Numerical Linear Algebra” is highly recommended.

References

[1] C.-K. Li and R.-C. Li. A note on eigenvalues of perturbed Hermitian matrices. Linear Algebra Appl., 395:183–190, 2005. [2] L.-H. Lim. Singular values and eigenvalues of tensors: a variational approach. In 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing., pages 129–132. IEEE, 2005.

1.6 Chebfun Dissertation Topics

Supervisor: Prof Nick Trefethen Contact: [email protected] Chebfun is an open-source algorithms and software project for numerical computation with 1D, 2D and 3D functions, based on the idea of overloading Matlab’s vectors and matrices to functions and operators. Chebfun can do almost anything in 1D (integration, optimization, rootfinding, differential equations,. . . ) and quite a bit in 2D and 3D too. A number of M.Sc. dissertations related to Chebfun have been written over the years. There are many possibilities, and we can tailor the project to the student’s interests and expertise in areas including differential equations, approximation theory, smooth random functions, interpolation, quadrature, rootfinding, linear algebra, complex analysis, optimization, and modelling. Here are two specific possibilities, but I would also be happy to talk about projects in other areas. Also, some of Yuji Nakatsukasa’s dissertation topics are Chebfun-related. (1) Boundary and interior layers. One of the best-developed areas of ordinary differential equations is boundary-layer theory, an area of asymptotic analysis of both mathematical and physical importance. Chebfun makes it easier to explore ODE boundary layers than ever before, but beyond Chapter 20 of Exploring ODEs [2], little has done to take advantage of this capability. It would be interesting to explore what can be done in this area with particular connection to a physical problem to be determined with input from appropriate OCIAM faculty. (2) Basins defined by level curves of smooth random functions. 2D smooth random functions (a case of Gaussian random fields) have been of interest in the sci-

7 ences since a paper by Longuet-Higgins in 1957. One aspect of particular interest are the basins deﬁned by their levels curves. In Chebfun, we can explore such things with com- mands like f = roots(randnfun2(.2)) followed by plot(f-c,’zebra’), axis equal or plot(roots(f-c)), axis equal, where c is a constant. Fascinating questions arise here including a phase transition as c passes through zero, see [3] (the cover article of the March 2019 SIAM Review).

References

[1] L.N. Trefethen. Approximation Theory and Approximation Practice, SIAM, 2013. [2] L.N. Trefethen, A. Birkisson, and T.A. Driscoll. Exploring ODEs, SIAM, 2017. (Freely available at https://people.maths.ox.ac.uk/trefethen/.) [3] S. Filip, A. Javeed, and L.N. Trefethen Smooth Random Functions, Random ODEs, and Gaussian Processes. SIAM Review, 61(1):185–205, 2019. (Available fromhttps://people.maths.ox.ac.uk/trefethen/papers.html.) [4] https://www.chebfun.org (especially the Examples collection)

1.7 Rational Functions Dissertation Topics

Supervisor: Prof Nick Trefethen Contact: [email protected] Rational functions have become an exciting area of numerical comptuation recently due to developments including lightning Laplace solvers, AAA and AAA-Lawson approximation, Floater-Hormann approximation, and model order reduction. There are many unanswered questions and unexplored algorithms in this area, including the two below. (1) Approximation theory for monopoles. Poles 1/(z − a) of rational functions are effectively “dipoles” in the physical sense, since they have a strength and an orientation (i.e., the residue has a modulus and an argument). But for approximating harmonic functions, it would seem more straightforward to work with monopoles, log |z − zk|. Very little seems to be known about either the theory or the algorithmic practice of such approximations. Can we develop some theory of existence, uniqueness, and characteri- zation? Can we devise a monopole analogue of lightning Laplace solvers? (2) Floater-Hormann approximation. One of the best ideas in computing with rational functions is Floater-Hormann approximation, involving a rational barycentric formula with precisely determined coefficients. (An adaptive-order Floater-Hormann approximation is the basis of the Chebfun ‘equi’ flag.) Yet much is not understood. What happens to F-H interpolation when the grid points are exponentially graded? Can the F-H idea be derived in a different fashion of the usual (polynomial “blending” functions) that gives a more intuitive and usable idea of why the method is so powerful? Why do rows of the Pascal triangle turn up in the F-H coefficients? Might these formulas be related to discretized contour integrals and even to the idea of hyperfunctions?

8 References

[1] M.S. Floater and K. Hormann. Barycentric rational interpolation with no poles and high rates of approximation. Numerische Mathematik, 107:315–331, 2007. [2] A. Gopal and L.N. Trefethen. Solving Laplace problems with corner singularities via rational functions. SIAM J. Numerical Analysis, 57(5):2074–2094, 2019. [3] Y. Nakatsukasa, O. S`ete,and L.N. Trefethen. The AAA algorithm for rational approximation. SIAM J. Sci. Comp, 40(3):A1494–A1522, 2018.

1.8 Low-Rank Plus Sparse Matrix Model and its Application

Supervisor: Prof Jared Tanner Contact: [email protected] (Additional consultation available from Simon Vary (mathematics)) Low-rank matrix approximation is a cornerstone of numerical linear algebra and data science. Such an approximation both reveals information about the dominant subspaces in which the matrix is active, and allows for numerous computational benefits ranging from the pseudo-inverse to reduced memory and computational complexity. However, traditional low-rank approximation suffers from not being robust to large entry perturbation and does not capture structure in matrices that do not align with the coordinate axes (e.g. the identity matrix is not low rank). The decomposition of a matrix into the addition of a low-rank matrix plus a sparse matrix overcomes both aforementioned limi- tations. Moreover, it allows for some novel applications such as automatic segmentation of moving objects in a static background in video sequences. In this project one will consider the low-rank plus sparse matrix model and its application for either denoising or deep neural network regularization. These are two distinct applications which are described below in separate paragraphs. Block matching 3D is an image denoising method by which: a) from an image, all patches, say 3 by 3 local regions, are formed to create many such small patches, b) these patches are then grouped in some manner and stacked to form a tensor, say 3 × 3 × n, c) each tensor is then approximated in a way, such as low-rank approximation, to reduce the noise, and d) the image is reformed by the denoised patches. This method is highly effective, and is considered state of the art for many applications. However, it suffers from the inability to remove extreme outliers or to handle patches of the image for which there are not other similar objects. One project is to adapt the above structure to have low-rank plus sparse as the model for step c) and in so doing overcome extreme outliers and potentially allow sharper denoising of regions where patches have few components. Such a project is largely computational, but can include application to medical imaging with a partner in cardiology or could include some theoretical development where one might prove approximation rates for images of a certain smoothness structure. The theoretical directions would naturally lead to a doctoral research project. Deep neural networks are formed of the repeated application of affine transforms (matrix vector product plus a vector) followed by non-convex activations. Not constraining the

9 network weight matrices allows for the best ability to ﬁt the desired goal of the network, but at the cost of potential overﬁtting and lack of rubustness to adversarial attacks. There is ongoing work to enforce that the matrices are approximately low-rank as a method to overcome both aforementioned issues. One project is to replace the low-rank structure by low-rank plus sparse which will allow for potentially substantially much greater approximation rate, while also greatly reducing the complexity of the network. This project would be largely computational.

References

[1] E. Candes, X. Li, Y. Ma, and J. Wright. Robust Principal Component Analysis? https://arxiv.org/abs/0912.3599 [2] A. Buades, B. Coll, and J. M. Morel A review of image denoising algorithms, with a new one https://epubs.siam.org/doi/10.1137/040616024 [3] A. Dubey, M. Chatterjee, and N. Ahuja. Coreset-Based neural network compression. http://openaccess.thecvf.com/content ECCV 2018/papers/Abhimanyu Dubey Coreset- Based Convolutional Neural ECCV 2018 paper.pdf

1.9 Eﬃcient Exemplar Selection for Representation Learning

Supervisor: Dr Vinayak Abrol Contact: [email protected]

Description of proposal: Extracting inherent patterns from large amount of data using decompositions of data matrix by a sampled subset of exemplars has found many applications in machine learning (ML). Recently, there is an active interest in developing computationally eﬃcient algorithms for adaptive exemplar sampling, i.e., sampling a n×l small number of columns of a matrix X ∈ R such that kX−PXk is close to kX−Xkk. In other words, the error between the target matrix X and its rank-k approximation Xk with respect to any unitarily invariant norm should be minimum. Here, P is the projection matrix of the sampled exemplars, and k < r = rank(X). However, this l problem is believed to be NP-hard, as one has to search over all possible k choices. The starting point of the project would be to explore and study existing exemplar selection algorithms. The main part of the dissertation could then focus either on developing an eﬃcient variant or certain applications.

Possible avenues of investigation:

• developing a parallel algorithm for ML tasks on GPUs

• working on developing theoretical convergence and performance bounds

• extension to streaming data in an online setting

10 • compression/pruning of deep neural networks for application on mobile devices

• importance sampling for learning ML models from large amount of redundant data

References

[1] D. Bernstein. Matrix Mathematics: Theory, Facts, and Formulas. Princeton Uni- versity Press, 2009. [2] A. Civril. Column subset selection problem is UG-hard. Journal of Computer and System Sciences, 80(4):849–859, 2014. [3] F. de Hoog and R. Mattheij. Subset selection for matrices. Linear Algebra and its Applications, 422(23):349–359, 2007. [4] A. Civril and M. Magdon-Ismail. Column subset selection via sparse approximation of SVD. Theoretical Computer Science, 421:1–14, 2012. [5] E. Elhamifar, G. Sapiro, and R. Vidal. See all by looking at a few: Sparse modeling for ﬁnding representative objects. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1600–1607, June 2012. [6] A. Eftekhari, R.A. Hauser, and Andreas Grammenos. MOSES: A Streaming Algo- rithm for Linear Dimensionality Reduction. arXiv:1806.013041 [7] V. Abrol, P. Sharma and A.K. Sao. Fast exemplar selection algorithm for matrix approximation and representation:A variant oASIS algorithm. DOI: 10.1109/ICASSP.2017.7952995 [8] Rajiv Khanna, Ethan R. Elenberg, Alexandros G. Dimakis, Sahand Neghaban, and Joydeep Ghosh. Scalable Greedy Feature Selection via Weak Submodularity. arXiv:1703.02723.

1.10 Numerical Solution of a Problem in Electrochemistry

Supervisor: Dr Kathryn Gillow Contact: [email protected] The basic idea of an electrochemical experiment is that a known potential is applied to a working electrode in a solution. This causes oxidation or reduction to take place at the electrode and in turn this means that a current (which can be measured) flows. The current depends on a number of physical parameters of the solution including the diffusion coefficient, the resistance of the solution and the rate of reaction. Mathematically the concentration of the chemicals is modelled using a reaction-con- vection-diffusion equation and for a planar electrode we can assume that one space dimension is enough for the model. The current is then a linear functional of the concentration. Solving for the current with given values of the parameters is known as the forwards problem. Of more interest is the inverse problem where an experimental current is given and and the parameters are to be calculated. Of course, the experimentally measured current includes noise due to measurement error and so we cannot expect to

11 recover the parameters exactly. Thus information about the likelihood of the recovered parameters being correct is helpful and can be provided by using Markov chain Monte Carlo methods. Such an approach has been used for the simplest problem in [3] and the purpose of this project is to extend this work to more interesting chemical reactions where there are more parameters to recover.

References

[1] D. J. Gavaghan and A. M. Bond. A complete numerical simulation of the techniques of alternating current linear sweep and cyclic voltammetry: analysis of a reversible process by conventional and fast Fourier transform methods. Journal of Electroana- lytical Chemistry, 480(1–2):133–149, 2000. [2] G. P. Morris, A. N. Simonov, E. A. Mashkina, R. Bordas, K. Gillow, R. E. Baker, D. J. Gavaghan and A. M. Bond. A Comparison of Fully Automated Methods of Data Analysis and Computer Assisted Heuristic Methods in an Electrode Kinetic Study of the Pathologically Variable [Fe(CN)(6)](3-/4-) Process by AC Voltammetry. Analytical Chemistry, 85(24):11780–11787, 2013. [3] D. J. Gavaghan, J. Cooper, A. C. Daly, C. Gill, K. Gillow, M. Robinson, A .N. Simonov, J. Zhang, and A. M. Bond. Use of Bayesian Inference for Parameter Recovery in DC and AC Voltammetry. ChemElectrochem, 5(6):917–935, 2018.

1.11 Nonconvex Optimisation

Supervisor: Prof Coralia Cartis Contact: [email protected] I am interested in methods for nonconvex optimization, in both their theoretical and numerical aspects. In particular, recent interests include derivative free optimization of expensive functions and dimensionality reduction techniques for very large scale problems such as those arising in machine learning. I am also interested in the complexity/global rate of convergence of nonconvex optimization, compressed sensing for signal processing and optimization on manifolds. I have one space available, subject to ﬁnding a project that is mutually of interest.

12 2 Biological and Medical Application Projects

2.1 Localisation of Turing Patterning in Discrete Heterogeneous Me- dia

Supervisors: Dr Andrew Krause and Prof Eamonn Gaﬀney Contact: [email protected] and [email protected]

Background and Problem: Turing’s chemical theory of morphogenesis posits that biological patterns, such as the jaguar spots shown in the first panel of the Figure, can arise due to the reaction and diffusion of two chemical substances [5]. While Turing’s model has given rise to a huge research enterprise in mathematics, chemistry, and biology, he was aware of how simplistic of a starting point his theory was, calling it, “a simplification and an idealization, and consequently a fal- sification.” In particular, the jaguar spots shown 0.7 0.68 are not simple repeated patterns typical of Turing- 0.66 type mechanisms, but patterns at multiple spa- 0.65 0.75 0.8 tial scales with qualitatively different structures at each scale. In his original paper, he notes that this reaction-diffusion mechanism gives rise to pat- 0.6 terns from a homogeneous state, but that most 0 0.5 1 organisms are developing “from one pattern into another, rather than from homogeneity into a pattern.” Recently, we have extended Turing’s original thesis to this case by analyzing spatially- heterogeneous reaction-diffusion equations [3]. We exploited the difference in the length- scales of subsequent patterns to utilize a WKBJ-asymptotic approach in order to derive local patterning criteria. The second panel of the figure demonstrates this, with a finer- scale pattern emerging beyond the Turing threshold given by the red line. This project will consider the discrete case of coupled ordinary-differential equations. Such a discrete setting was already investigated in Turing’s original work [5], and has numerous motivations from discrete cellular systems, to chemostats, and niche formation in spatial ecology. Such a setting has been studied in some cases in the literature [6, 4], but the localization of patterns on top of a heterogeneous steady state has not been investigated.

Approach and Prerequisites: The student should have familiarity and interest in some basic methods of perturbation theory, as well as an interest in exploring discrete and continuum analogs of classical models. It is assumed that the student will be able to solve systems of ordinary diﬀerential equations numerically, and be able to analytically

13 study the instability of steady states in such systems. We refer to [2], or the Part C notes on Perturbation Methods for a brief introduction to WKBJ in general, and to [1] for an approach to WKBJ theory for discrete systems.

Project Goals: It is expected that the student will analyze reaction-diﬀusion systems on heterogeneous 1-D lattices ﬁrst, both numerically and analytically. Time permit- ting, extensions to lattices in higher dimensions, or irregular graphs, will be pursued. The student should have a good grasp on the relevant literature, being able to explain motivating examples in developmental biology and ecology.

References

[1] R.B. Dingle and G.J. Morgan. WKB methods for difference equations I. Applied Scientific Research, 18(1):221–237, 1968. [2] E.J. Hinch. Perturbation Methods. Cambridge Texts in Applied Mathematics, Cam- bridge University Press, 1991. [3] Andrew L. Krause, Václav Klika, Thomas E. Woolley, and Eamonn A. Gaffney. From one pattern into another: Analysis of Turing patterns in heterogeneous domains via WKBJ. arXiv preprint arXiv:1908.07219, 2019. [4] Nathan Tompkins, Ning Li, Camille Girabawe, Michael Heymann, G. Bard Ermen- trout, Irving R. Epstein, and Seth Fraden. Testing Turing’s theory of morphogenesis in chemical cells. Proceedings of the National Academy of Sciences, 111(12):4397– 4402, 2014. [5] A.M. Turing. The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237(641):37–72, 1952. [6] Matthias Wolfrum. The Turing bifurcation in network systems: Collective patterns and single differentiated nodes. Physica D: Nonlinear Phenomena, 241(16): 1351– 1357, 2012.

2.2 Distributed Delay in Reaction-Diﬀusion Systems

Supervisors: Dr Andrew Krause and Prof Eamonn Gaﬀney Contact: [email protected] and [email protected]

Background and Problem: Reaction-diﬀusion systems arise throughout biology, and notably in Turing’s theory of morphogenesis [7]. In cellular signalling, instantaneous reactions are an approximation, as gene expression takes time once a signal has entered a cell [2]. Such time delays have been investigated in the context of both cellular oscillations and Turing-type pattern formation [4, 6, 3, 5]. In general, such delays can completely prevent the initiation of pattern formation, induce oscillations, or radically increase the time taken for a pattern to grow. Hence, it has been suggested that such

14 time delays due to gene expression provide an obstruction to applying Turing’s theory to real systems. One simplification in the above analysis is the use of a constant time delay. In real- ity, there is a large amount of stochasticity in gene transcription and translation which amounts to a distribution of delay times at the macroscopic level [1]. Such a distributed delay leads to mean-field models which are integro-partial differential equations, and hence their mathematical analysis is substantially more involved than even delayed partial differential equations. This project will consider such distributed delay, and determine how it differs from the case of constant delay used in the above studies. In particular, we aim to investigate if this more realistic model of gene expression allevi- ates (or worsens) some of the problems suggested above in applying Turing’s theory to morphogens interacting via cellular signalling.

Approach and Prerequisites: The models of interest in this project will be analysed via numerical methods, and so a student should be capable and interested in implementing a variety of such schemes. The project will begin with an analysis purely of a kinetic reaction-diffusion model incorporating distributed delay. The student should compare the classical ordinary differential equation model, the constant delayed model, and the distributed delay model to build some intuition for how each of these systems behaves in a spatially homogeneous environment. From there, discretization of a spatial variant of each of these will be pursued. Some familiarity with the biological questions and mathematical frameworks (delay differential equations) is helpful, but this can be developed during the project.

Project Goals: The dissertation should be able to convey the qualitative impact of these delay-induced dynamics in the diﬀerent model settings, ideally culminating in answering the question regarding Turing patterning in real cells. A canonical set of model kinetics will be considered, but this can be extended as time permits.

References

[1] Dmitri Bratsun, Dmitri Volfson, Lev S. Tsimring, and Jeff Hasty. Delay-induced stochastic oscillations in gene regulation. Proceedings of the National Academy of Sciences, 102(41):14593–14598, 2005. [2] Ting Chen, Hongyu L. He, and George M. Church. Modeling gene expression with differential equations. In Biocomputing’99, pages 29–40. World Scientific, 1999. [3] E.A. Gaffney and S. Seirin Lee. The sensitivity of Turing self-organization to biological feedback delays: 2D models of fish pigmentation. Mathematical medicine and biology: a journal of the IMA, 32(1): 57–79, 2013. [4] E.A. Gaffney and N.A.M. Monk. Gene expression time delays and Turing pattern formation systems. Bulletin of mathematical biology, 68(1):99–130, 2006.

15 [5] E.A. Gaffney, F. Yi, and Sungrim Seirin Lee. The bifurcation analysis of Turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete and Continuous Dynamical System-Series B, 22(2), 2016. [6] S. Seirin Lee, E.A. Gaffney, and R.E. Baker. The dynamics of Turing patterns for morphogen-regulated growing domains with cellular response delays. Bulletin of mathematical biology, 73(11):2527–2551, 2011. [7] A.M. Turing. The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237(641):37–72, 1952.

2.3 Aspects of Flagellate Swimmer Dynamics and Mechanics

Supervisor: Prof Eamonn Gaﬀney Contact: [email protected] (Doctoral student Ben Walker will also be available for consultation.)

Background and Problem: Eukaryotic flagella and cilia are ubiquitous slender ap- pendages that function as microbiological fluid actuators, enabling sperm swimming and egg transport in mammalian reproduction, mucociliary clearance within the lung, the vir- ulence of numerous parasitic pathogens such as the Kinetoplastids, and transport within the cerebrospinal system. Ciliary and flagellar motion is driven by a remarkable slender structure, known as the axoneme, which consists of a central pair of microtubules (molecular-scale filaments) on the axoneme centreline, surrounded by nine concentric pairs of microtubule doublets. Molecular motors bridge the adjacent outer microtubule doublets and their differential contraction drives shear and bending of the axoneme, and hence the movement of the whole structure of the flagellum or cilium. In particular, a waveform emerges from a balance of viscous drag from the medium surrounding the cell, structural restoring forces and the forces exerted by the molecular motors, leading to fluid actuation and, for free cells, propulsion. This propulsion can be qualitatively understood from Newton’s third law on considering a sperm that is held fixed by its head: the waveform travelling away from the cell body then imparts momentum to the surrounding fluid in the wave direction; this results in an equal but opposite total force on the flagellum (sperm tail), which would act to drive the cell forward if it were not fixed and hence the cell swims forward in the absence of this fixing.

Project Goals: In this project, our objective will be to support research to further understand the mechanical principles of axoneme based cell motility and ﬂuid actuation, in particular by supporting the extraction of statistical and mechanical measures from microscopy movies of swimming cells.

Approach and Pre-requisites: A fundamental pipeline is already in place for the basic analysis of movies of ﬂagellate swimmers. Developing the extraction of statistical measures will be one objective and reliant on data processing. A further objective will

16 be to extract mechanical measures, which will require the application of the theory of Stokes flows and elastic filament mechanics to movie data. The balance between these two objectives will be contingent on student preference, though the mechanical measures are anticipated to be the more substantive element of the project. Experience in Scientific Computing will be required and familiarity with MATLAB would be beneficial. In addition experience with viscous flow theory at the level of a first course (e.g. at the level of module B5.3, Viscous Flow) would be strongly recommended. Experience of solid mechanics, image processing and data analysis would be beneficial but is certainly not necessary and can be developed as required during the project.

References

[1] E. Lauga and T.R. Powers. The Hydrodynamics of Swimming Microorganisms. Rep. Prog. Phys., 72:096601, 2009. [2] E.A. Gaffney, H. Gadêlha,D.J. Smith, J.R. Blake, and J.C. Kirkman-Brown. Mam- malian Sperm Motility: Observation and Theory. Annual Reviews in Fluid Mechan- ics, 43:501–528, 2011. [3] B.J. Walker, S. Phuyal, K. Ishimoto, C. Tung, and E.A. Gaffney. Computer Assisted Waveform Analysis and the Beats of Bovine Spermatozoa. Preprint, available on request.

2.4 Learning Time Dependence with Minimal Time Information

Supervisors: Dr Aden Forrow Contact: [email protected] In many biological contexts, collecting accurate time series measurements is slow, diffi- cult, or impossible. Gene expression patterns, for example, are measured at the largest scale by breaking apart cells to release their RNA for counting; techniques that produce time series from the same samples have much lower throughput. Similar issues arise in protein folding, where time-dependent measurements come from expensive molecular dynamics simulations while experimental techniques like cryo-electron microscopy can provide only snapshots of a population. A natural question, then, is how to learn from large-scale time-independent measurements together with limited time-resolved data. A summer project could make progress on a piece of this problem. One possibility is constructing a model system where samples with and without time can be naturally combined. The simplest type of model to start with is a discrete Markov chain; continuous dynamical systems would be more appropriate in contexts like cell differentiation. Alternatively, a student could adapt currently existing methods that perform well in equilibrium settings [2] to handle small deviations from equilibrium. Many relevant systems, such as the cell cycle, are not at equilibrium in the appropriate sense [3]. For either of these project options, knowledge of differential equations, stochastic simulation, and parameter inference, along with scientific programming experience, will be helpful.

17 The precise goals would be adapted to suit the student’s background and interest. By the end of the project, we would ideally have a computational pipeline that in some simple setting uses time-dependent and time-independent data to perform better than methods using only one type of information.

References

[1] C. Weinreb, S. Wolock, B.K. Tusi, M. Socolovsky, and A.M. Klein. Fundamental limits on dynamic inference from single-cell snapshots. PNAS, 115(10):E2467–E2476, 2018. https://doi.org/10.1073/pnas.1714723115 [2] P. Pearce, F.G. Woodhouse, A. Forrow, A. Kelly, H. Kusumaatmaja, and J. Dunkel. Learning dynamical information from static protein and sequencing data. Nat. Comm., 2019. https://doi.org/10.1038/s41467-019-13307-x [3] C. Li, and J. Wang. Landscape and ﬂux reveal a new global view and physical quantiﬁcation of mammalian cell cycle. PNAS, 111(39):14130–14135, 2014. https://doi.org/10.1073/pnas.1408628111

2.5 Surrogate Modelling for Particle Inﬁltration into Tumour Spheroids

Supervisors: Dr Joshua Bull and Prof Helen Byrne Contact: [email protected] and [email protected] Tumour spheroids are compact clusters of tumour cells grown in the lab to characterise the dynamics and response to treatment of different cancer cell lines. Nutrients and oxygen present in the culture medium diffuse into the spheroids and are consumed by the tumour cells, providing them with the energy needed for cell proliferation. As the spheroid grows, nutrient levels at its centre fall, causing the cells first to stop proliferating and then, if nutrient levels fall further, to die and form a necrotic core. The mechanical pressure created by cell proliferation on the spheroid boundary and cell death and degradation at its centre drives an advective flow of tumour cells from the spheroid rim towards its core. Evidence for this advective flow comes from experiments in which in- ert microbeads and radiolabelled cells, added to the surface of tumour spheroids, were observed to be passively advected towards the spheroid centre [2]. Dorie et al’s experiments have been modelled using continuum PDE models [3] and spatially resolved agent-based models [1]. The aim of this project is to undertake a systematic comparison of the two modelling approaches by comparing model predictions of quantities of interest such as where the peak of the microbead distribution falls and how quickly it travels. Of particular interest will be establishing conditions under which the more detailed and computationally-expensive agent-based model (ABM) can be approximated by the coarser-grained PDE model, and the extent to which predictions about tumour composition can be extracted from observations of the infiltrating particles.

18 Figure 1: Snapshot of an agent-based model simulation of passive migration in a tumour spheroid. Yellow cells are radiolabelled tumour cells which are carried towards the necrotic core via advection. Colours — purple: proliferating cells; blue: quiescent cells; green: hypoxic cells; grey: necrotic cells.

References

[1] J.A. Bull, T. Quaiser, F. Mech, S.L. Waters, and H.M. Byrne, H.M. Mathemat- ical modelling reveals cellular dynamics within tumour spheroids. 2019, Preprint available. [2] M.J. Dorie, R.F. Kallman, D.F. Rapacchietta, D. Van Antwerp, and Y.R. Huang. Migration and internalization of cells and polystyrene microspheres in tumor cell spheroids. Experimental Cell Research, 141(1):201–209, 1982. https://doi.org/10.1016/0014-4827(82)90082-9 [3] K.E. Thompson and H.M. Byrne. Modelling the internalization of labelled cells in tumour spheroids. Bulletin of Mathematical Biology, 61(4):601–623, 1999. https://doi.org/10.1006/bulm.1999.0089

2.6 Mathematical Modelling of polyelectrolyte hydrogels for Applica- tion in Regenerative Medicine

Supervisors: Prof Sarah Waters, Prof Andreas M¨unch and Dr Matthew Hen- nessy

19 Contact: [email protected], [email protected] and [email protected] In their 2008 paper, Mason and Dunhill state “Regenerative medicine replaces or regen- erates human cells, tissues or organs, to restore or establish normal function” (Mason & Dunhill 2008). When artificially engineering cell-based regenerative medicine projects, cells are provided support from sophisticated biomaterials, able to provide the required biomechanical and biochemical cues to the embedded cells. Hydrogels are commonly used as such biomaterials, as they mimic many of the properties of native extracellular matrix (ECM) surrounding cells in vivo. In particular, ECM consists of networks of polymers (e.g. collagen) and charged proteins bathed in interstitial fluid, and is hence a polyelectrolyte hydrogel. As well as finding applications in regenerative medicine, synthetic polyelectrolyte hydrogels are currently employed for a wide range of applications, such as drug delivery, biomedical devices and soft robotics. Of particular interest across all these areas is the phenomenon of swelling of these synthetic gels, and the subsequent degradation of these materials. In this project we will develop a continuum mathematical model of these processes, using a classical non-equilibrium thermodynamics approach. The resulting equations will be solved using a combination of analytical and numerical methods. We will also explore the application of these models to experimental regenerative medicine approaches. This project will be in collaboration with Professor Barbara Wagner (WIAS, Berlin) and Giulia Celora (WCMB).

References

[1] T. Bertrand, J. Peixinho, S. Mukhopadhyay, and C.W. MacMinn. Dynamics of swelling and drying in a spherical gel. Physical Review Applied, 6(6):064010, 2016. [2] A.D. Drozdov and J.C. Christiansen. Modeling the eﬀects of pH and ionic strength on swelling of polyelectrolyte gels. The Journal of Chemical Physics, 142(11):114904, 2015. [3] R. Abi-Akl, E. Ledieu, T.N. Enke, O.X. Cordero, and T. Cohen. Physics-based prediction of biopolymer degradation. Soft Matter, 15(20):4098–4108, 2019

20 3 Physical Application Projects

3.1 Evaporation-Driven Instabilities in Complex Fluids

Supervisor: Dr Matthew Hennessy Contact: [email protected]

Background and problem statement. Complex ﬂuids composed of a volatile solvent (e.g. water) and non-volatile components (e.g. nanoparticles) undergo a complicated drying process, as shown in Fig. 2. Removal of solvent by evaporation transforms the ﬂuid from a viscous liquid into a poroelastic solid. The generation of elastic stress during drying can lead to mechanical instabilities such as buckling, fracture, and delamination. This project will focus on using mathematical modelling to understand the evaporation- driven formation of poroelastic network and the onset of mechanical instabilities.

Figure 2: Drying of a drop of blood results in a striking fracture pattern [1].

Approach and techniques. The project will begin by reviewing the theory of poroelastic materials [2] and relevant experimental papers [3]. A nonlinear poroelastic model will then be developed for an experimental system. A recent example consists of a drop of fluid that is confined between two glass slides which fractures during drying [4]. Asymp- totic methods will be used to reduce the governing equations. Analytical or numerical solutions of the reduced model will be used to study the evolution of the solidification front and elastic stresses within the solid. Conditions for the onset of instability can be obtained using linear stability theory. A poroelastic-fracture model can be developed and numerically simulated using finite elements [5]. The results of these studies will be compared with experimental data.

Outcomes: The outcome of this project will be a suite of models that capture the poromechanics of evaporating complex ﬂuids and new understanding of their rich behaviour.

21 References

[1] B. Sobac and D. Brutin. Structural and evaporative evolutions in desiccating sessile drops of blood. Phys Rev E, 84, 2011. [2] C. W. MacMinn, E. R. Dufresne, and J. S. Wettlaufer. Fluid-driven deformation of a soft granular material. Phys Rev X, 5(1), 2015. [3] F. Giorgiutti-Dauphin and L. Pauchard. Drying drops. Euro Phys J E, 41(3), 2018. [4] A. Bouchaudy and J.-B. Salmon. Drying-induced stresses before solidification in col- loidal dispersions: in situ measurements. Soft Matter, 15(13), 2019. [5] L. Böger,M.-A. Kiep, C. Miehe. Minimization and saddle-point principles for the phase-field modeling of fracture in hydrogels. Comput Mater Sci, 138, 2017.

3.2 Phase Separation in Swollen Hydrogels

Supervisors: Dr Matthew Hennessy and Prof Andreas M¨unch Contact: [email protected] and [email protected]

Background and problem statement. Hydrogels are soft materials that are used in a large number of applications ranging from tissue engineering to 3D printing. A hydrogel can be envisioned as a network of connected springs called polymers. These polymers are highly deformable and hydrophilic, which leads to hydrogels being extremely absorbent. Thus, when exposed to a liquid such as water, hydrogels undergo substantial swelling and can increase their volume by more than a factor of 100. Hydrogels are particularly interesting because their behaviour is dictated by a combination of thermodynamics, elasticity, and ﬂuid mechanics. Furthermore, an instability known as phase separation can be triggered by subjecting a swollen hydrogel to an ex- ternal stimulus, e.g. a change in temperature, acidity, or electric ﬁeld. When phase separation occurs, a uniformly swollen hydrogel will spontaneously separate into two co-existing phases, one of which will be highly swollen and the other weakly swollen (see Fig. 3). If properly controlled, phase separation can be used to induce tuneable shape changes in materials. However, phase separation in hydrogels remains poorly understood and many experimental observations cannot be explained using conventional models such as the Cahn–Hilliard equation. This project will focus on developing the mathematical theory of phase separation in hydrogels using modelling, asymptotic analysis, and numerical simulations.

Approach and techniques. The project will begin by reviewing models of phase separation [2] and hydrogel swelling [3]. An extended hydrogel model will then be formulated which accounts for ﬂuid transport within the gel, the (nonlinear) elastic response of the gel, and phase separation [4]. The conditions for phase separation will be determined through a linear stability analysis of the governing equations. The nonlinear behaviour of the hydrogel will be explored using asymptotic methods, tools from dynamical systems theory (e.g. phase-plane analysis), and/or numerical simulations of the full model. The

22 Figure 3: Phase separation in a hydrogel cylinder leads to co-existing phases that are highly swollen and weakly swollen (collapsed) [1]. theoretical results will be interpreted in the context of the recent experiments described in [5].

Outcomes. This project will provide new insights into dynamics of phase separation in hydrogels and how this is inﬂuenced by the physical properties and geometry of the hydrogel.

References

[1] A. Suzuki and T. Ishii. Phase coexistence of neutral polymer gels under mechanical constraint. J Chem Phys, 110 (4), 1999. [2] J. W. Cahn. Phase separation by spinodal decomposition in isotropic systems. J Chem Phys, 42 (1), 1965. [3] T. Bertrand, J. Peixinho, S. Mukhopadhyay, and C. W. MacMinn. Dynamics of swelling and drying in a spherical gel. Phys Rev Appl, 6 (6), 2016. [4] W. Hong and X. Wang. A phase-ﬁeld model for systems with coupled large deformation and mass transport. J Mech Phys Solids, 61 (6), 2013. [5] R. W. Style et al., Liquid-liquid phase separation in an elastic network. Phys Rev X, 8 (1), 2018.

3.3 Pattern Formation in Polymers

Supervisor: Prof Andreas M¨unch Contact: [email protected] Polymer systems are made of diﬀerent components and therefore typically undergo spon- taneous pattern formation at the micro- and nano-scale. These structures determine the

23 properties of the polymer and are crucial for technological applications or for understanding function in biology. A continuum theory that captures the structure of these polymers is the so-called self-consistent ﬁeld theory (SCFT). However the full theory can only be solved numerically. To obtain a ﬁrst analytical glimpse of the onset of instabilities or pattern formation, the random phase approximation of the full SCFT has been widely used, for example, to predict when the pattern formation occurs. In this project, we will develop a systematic asymptotic approach to this approximation for a polymer system and compare this with numerical results.

24 4 Data Science

4.1 Computational Topology for Analysing Neuronal Branching Mod- els and Data

Supervisors: Prof Heather Harrington and Dr Bernadette Stolz-Pretzer Contact: [email protected] and [email protected] Topological data analysis is a vibrant field of computational mathematics, rich with theory combined with statistics, optimisation and data. A particularly successful application field is neurotopology, with the brain being a complex system. Different levels of complexity have been analysed, ranging from structural brain networks to functional brain networks. Here we aim to analyse a smaller structure — the branching of neurons — using a method developed by Kanari et al. Experimentally, data of these dendritic structures in flies has been obtained from the lab of Joe Howard (Yale). This group have also developed mathematical models (agent based models). They would like methods for characterising the geometry of such complex biological data and models. One such method we will test is topological data analysis. The student will learn persistent homology (theory and practical computation), and test experimental input data as well as model input data. Then the student will implement agent based model of the system, simulate different conditions/parameters, and test whether data from different parameters of the model give different topological (and model) classifications. To do this, they will learn how to interpret the output of persistent homology statistically, e.g., using output summaries such as persistence images which can be used as input into statistical and machine learning methods. Finally, this methodology will be applied to the experimental dataset.

References

[1] Lida Kanari, Pawel Dlotko, Martina Scolamiero, Ran Levi, Julian Shillcock, Kathryn Hess, Henry Markram. A Topological Representation of Branching Neuronal Mor- phologies. Neuroinformatics 16: 3–13, 2017. [2] Nina Otter, Mason A. Porter, Ulrike Tillmann, Peter Grindrod, and Heather A. Harrington. A roadmap for the computation of persistent homology. EPJ Data Science, 6:17, 2017. [3] Ann E. Sizemore, Jennifer E. Phillips-Cremins, Robert Ghrist, and Danielle S. Bas- sett. The importance of the whole: Topological data analysis for the network neuro- scientist. Network Neuroscience, 3(3):656–673, 2019

25 5 Research Interests of Academic Staﬀ

• Professor Ruth Baker — Periodic pattern formation in developmental biology, domain growth, biological oscillators, stochastic modelling of cell motility.

• Professor Chris Breward — Fluid mechanics, surfactants, modelling biological and industrial systems.

• Professor Helen Byrne — Mathematical biology, continuum and multiscale modelling of biomedical systems.

• Professor Coralia Cartis — Algorithm design, analysis and implementation for linear and nonlinear optimisation, convex and nonconvex problems.

• Professor Jon Chapman — Modelling, asymptotics and diﬀerential equations applied to ﬂuid and solid mechanics including applications in medicine and biology.

• Dr Mohit Dalwadi — Cleaning and decontamination, mathematical modelling of industrial problems, tissue engineering, singular perturbation theory, synthetic biology, viscous ﬂow.

• Professor Paul Dellar — Lattice Boltzmann methods, kinetic theory, Hamiltonian and geophysical ﬂuid dynamics, scientiﬁc computation, magnetohydrodynamics.

• Professor Radek Erban — Applied mathematics, mathematical biology, multiscale modelling, partial diﬀerential equations, stochastic simulation algorithms, gene regulatory networks, chemotaxis, collective animal behaviour.

• Professor Doyne Farmer — Economics, including agent-based modelling, ﬁnancial instability and technological progress.

• Professor Patrick Farrell — automatic derivation of adjoint models and applications, adaptive mesh discretisations.

• Professor Andrew Fowler — Environmental and geophysical problems, dynamical systems, medical applications.

• Professor Eamonn Gaffney — Modelling tumours and chemotherapy scheduling, pattern formation, reaction diffusion systems, microbiological fluid dynamics, models of cell movement, signalling and interaction.

• Dr Kathryn Gillow — Numerical methods for problems arising in biology and electrochemistry, in particular adaptive ﬁnite element methods, high order methods and methods for inverse problems.

• Professor Alain Goriely — Solid mechanics, morphogenesis, growth, dynamical systems.

• Professor Ian Griffiths — Hydrodynamics with application to physicochemical applications; water purification; glass manufacture; slow viscous flow; surfactant systems; asymptotic analysis.

26 • Professor Peter Grindrod — Dynamically evolving networks; dynamical systems and DDEs; inference and forecasting problems; real time recognition of anomalies within vast communications data sets; applications of mathematics to social media, digital media and marketing, the digital economy and social norms and attitudes.

• Professor Heather Harrington — Algebraic systems biology; inverse problems; computational biology; information processing in biological and chemical systems.

• Professor Raphael Hauser — Numerical optimisation, applied probability, operations research.

• Dr Matthew Hennessy — Interdisplinary problems involving continuum mechanics, heat and mass transport, and interfacial phenomena.

• Professor Ian Hewitt — Mathematial modelling, applications to geoscience, ﬂuid mechanics, solid mechanics.

• Professor Peter Howell — Modelling, asymptotics and diﬀerential equations, applied to ﬂuid and solid dynamics.

• Professor Sam Howison — Mathematical finance, free boundary problems in heat flow and fluid dynamics, superconductivity.

• Dr Andrew Krause — Nonlinear dynamical systems, with applications to mathematical biology and other areas.

• Professor Renaud Lambiotte — Complex systems, dynamics on networks, temporal networks.

• Professor Philip Maini — Mathematical and computational modelling of temporal and spatiotemporal phenomena in the life sciences, including developmental biology, wound healing, cancer biology and bacterial chemotaxis.

• Professor Irene Moroz — Nonlinear geophysical ﬂuid dynamics, wavelets, pre- dictability, dynamical systems, voice morphing, nonlinear time series analysis, tracking, data assimilation.

• Professor Derek Moulton — Problems relating to mechanical biology and physi- ology, growth and pattern formation, morphoelasticity, and elastic mechanisms in nature.

• Professor Andreas Münch — Nano and microfluidics, capillary interfaces, asymptotics, scientific computing.

• Professor Yuji Nakatsukasa — Rational approximation theory, numerical linear algebra, especially matrix eigenvalue problems.

• Professor James Oliver — Free and moving boundary problems in ﬂuid dynamics and biology; splashing and jet impact, cell motility, biomechanics.

• Professor Colin Please — Modelling of physical phenomena arising in practical problems, primarily in engineering and bio-science.

27 • Dr Anna Seigal — Tensors and multilinear algebra, applied algebraic geometry, algebraic statistics.

• Dr Priya Subramanian — Understanding physical mechanisms that govern spatiotemporal patterns and emergent behaviours that occur in convective/shear ﬂows of ﬂuids, in the motion of active organelle laments on motility assays and in thermo- acoustic interactions.

• Professor Endre Süli— Partial differential equations and their numerical analysis, and the mathematical theory of finite element methods for nonlinear PDEs.

• Professor Jared Tanner — Design, analysis and application of numerical algorithms for information inspired applications in signal and image processing.

• Professor Nick Trefethen — Numerical algorithms, especially “Chebyshev technol- ogy” and other tools for numerical computation with functions.

• Dr Carolina Urzua Torres — Boundary integral equations, wave scattering, preconditioning.

• Professor Dominic Vella — Surface tension, thin elastic objects, ﬂow in porous media.

• Professor Sarah Waters — Mathematical medicine and biology, bioﬂuid mechanics, biomechanics, tissue engineering.

• Professor Andy Wathen — Numerical analysis of methods for partial diﬀerential equations and numerical linear algebra.