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Recommended modules: Functional Analysis, Partial Differential EquationsReferences:!$1$! Larman, D. 72(1972), 205--207Masoumeh DashtiProject 1: Metrics on the space of probability measures (Dr M. Dashti)Studying the convergence properties of sequences of probability measures comes up in many applications (for example in the study of approximations of probability measures and stochastic inverse problems).

In such problems, it is of course important to choose an appropriate metric on the space of the probability measures. This project consists of learning about some of the important metrics on the space of probability measures (for example: Hellinger, Prokhorov and Wasserstein), and studying the relationship between them.

We also look at convergence properties of some sampling techniques. Key words: probability metrics, rates of convergence, Bayesian inverse problemsRecommended modules: Introduction to Probability, Measure and Integration.

(2002) On choosing and bounding probability metrics. Project 2: Inverse problems: classical and Bayesian approach (Dr M. Dashti)Consider the problem of finding the initial temperature field of a one dimensional heat equation from (noisy) measurements of the temperature function at a positive time.

This is an example of an inverse problem (considering the underlying heat equation, given initial temperature field, as the direct problem). Such problems where the function of interest cannot be observed directly, and has to be obtained from other observable quantities and through the mathematical model relating them, appears in many practical situations.

Inverse problems in general do not satisfy Hadamard's conditions of well-posedness: for example in the case of the above inverse heat problem, the solution (here the initial field) does not depends continuously on the temperature function at a positive time. We can, however, obtain a reasonable approximation of the solution in a stable way by regularizing the problem using a priori information about the solution.

In this project, we will study classical regularization methods, and also the Bayesian approach to regularization in the case of statistical noise. Key words: Inverse problems, Tikhonov regularization, Bayesian regularizationRecommended modules: Partial differential equations, Functional analysis, Probability and statistics, Measure and Integration.

(2000) Regularization of inverse problems, Kluwer Academic Publishers. (2010) Inverse problems: a Bayesian perspective, 19, 451--559. Project 3: Conditional regularity of the Navier-Stokes equations in terms of pressure (Dr M.

Dashti)We start by studying Leray-Hopf weak solutions of the three dimensional Navier-Stokes equations which are known to exit globally (for all positive times). The strong solutions are only known to exist locally.

There are, however, results which show the global existence of strong solutions under extra conditions on the velocity field or pressure (conditional regularity results). In this direction, we will study Serrin's conditional regularity result and then examine similar conditions in terms of the pressure field.

Key words: Navier-Stokes equations, Regularity theoryRecommended modules: Partial differential equations, Functional analysis, Measure and Integration. (2001) Regularity criterion in terms of pressure for the NavierStokes equations, Nonlinear AnalysisArchive for Rational Mechanics and Analysis, 9, 187-195.

Andrew DuncanProject 1: Approximate Bayesian Computation (Dr A. Duncan)In Bayesian inference, the objective is to sample from the posterior distribution !$$\pi(\theta|X) = \frac \pi(X|\theta) \pi 0(\theta) \pi(X)$$! of the parameter !$\theta$! given the observed data !$X$!.

In many applications, the likelihood !$\pi(X|\theta)$! has no closed form expression, or is simply too expensive to calculate, however it is relatively cheap to generate samples of $X$ for a given parameter value !$\theta$!. Approximate Bayesian Computation (ABC) is a computational technique which can be applied in such situations.

Applications of ABC can be found in genetics and in the analysis of complex stochastic systems. The aim of the project is to understand and summarise the article by Beaumont, to implement the ABC method on a computer, and to perform numerical experiments with the resulting program.

This project is mostly computational, but also has some theoretical aspects. Key words: Bayesian Inference, intractible likelihoods, big data.

Recommended modules: Probability Models, Random Processes, Numerical Differential Equations, Statistics. References:!$1$! Mark A Beaumont, Wenyang Zhang, and David J Balding.

Approximate bayesian computation in population genetics. Non-linear regression models for approximate bayesian computation.

Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate Bayesian computation. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 74(3):419-474, 2012.

Project 2: Optimal Tuning of Metropolis-Hastings Algorithms (Dr A. Duncan)In many applications, we are interested in generating samples from a probability distribution !$\pi(x) = \frac 1 Z \widetilde \pi (x)$! which will typically be known only up to a normalization constant !$Z$!, so that only the unnormalized density !$\widetilde \pi$! is generally available.

This is typically the case:in statistical physics, where given a Hamiltonian !$H:S \rightarrow \mathbb R$!, we wish to sample from the associated Boltzman distribution !$\pi \beta(x) \propto \exp(-\beta H(x))$! which describes the system at inverse temperature !$\beta$!. in Bayesian statistics, where the posterior distribution !$\pi(\theta \, | \, X) \propto \pi(X \, | \,\theta)\pi 0(\theta)$! is again defined up to a normalization constant.

Given the unnormalized density !$\widetilde \pi$!, the Markov Chain Monte Carlo (MCMC) approach to sampling involves constructing a Markov chain !$(\theta n) n \in \mathbb N$! which possesses !$\pi$! as the unique stationary distribution. One such approach is the \textbf Metropolis-Hastings algorithm which as discovered by Metropolis and co-authors in 1953 and described in their celebrated paper 2 .

This method has enjoyed huge popularity throughout many areas of science, mainly due to its ease of implementation and wide applicability. The general idea of the scheme is that at step !$n$!, given !$\theta n$! a \textbf proposal !$y$! is general for !$\theta n+1$!, sampled from a proposal distribution.

This proposal !$y$! is then accepted with probability dictated by the \textbf acceptance probability which will depend on !$\widetilde \pi$!, in which case !$\theta n+1 = y$!. Otherwise the proposal is rejected and !$\theta n+1 = \theta n$!.

A good choice of the proposal distribution is crucial to the success of the Metropolis-Hastings scheme, and a natural question is how to choose the proposal distribution optimally to maximize the performance of the algorithm. Before the 90's, the tuning of the proposals was almost invariably performed by trial and error, with a number of rules of thumb being applied.

It came as a surprise when it was proved by Roberts, Gelman and Gilks that, under certain assumptions, it is optimal to accept a proportion of only 23% of the proposed moves. This means that for optimality the chain must stay still 77\% of the time, which might seem counterintuitive when the goal is to obtain a chain converging fast to its stationary distribution.

The seminal result described in 3 is only valid for random walk proposals with a very specific target distribution, and described the behaviour of the algorithm at stationarity: different authors have subsequently tried to relax the different assumptions needed to the application of the 23% rule. The aim of this project is to understand and summarize the paper by Roberts, Gelman and Gilks and some of its extensions, to implement a Metropolis-Hastings scheme and numerically test the conclusions of that paper, and possibly to investigate some new extensions.

This project is mostly analytical but there will also be a computational component. Key words: Bayesian Inference, Markov Chain Monte Carlo, Diffusion Limits, Stochastic Differential EquationsRecommended modules: Probability Models, Random Processes, Numerical Differential Equations, Statistics.

References:!$1$! Robert, Christian, and George Casella. !$2$! Nicholas Metropolis, Arianna W Rosenbluth, Marshall N Rosenbluth, Augusta H Teller, and Edward Teller.

Equation of state calculations by fast computing machines. (The journal of chemical physics), 21(6), 1087-1092, 1953!$3$! Gareth O Roberts, Andrew Gelman, Walter R Gilks, et al.

Weak convergence and optimal scaling of random walk metropolis algorithms. (The annals of applied probability) 7 (1), 110-120, 1997.

Project 3: Stochastic Homogenization of Nonlinear PDEs (Dr A. Duncan)Many applications, such as porous media or composite materials, involve heterogeneous media described by partial differential equations with coefficients that randomly vary on a small scale.

On macroscopic scales (large compared to the dimension of the heterogeneities) such media often show an effective behavior. Typically that behavior is simpler, since the complicated, random small scale structure of the media will be averaged out'' on large scales.

Indeed, in many cases the effective behavior can be described by a deterministic, macroscopic model with constant coefficients. This process of coarse graining the inhomogenieties of the medium is called homogenization .

Mathematically, it means that the replacement of the original random equation by one with certain constant, deterministic coefficients is a valid approximation in the limit when the ratio between macroscale and microscale tends to infinity. \\\\ The aim of this project is to understand and review the theory of stochastic homogenization for linear and nonlinear elliptic PDEs with random coefficients, starting from selected chapters from 1 .

The main aim is to understand the concepts of compensated compactness and the div-curl lemma 2,3 , Weyl's decomposition, convex analysis 4 and their application to nonlinear PDEs 5 . \\\\Key words: Weak Convergence, Homogenization, Random Media, Div-Curl LemmaRecommended modules: Probability Models, Random Processes, Partial Differential Equations, Functional Analysis, Measure Theory.

References:!$1$! Vasilii Jikov, Sergei M Kozlov, and Olga Arsen'evna Oleinik. Homogenization of differential operators and integral functionals .

Springer Science \& Business Media, 2012!$2$! Franois Murat. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 5(3), 489-507, 1978Nonlinear analysis and mechanics, Heriot-Watt symposium, volume 4, pages 136--211. Scale-integration and scale-disintegration in nonlinear homogenization.

Calculus of Variations and Partial Differential Equations36(4), 565-590, 2009Project 1: Astronomical Geometry (Dr R. Fenn)This would be of interest to those with a liking for astronomy. Topics could include; fixing the stars in the sky and spherical geometry, sidereal time, planetary motion, parallax, refraction and the solution of the many body problem.

Key words: position of stars and planets, spherical geometry, calendar timeRecommended modules: Linear Algebra, Foundations of Analytical Skills, AnalysisReferences:!$2$! Fenn, R. Fenn)The problem of distinguishing two knots in a piece of string has occupied mathematicians since the beginning of the twentieth century.

A convenient method is to assign polynomials to all knots by some means and to distinguish the knots (hopefully) by showing that their polynomials are different. The first polynomial was discovered by Alexander in 1928.

More recently a whole clan of different polynomials have been discussed. Although these do not distinguish all knots, they do however show that many simple knots are different.

Key words: knots, geometric topology, polynomialsRecommended modules: Linear Algebra, Foundations of Analytical Skills, AnalysisReferences: !$1$! Alexander, J. W Topological invariants of knots and links, Trans.

R A polynomial invariant for knots via von Neumann algebras, Bull. C The new polynomial invariants of knots and links, Maths Magazine 61 (1988), pp 3-23. Nicos GeorgiouProject 1: Optimal sequence alignments (Dr N.

Georgiou)Consider three words in di erent languages, for example Henry, Enrico and Heinrich. The alphabet used to construct these words is the Latin alphabet and just by looking at them we are convinced that the words look “the same”.

One way to make this rigorous is to “align” the words so that similar letters line up. For example,Hoand then introduce a score function for which aligned letters gain a score +1, mismatched letters (like h and o in the last two words) get penalised by −a and gaps (denoted by underscores) are penalised by −b.

An alignment is called optimal if it achieves the highest possible score from all possible alignments between the words and a high score indicates a higher probability that the words are “similar”. This can be used for example when comparing DNA sequences of two species.

While the example above is deterministic, one can add randomness to this by creating two words over any nite alphabet with random letters, and then trying to nd the optimal score and alignment, as well as the behaviour of these scores when the alphabet size k, costs a, b and size n change. The project has several aspects suitable for various forms of an MSc thesis:(1) Theoretical aspects: There exists a vast literature on the longest common subsequence (LCS) of words from a nite alphabet.

This is only the case that the gap penalty is 0, but already the project can be only on the LCS. E cient algorithms computing optimal alignments and optimality regions (what are those?:-)) are scarce and buried in mathematical biology books and journals. (3) The problem is a window to an area of mathematics called “Algebraic Statistics” that can serve as an umbrella to the thesis.

Several other problems are analysed in the area with similar techniques. Key words: Sequence alignment, global alignment, optimality regions, multiple sequence alignments, algebraic statistics.

Georgiou) Consider a collection of independent Bernoulli random variables !$\ X v\ v\in \mathbb Z 2$! with !$\mathbb P(X v=1)=p=1-q$! and interpret the event that !$X v=1$! as the event of having site !$v$! as marked. ,n\ $! we can define the random variable !$L(m,n)$! that denotes the maximum possible number of marked sites that one can collect along a path from !$(1,1)$! to !$(m,n)$! that is strictly increasing in both coordinates. It is possible that there is more than one optimal path, and any such path is called a Bernoulli longest increasing path (BLIP). ' The random variables !$L(m,n)$! satisfy a certain property, called subadditivity. By Kingman's Subadditive Ergodic Theorem one can prove !$n Part of the project will be to prove the closed formula for !$\Psi(x,y)$! given by\begin equation \Psi(x,y) =\left\ \begin array lll x, & \textrm if x< py \\ \displaystyle \frac 2\sqrt pxy -p(x+y) q , & \textrm if p

-1 y\geq x\geq py \\ y, &\textrm if y< px \end array \right. \end equation There is a vast literature in statistical physics that studies this model as a simplified alternative to the hard longest common subsequence (LCS) model (see other projects).

Key words: Longest increasing path, Hammersley process, totally asymmetric simple exclusion process, corner growth model, last passage percolation, subadditive ergodic theoremPeter GieslProject 1: Calculating Basins of Attraction with Radial Basis Functions (BSc, MMath, MSc)(Dr P Giesl)This project is concerned with a discrete dynamical system, given by the iterations !$x n+1 =f(x n)$! with given function !$f\colon\mathbb R a fixed point) consists of all solutions that approach the attractor as !$n$! goes to infinity. It is difficult to find the attractors and their basins of attraction in a general system. For one attractor, we can use a Lyapunov function !$V\colon \mathbb R

d\to \mathbb R$!, which decreases along solution of the dynamical system, to find its basin of attraction. When considering several attractors and their basin of attraction, we can define a complete Lyapunov function, which decreases along solutions everywhere apart from the attractors (and repellers, in fact the so-called chain-recurrent set), where it is constant. We consider a function which satisfies the equation !$$V'(x)=V(f(x))-V(x)=-1,$$! i. Such a function actually is not defined on the attractors, but we can still seek to approximate this function using Radial Basis Functions, a numerical approximation method. The method to approximate a complete Lyapunov function using Radial Basis Functions has been introduced for differential equations, but not yet for discrete dynamical systems. This project will mainly consist of programming this method in MATLAB and trying it out on lots of examples. Key words: discrete dynamical systems, complete Lyapunov function, approximation, MATLAB, Radial Basis Functions. Recommended modules: Dynamical Systems, (Introduction to Mathematical Biology)References: !$ 1 $! Jack Hale \& Huseyin Kocak: Dynamics and Bifurcations, Springer 1991. !$ 2 $! Charles Conley: Isolated invariant sets and the Morse index. CBMS Regional Conference Series in Mathematics, 38. Fasshauer: Meshfree approximation methods with MATLAB. !$ 4 $! Peter Giesl: On the determination of the basin of attraction of discrete dynamical systems. !$ 5 $! Carlos Argaez, Peter Giesl \& Sigurdur Hafstein: Analysing Dynamical Systems - Towards Computing Complete Lyapunov Functions. ### My wife's social science phd is as confusing as my maths dissertation Project 2: Calculating the dimension of attractors with Radial Basis Functions (only MMath and MSc) (Dr P Giesl)This project is concerned with an ordinary differential equation \begin eqnarray \dot x &=&f(x)\label ODE \end eqnarray where !$x\in\mathbb R

A (global) attractor !$A$! of a dynamical system is a compact, invariant set that attracts all bounded sets.

Its (Hausdorff) dimension can be bounded by !$\dim H A Radial Basis Functions have been used to compute Lyapunov functions, given an equation for its orbital derivative. Here, we will need to either find a suitable equation or solve the inequality. This project will consist of understanding the problem, deriving an equation, programming this method in MATLAB and trying it out on examples, such as the Lorenz attractor. Key words: dynamical systems, attractor, dimension, Lyapunov function, approximation, MATLAB, Radial Basis FunctionsRecommended modules: Dynamical Systems, (Introduction to Mathematical Biology)References: !$ 1 $! Paul Glendinning: Stability, instability and chaos: an introduction to the theory of nonlinear differential equations. Fasshauer: Meshfree approximation methods with MATLAB. !$ 3 $! Peter Giesl: Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Mathematics, Springer 2007. Gelfert: Dimension estimates in smooth dynamics: a survey of recent results Ergod. Boichenko: Lyapunov's direct method in the estimation of the Hausdorff dimension of attractors. Shepeljavyi: Frequency methods in oscillation theory. James HirschfeldProject 1: Algebraic Geometry (Professor J. Hirschfeld)Given one or more polynomials in several indeterminates, what do their set of common zeros look like? Curves and surfaces are typical examples. This topic examines the basic theory of such objects. It can be done both at an elementary level and at a more sophisticated level. The material of the Term 7 course on Ring Theory would be handy. Recommended modules: Galois Theory, Coding TheoryReferences: !$ 2 $! Hirschfeld, J. Projective Geometries over a Finite Field Oxford University Press, 1998. Istvan KissMathematical epidemiology is the study of the spread of diseases, in space and time, with the objective to trace factors that are responsible for, or contribute to, their occurrence 1-6 . Mathematical models are frequently used in real applications (e. control of Childhood disease, Foot-and-Mouth disease and Pandemic Influenza outbreak) with the aim to predict the time course of an epidemic and to determine the efficacy of various control strategies such as vaccination and (S), infected and infectious (I), and recovered or removed (R). In these basic but fundamental models, susceptible or healthy individuals can become infected upon contact with infected individuals and these can then recover and become susceptible again or become immune or removed with no further impact on the epidemic. In this context the following projects are proposed:Project 1: Adaptive networks and their impact on epidemic models (Dr I. Kiss)We will consider a pairwise model that allows to capturing epidemic dynamics on a network that evolves in time 8 . Namely, the network has a fixed set of edges that can become deactivated and re-activated, as a possible response by individuals who try to avoid infection. The aim of this project is to formulate an SIS (susceptible-infected-susceptible) pairwise model and to analyse this both analytically and numerically in order to determine the epidemic threshold, disease prevalence and to characterise the interaction between disease and network dynamics. The project will involve model formulation and the derivation of differential equations to construct the pairwise model, as well as analytical and numerical analysis of the resulting model. Key words: networks, ordinary differential equations, dynamical systems, simulation, stochastic processes, MatlabRecommended modules: Mathematical Modelling, Mathematics in Everyday Life, Computing with Matlab, Applied Mathematics, Probability and Statistics, Introduction to Mathematical Biology, Random ProcessesProject 2: Model reduction techniques for epidemic dynamics on networks (Dr I. Kiss)This project will focus on various types of stochastic epidemic models 7, 9 based on the classic SIS (susceptible-infected-susceptible) and SIR (susceptible-infected-removed) models. In this project we aim to derive exact models by exploring the symmetries of the network in term of the networks automorphism group. We will start from simple toy networks, with extension to more realistic networks, and we will formulate ordinary differential equation models that are related to the Kolmogorov forward equations corresponding to a continuous time Markov Chain. The project will involve model formulation, numerical solution to the formulated model, as well as comparison to simulation results. Key words: probability, stochastic processes, Markov Chain, Kolmogorov equations, automorphism, ODEs, MatlabRecommended modules: Mathematical Modelling, Mathematics in Everyday Life, Computing with Matlab, Applied Mathematics, Probability and Statistics, Introduction to Mathematical Biology, Random ProcessesProject 3: Non-markovian epidemic models on networks (Dr I. Kiss)This project will aim to develop some basic models where the infection and/or recovery process is non-markovian 10 . While this makes the modeling and the mathematical analysis more challenging, it provides a more realistic model compared to classic models where the infection and recovery processes are all Poisson. This is relatively new area with some results, but with much scope for further progress. Model development and analysis will be accompanied by numerical simulation. Key words: probability, networks, simulation, stochastic processes, MatlabRecommended modules: Mathematical Modelling, Mathematics in Everyday Life, Computing with Matlab, Applied Mathematics, Probability and Statistics, Introduction to Mathematical Biology, Random ProcessesReferences for all three projects: !$ 1 $! Roberts, M. (2008) Modeling infectious diseases in animals and humans. (2003) Essential Mathematical Biology: Infectious Diseases. (2000) Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Cambridge Studies in Mathematical Biology, Cambridge University Press. (2011) Exact epidemic models on graphs using graph-automorphism driven lumping. Simon (2012) Modelling approaches for simple dynamic networks and applications to disease transmission models. Wilkinson (2013) Exact deterministic representation of Markovian SIR epidemics on networks with and without loops. van de Bovenkamp (2013) Non-Markovian infection spread dramatically alters the susceptible-infected-susceptible epidemic threshold in networks. Konstantinos KoumatosProject 1: A variational approach to microstructure formation in materials: from theory to design of smart materials (Dr K. Koumatos)From the prototypical example of steel to modern day shape-memory alloys, materials undergoing martensitic transformations exhibit remarkable properties and are used in a wide range of applications, e. as thermal actuators, in medical devices, in automotive engineering and robotics. The properties of these materials, such as the toughness of steel or Nitinol being able to remember its original shape, are related to what happens at small length scales and the ability of these materials to form complex microstructures. Hence, understanding how microstructures form and how they give rise to these properties is key, not only to find new applications, but also to design new materials. A mathematical model, developed primarily in the last 30 years 1,2,3 , views microstructures as minimizers of an energy associated to the material and has been very successful in explaining many observables. In fact, it has been successful even in contributing to the design of new smart materials which exhibit enhanced reversibility and low hysteresis, properties which are crucial in applications. In this project, we will review the mathematical theory - based on nonlinear elasticity and the calculus of variations - and how it has been able to give rise to new materials with improved properties. Depending upon preferences, the project can be more or less technical. Key words: microstructure, energy minimisation, elasticity, calculus of variations, non-convex variational problemsRecommended modules: Continuum Mechanics, Partial Differential Equations, Functional Analysis, Measure and IntegrationReferences: !$ 1 $! J. Ball, Mathematical models of martensitic microstructure, Materials Science and Engineering A 378, 61--69, 2004!$ 2 $! J. James, Fine phase mixtures as minimizers of energy, Archive for Rational Mechanics and Analysis 100 (1), 13--52, 1987!$ 3 $! K. Bhattacharya, Microstructure of martensite: why it forms and how it gives rise to the shape-memory effect, Oxford University Press, 2003!$ 4 $! X. James, Study of the cofactor conditions: conditions of supercompatibility between phases, Journal of the Mechanics and Physics of Solids 61 (12), 2566--2587, 2013!$ 5 $! S. Muller, Variational models for microstructure and phase transitions, in Calculus of variations and geometric evolution problems, 85--210, 1999Project 2: Polyconvexity and existence theorems in elasticity (Dr K. Koumatos)The equilibrium problem of nonlinear elasticity can be formulated as that of minimising an energy functional of the form !$$\mathcal E(u) = \int \Omega W(\nabla u(x))\,dx,$$! subject to appropriate boundary conditions on !$\partial\Omega$!, where !$\Omega\subset \mathbb R

n$! represents the elastic body at its reference configuration and !$u:\Omega\to \mathbb R n$! is a deformation of the body mapping a material point !$x\in \Omega$! to its deformed configuration !$u(x)\in \mathbb R

The function !$W$! is the energy density associated to the material and physical requirements force one to assume that !$$W(F) \to \infty, \mbox as \det F\to0 \tag \ast$$! As the determinant of the gradient expresses local change of volume, the conditions above translate to the requirement of infinite energy to compress a body to zero volume as well as the requirement that admissible deformations be orientation-preserving.

It turns out that (!$\ast$!) is incompatible with standard conditions required on !$W$! to establish the existence of minimisers in the vectorial calculus of variations. In this project, we will review classical existence theorems as well as the seminal work of J.

Ball 1 proving existence of minimisers for !$\mathcal E$! and energy densities !$W$! that are !$\it polyconvex$! and fulfil condition (!$\ast$!). Such energies cover many of the standard models used in elasticity.

Key words: nonlinear elasticity, polyconvexity, quasiconvexity, existence theories, determinant constraintsRecommended modules: Continuum Mechanics, Partial Differential Equations (essential), Functional Analysis (essential), Measure and IntegrationReferences:!$1$! J. Ball, Convexity conditions and existence theorems in elasticity, Archive for Rational Mechanics and Analysis 63 (4), 337--403, 1977!$2$! B. Dacorogna, Direct methods in the calculus of variations, volume 78, Springer, 2007Project 3: Compensated compactness and existence theory in conservation laws via the vanishing viscosity method (Dr K.

Koumatos)Existence of solutions to nonlinear PDEs often relies in the following strategy: construct a suitable sequence of approximate solutions and prove that, up to a subsequence, the approximations converge to an appropriate solution of the PDE. A priori estimates coming from the PDE itself typically allow for convergence of the approximation to be established in some weak topology which, however, does not suffice to pass to the limit under a nonlinear quantity.

This loss of continuity with respect to the weak topology is a great obstacle in nonlinear problems. Murat (see 3 for a review) introduced a remarkable method, referred to as compensated compactness, which gives conditions on nonlinearities !$Q$! that allow one to establish the implication: !$$V j \rightharpoonup V \Longrightarrow Q(V j) \rightharpoonup Q(V)\tag \ast$$! under the additional information that the sequence !$V j$! satisfies some differential constraint, e.

the !$V j$!'s could be gradients, thus satisfying the constraint !$\rm curl \, V j = 0$!.

Note that (!$\ast$!) is not true in general and it is the additional information on !$V j$! that `compensates'' for the loss of compactness. In this project, we will review the compensated compactness theory and investigate its consequences on the existence theory for scalar conservation laws in dimension 1 via the vanishing viscosity method.

In particular, we will use the so-called div-curl lemma to prove that a sequence !$u \varepsilon$! verifying \begin align* \partial t u

\varepsilon\\ u(\cdot,t = 0) & = u 0 \end align* converges in an appropriate sense as $\varepsilon\to0$ to a function $u$ solving the conservation law \begin align* \partial t u + \partial x f(u) & = 0\\ u(\cdot,t = 0) & = u 0. \end align* Key words: compensated compactness, div-curl lemma, weak convergence, oscillations, convexity, wave cone, conservation laws, vanishing viscosity limitRecommended modules: Continuum Mechanics, Partial Differential Equations (essential), Functional Analysis (essential), Measure and Integration (essential)References:!$1$! C.

Dafermos, Hyperbolic conservation laws in continuum physics, Springer, 2010!$1$! L.

Evans, Weak convergence methods for nonlinear partial differential equations, American Mathematical Society, 1990!$1$! L.

Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics: Heriot-Watt symposium, 136--212, 1979Project 4: On the Di Perna-Lions theory for transport equations and ODEs with Sobolev coefficients (Dr K. Koumatos)For !$t\in \mathbb R$!, consider the system of ordinary differential equations !$$\frac d dt X(t) = b(X(t)),\quad X(0) = x\in \mathbb R \tag !\ast!$$! The classical Cauchy-Lipschitz theorem (aka Picard-Lindel\"of or Picard's existence theorem) provides global existence and uniqueness results for (!$\ast$!) under the assumption that the vector field !$b$! is Lipschitz.

fluid mechanics, kinetic theory) the Lipschitz condition on !$b$! cannot be assumed as a mere Sobolev regularity seems to be available. In pioneering work, Di Perna and Lions 2 established existence and uniqueness of appropriate solutions to (!$\ast$!) under the assumption that !$b\in W 1,1 \tiny\rm loc$!, a control on its divergence is given and some additional integrability holds. In this project, we will review the elegant work of Di Perna and Lions.

Remarkably, their proof of a statement concerning ODEs is based on the transport equation (a partial differential equation) !$$\partial t u(x,t) + b(x)\cdot \rm div \, u(x,t) = 0, \quad u(x,0) = u 0(x)$$! and the concept of renormalised solutions introduced by the same authors. The relation between (!$\ast$!) and the transport equation lies in the method of characteristics which states that smooth solutions of the transport equation are constant along solutions of the ODE, i.

$$!Key words: ODEs with Sobolev coefficients, DiPerna-Lions, transport equation, renormalised solutions, continuity equationRecommended modules: Continuum Mechanics, Partial Differential Equations (essential), Functional Analysis (essential), Measure and Integration (desirable)References:! 1 ! C. De Lellis, Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio, Seminaire Bourbaki 972, 2007! 2 ! R. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae, 98, 511--517, 1989! 3 ! L. Evans, Partial Differential Equations, American Mathematical Society, 1998Yuliya Kyrychko and Konstantin BlyussProject 1: Synchronisation of Coupled Spatially Extended Systems. Blyuss)The project will look into the ways spatially extended systems can be coupled and synchronized using time delay. This is an interesting problem which arises in numerous applications including lasers, chemical reactions and some engineering experiments. The effects of time delay and its interactions with spatial extent will be studied using analytical methods and numerical simulations. This project will require strong programming skills. Key words: synchronisation, time delay, spatially-extended systemsRecommended modules: Computing with Matlab (G5084), Applied Mathematics (G5097), Mathematical Modelling (G5083). Kurths, Synchronization: a universal concept in nonlinear sciences, Cambridge University Press, 2001. Wu, Theory and applications of partial functional differential equations, Springer, 1996. Omar LakkisProject 1: Geometric Motions and their Applications (Dr O. Lakkis)Geometric constructs such as curves, surfaces, and more generally (immersed) manifolds, are traditionally thought as static objects lying in a surrounding space. In this project we view them instead as moving within the surrounding space. While Differential Geometry, which on of the basis of Geometric Motions, is a mature theory, the study of Geometric Motions themselves has only really picked-up in the late seventies of the past century. This is quite surprising given the huge importance that geometric motions play in applications which range from phase transition to crystal growth and from fluid dynamics to image processing. Here, following the so-called classical approach, we learn first about some basic differential geometric tools such as the mean and Gaussian curvature of surfaces in usual 3-dimensional space. We then use these tools to explore a fundamental model of geometric motions: the Mean Curvature Flow. We review the properties of this motion and some of its generalisations. We look at the use of this motion in applications such as phase transition. This project has the potential to extend into a research direction, depending on the students will and ability to pursue this. One way of performing this extension would be to implement computer code simulating geometric motions and analysing the algorithms. Key words: Parabolic Partial Differential Equations, Surface Tension, Geometric Measure Theory, Fluid-dynamics, Growth Processes, Mean Curvature Flow, Ricci Flow, Differential Geometry, Phase-field, Level-set, Numerical AnalysisRecommended modules: Finite Element Methods, Measure and Integration, Numerical Linear Algebra, Numerical Differential Equations, Intro to Math Bio, Applied Whatever Modelling. , Thermomechanics of evolving phase boundaries in the plane. The Clarendon Press, Oxford University Press, New York, 1993. ISBN 0-19-853694-1! 2 ! Huisken, Gerhard, Evolution Equations in Geometry, in Mathematics unlimited-2001 and beyond, 593-604, Springer, Berlin, 2001. ! 3 ! Spivak, Michael, A Comprehensive Introduction to Differential Geometry. ISBN 0-914098-83-7! 4 ! Struwe, Michael, Geometric Evolution Problems. Nonlinear Partial Differential Equations in Differential Geometry (Park City, UT, 1992), 257-339, IAS/Park City Math. Project 2: Stochastic differential equations: computation, analysis and modelling (Dr O. Lakkis)Stochastic Differential Equations (SDEs) have become a fundamental tool in many applications ranging from environmental risk management to mechanical failure control and from neurobiology to financial analysis. While the need for effective numerical solutions of SDEs, which are differential equations with a probabilistic (uncertain) data, closed form solutions are seldom available. This project can be specialised, according to the student's tastes and skills into 3 different flavours: (1) Analysis/Theory, (2) Analysis/Computation, (3) Computational/Modelling. (1) We explore the rich theory of stochastic processes, stochastic integration and theory (existence, uniqueness, stability) of stochastic differential equations and their relationship to other fields such as the Kac-Feynman Formula (related to quantum mechanics and particle physics), or Partial Differential Equations and Potential Theory (related to the work of Einstein on Brownian Motion), stochastic dynamical systems (large deviation) or Kolmogorov's approach to turbulence in fluid-dynamics. Prerequisites for this direction are some knowledge of probability, stochastic processes, partial differential equations, measure and integration and functional analysis. (2) We review the basics of SDEs and then look at a practical way of implementing algorithms, using any one of Octave/Matlab/C/C++, that give us a numerical solution. In particular, we learn about pseudorandom numbers, Monte-Carlo methods, filtering and the interpretation of those numbers that our computer produces. Although not a strict prerequisite, some knowledge of probability, ordinary differential equations and their numerical solution will be useful. (3) We look at practical models in environmental sciences, medicine or engineering involving uncertainty (for example, the ideal installation of solar panels in a region where weather variability can affect their performance). We study these models both from a theoretical point of view (connecting to their Physics) and we run simulations using computational techniques for stochastic differential equations. The application field will be emphasised and must be clearly to the student's liking. (Although very interesting as a topic, I prefer not to deal with financial applications. ) The prerequisites are probability, random processes, numerical differential equations and some of the applied/modelling courses. Key words: Stochastic Differential Equations, Scientific Computing, Random Processes, Probability, Numerical Differential Equations, Environmental Modelling, Stochastic Modelling, Feynman-Kac Formula, Ito's Integral, Stratonovich's Integral, Stochastic Calculus, Malliavin Calculus, Filtering. Recommended modules: Probability Models, Random Processes, Numerical Differential Equations, Partial Differential Equations, Introduction to Math Biology, Fluid-dynamics, Statistics. Evans, An Introduction to stochastic differential equations. Lecture notes on authors website (google: Lawrence C Evans). Gardiner, Handbook of stochastic methods for physics, chemistry and the natural sciences. Schurz, Numerical solution of SDE through computer experiments. Stuart, MCMC methods for sampling function space, ICIAM2007 Invited Lectures (R. Doob, Classical potential theory and its probabilistic counterpart, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1984 edition. ISBN 3-540-41206-9Anotida MadzvamuseProject 1: Open problems in experimental sciences: Bringing mathematics closer to life (Dr A. Madzvamuse)This research seeks to study open topical problems in experimental sciences (biochemistry, biomedical engineering, genetics, cell biology, etc. By analysing experimental observations, the challenge is to derive new mathematical models from first principles that will describe both qualitatively and quantitatively these observations. In most cases, numerical approximate solutions to the analytical solutions will be sought. A candidate researcher should be one who is fascinated with the idea of applying mathematics to non-standard research questions emanating from experimental sciences 1 Oct 2017 - Her years of toil are neatly packaged in a 300-page dissertation on… no doubt gleaned, dinner talk in the Mubeen household can get intense (and yes, a tad tedious). But the language of mathematics can also confound.. ### Dissertation - info.maths Furthermore, good computing skills are an added advantage. In summary, the research entails identification of an appropriate research topic (in collaboration with the supervisor(s)), literature review, mathematical modelling, analysis of the models, numerical computations, solution visualisations, model validation and refinement. Finally, a project report will be written ideally using !LaTeX!. If the research results are ground-breaking, article publication is the ultimate goal. For further information, see Junior Research Fellowship Posters on level 5, Pevensey 3, between Offices 5C14 and 5C15, these give a flavour of the kind of research excellence expected. Key words: Modelling, numerical analysis, applications, conservation lawsRecommended modules: Introduction to Mathematical Biology, Dynamical Systems. Charalambos MakridakisProject 1: Mesh adaptivity and artificial diffusion in hyperbolic problems - 2 Projects (Prof. Makridakis)We are interested in finite element, finite difference schemes and adaptive strategies for the approximation of nonlinear hyperbolic systems. It is known that finite elements are not a very popular choice for computing singular solutions of hyperbolic problems. When applied directly to the system they will result computational solutions with oscillatory character close to shocks and/or not convergent approximations. This well known phenomenon is related to the fact that direct finite element discretizations behave like dispersion approximations. Similar behavior is observed in the study of related dispersive difference schemes approximating conservation laws. To overcome this difficulty in using standard schemes, several modifications have been proposed in the literature by adding artificial viscosity and / or extra stabilization terms in the schemes. The higher order versions of these methods are complicated and with poor theoretical backup. Recently many of these schemes have been tested with various mesh adaptation methods. Our motivation is to consider schemes designed to be used in conjunction with appropriate mesh refinement. Mesh (re-)distribution influences not only the accuracy of the scheme but also its stability behavior. We are interested toconsider finite element schemes for hyperbolic problems designed to be used with mesh adaptivity (Project 1). discuss adaptive strategies for shock computations based on estimator functionals or a posteriori error control (Project 2). conclude to observations for the behavior of the schemes. Classical dispersive-type finite element and finite difference schemes could be also considered. (Project 1, 2)Depending on the project, a solid background in Analysis and Real Analysis is required and/or excellent computer programming skills. Project 2: Atomistic and continuum modelling in crystalline materials (Prof. Makridakis)Modern multi-scale methods for the simulation of materials introduce several coupling mechanisms of the atomistic and the continuum descriptions aiming at the design of methods of "atomistic" accuracy with "continuum" cost. To understand these mechanisms and the behavior of the coupled models is a challenge both from the modeling point of view and from the computational perspective. It is known, for example, that ad-hoc coupling of models may lead to undesirable computational artifacts. The development and study of the mathematical foundations of coupled multiscale models therefore seems necessary. An important class of problems in multiscale modeling of crystalline materials concerns atomistic-to-continuum couplings in lattices. One of the most well-known methods in this direction is the so-called quasicontinuum method and its variants. Typically, in these methods, in regions of interest in the material (strong deformations, defects) the atomistic model is kept, while in regions of smooth deformations the atomistic model is replaced with a continuum model which, in turn, is discretized by finite elements. The main issue that arises in these methods is the proper matching of information across scales. Ad hoc coupling of atomistic and continuum energies often result in numerical artifacts at the interface between the atomistic and continuum regions; these are known as ghost forces. We are interested in studying energy-based couplings that are free of ghost forces in one, two- and three-dimensional crystal lattices with pair potentials, allowing interactions of finite but otherwise arbitrarily long range. Modern tools from Numerical Analysis become instrumental in this study. A solid background in Analysis and Real Analysis is required and/or excellent computer programming skills. Michael MelgaardProject 1: Spectral and scattering properties of Quantum Hamiltonians (Prof. Melgaard)Quantum Operator Theory concerns the analytic properties of mathematical models of quantum systems. Its achievements are among the most profound and most fascinating in Quantum Theory, e. , the calculation of the energy levels of atoms and molecules which lies at the core of Computational Quantum Chemistry. Among the many problems one can study, we give a short list:The atomic Schr dinger operator (Kato's theorem and all that);The periodic Schr dinger operator (describing crystals);Scattering properties of Schr dinger operators (describing collisions etc);Spectral and scattering properties of mesoscopic systems (quantum wires, dots etc);Phase space bounds (say, upper bounds on the number of energy levels) with applications, e. Key words: differential operators, spectral theory, scattering theory. Recommended modules: Functional Analysis, Measure and Integration theory, Partial Differential Equations. References: Stationary partial differential equations Vol. Project 2: Variational approach to Kohn-Sham models with magnetic fields (Prof. Melgaard)Quantum Mechanics (QM) has its origin in an effort to understand the properties of atoms and molecules. Its first achievement was to establish the Schr dinger equation by explaining the stability of the hydrogen atom; but hydrogen is special because it is exactly solvable. When we proceed to a molecule, however, the QM problem cannot be solved in its full generality. In particular, we cannot determine the solution (i. , the ground state) to !H =E !, where !H! denotes the Hamiltonian of the molecular system, ! ! is the wavefunction of the system, and !E! is the lowest possible energy. This problem corresponds to finding the minimum of the spectrum of !H! or, equivalently, !$$E= \inf \ \, \mathcal E 2 ! and !\mathcal H ! is the variational (Hilbert) space. For systems involving a few (say today six or seven) electrons, a direct Galerkin discretization is possible, which is known as Full CI in Computational Chemistry. For larger systems, with !N! electrons, say, this direct approach is out of reach due to the excessive dimension of the space ! 3N ! on which the wavefunctions are defined and the problem has to be approximated. Quantum Chemistry (QC), as pioneered by Fermi, Hartree, L wdin, Slater, and Thomas, emerged in an attempt to develop various ab initio approximations to the full QM problem. The approximations can be divided into wavefunction methods and density functional theory (DFT). For both, the fundamental questions include minimizing configuration, divided into Question I (i) necessary and sufficient conditions for existence of a ground state (=a minimizer), and Question I (ii) uniqueness of a minimizer, and Question II, necessary and sufficient conditions for multiple (nonminimal) solutions (i. A magnetic field has two effects on a system of electrons: (i) it tends to align their spins, and (ii) it alters their translational motion. The first effect appears when one adds a term of the form !-e m -1 s \cdot \mathcal B ! to the Hamiltonian, while the second, diamagnetic effect arises from the usual kinetic energy !(2m) Here ! \mathbf p ! is the momentum operator, !\mathcal A ! is the vector potential, !\mathcal B ! is the magnetic field associated with !\mathcal A !, and ! s ! is the angular momentum vector. Within the numerical practice, one approach is to apply a perturbation method to compute the variations of the characteristic parameters of, say, a molecule, with respect to the outside perturbation. It is interesting to go beyond and consider the full minimization problem of the perturbed energy. In Hartree-Fock theory, one only takes into account the effect (ii), whereas in nonrelativistic DFT it is common to include the spin-dependent term and to ignore (ii) and to study the minimization of the resulting nonlinear functional, which depends upon two densities, one for spin "up" electrons and the other for spin "down" electrons. Each density satisfies a normalisation constraint which can be interpreted as the total number of spin "up" or "down" electrons. The proposed project concerns the above-mentioned problems within the context of DFT in the presence of an external magnetic field. More specifically, molecular Kohn-Sham (KS) models, which turned DFT into a useful tool for doing calculations, are studied for the following settings:Recent results on rigorous QC are found in the references. As a first step towards systems subject to a magnetic field, Question I(i) is addressed for the unrestricted KS model, which is suited for the study of open shell molecular systems (i. , systems with a odd number of electrons such as radicals, and systems with an even number of electrons whose ground state is not a spin singlet state). The aim is to consider the (standard and extended) local density approximation (LDA) to DFT. The spin-polarized KS models in the presence of an external magnetic field with constant direction are studied while taking into account a realistic local spin-density approximation, in short LSDA. Key words: differential operators, spectral theory, scattering theory. Recommended modules: Functional Analysis, Measure and Integration theory, Partial Differential Equations. References: Stationary partial differential equations Vol. Project 3: Resonances for Schr dinger and Dirac operators (Prof. Melgaard)Resonances play an important role in Chemistry and Molecular Physics. in reactive scattering, state-to-state transition probabilities and photo-dissociation, and give rise to long-lived states well above scattering thresholds. The aim of the project is carry out a rigorous mathematical study on the use of Complex Absorbing Potentials (CAP) to compute resonances in Quantum Dynamics. In a typical quantum scattering scenario particles with positive energy arrive from infinity, interact with a localized potential !V(x)! whereafter they leave to infinity. The absolutely continuous spectrum of the the corresponding Schr dinger operator !T(\hbar)=-\hbar 2 D+V(x)! coincides with the positive semi-axis. Nevertheless, the Green function !G(x,x'; z)= \langle x | (T(\hbar)-z) -1 | x \rangle! admits a meromorphic continuation from the upper half-plane !\ \, \rm Im \, z >0 \,\ ! to (some part of) the lower half-plane !\ \, \rm Im \, z< 0 \,\ !. Generally, this continuation has poles !z k =E k -i k /2!, ! k >0!, which are called resonances of the scattering system. The probability density of the corresponding "eigenfunction" ! k (x)! decays in time like !e -t k / \hbar !, thus physically ! k ! represents a metastable state with a decay rate ! k / \hbar! or, re-phrased, a lifetime !\tau k =\hbar / k !. In the semi-classical limit !\hbar \to 0!, resonances !z k ! satisfying ! k =\mathcal O (\hbar)! (equivalently, with lifetimes bounded away from zero) are called "long-lived". Physically, the eigenfunction ! k (x)! only make sense near the interaction region, whereas its behaviour away from that region is evidently nonphysical (Outgoing waves of exponential growth). As a consequence, a much used approach to compute resonances approximately is to perturb the operator !T(\hbar)! by a smooth absorbing potential !-iW(x)! which is supposed to vanish in the interaction region and to be positive outside. The resulting Hamiltonian !T W (\hbar):=T(\hbar)-iW(x)! is a non-selfadjoint operator and the effect of the potential !W(x)! is to absorb outgoing waves; on the contrary, a real-valued positive potential would reflect the waves back into the interaction region. In some neighborhood of the positive axis, the spectrum of !T W (\hbar)! consists of discrete eigenvalues !\tilde z k ! corresponding to !L As mentioned above, the CAP method has been widely used in Quantum Chemistry and numerical results obtained by CAP are very good. The drawback with the use of CAP is that there are no proof that the correct resonances are obtained. (This is in stark contrast to the mathematically rigorous method of complex scaling). In applications it is assumed implicitly that the eigenvalues !\tilde z k ! near to the real axis are small perturbations of the resonances !z k ! and, likewise, the associated eigenfunctions !\widetilde k !, ! k (x)! are close to each other in the interaction region. Stefanov (2005) proved that very close to the real axis (namely, for !| \rm Im \, \tilde z k | =\mathcal O (\hbar n )! provided !n! is large enough), this is in fact true. Stefanov's proof relies on a series of ingenious developments by several people, most notably Helffer (1986), Sj strand (1986, 1991, 1997, 2001, 2002), and Zworski (1991, 2001). The first part of the project would be to understand in details Stefanov's work 2 and, subsequently, several open problems await. Key words: operator and spectral theory, semiclassical analysis, micro local analysis. Recommended modules:Functional Analysis, Measure and Integration theory, Partial Differential Equations. Melgaard, Complex absorbing potential method for Dirac operators. Stefanov, Approximating resonances with the complex absorbing potential method, Comm. Project 4: Critical point approach to solutions of nonlinear, nonlocal elliptic equations arising in Astrophysics (Prof. Melgaard)The Choquard equation in three dimensions reads:!\begin equation \tag* (0. 2 (y) W(x-y) \, dy \right) u(x) = -l u , \end equation $$! where !W! is a positive function. 2 \, dx dy,$$!which, in turn, arises from an approximation to the Hartree-Fock theory of a one-component plasma when !$W(y) =1/ | y |$! (Coulomb case).

Lieb (1977) proved that there exists a unique minimizer to the constrained problem !$E \rm NR (\nu) = \inf \ \, \mathcal E (u) \, : \, u \in \mathcal H The mathematical difficulty of the functional is caused by the minus sign in !$\mathcal E

\rm NR $!, which makes it impossible to apply standard arguments for convex functionals. Lieb overcame the lack of convexity by using the theory of symmetric decreasing functions. Later Lions (1980) proved that the unconstrained problem (0. For the constrained problem, seeking radially symmetric, normalized functions !$\| u \| L 2 =+1$!, or more generally, seeking solutions belonging to:!$$\mathcal C N = \ \, \in \mathcal H \rm r 2 =N \, \ ,$$! the situation is much more complicated and conditions on !$W$! are necessary. In the Coulomb case, Lions proves that there exists a sequence !$(l j , u j )$!, with !$l j >0$!, and !$u j $! satisfies !$ (0.

1) $! (with !$l=l j $!) and belongs to !$\mathcal C 1 $!We may replace the negative Laplace operator by the so-called quasi-relativistic operator, i. 2 -m$!; this is the kinetic energy operator of a relativistic particle of mass !$m \geq 0$!.

It is defined via multiplication in the Fourier space with the symbol !$\sqrt k 2 -m$!, which is frequently used in relativistic quantum physics models as a suitable replacement of the full (matrix valued) Dirac operator. The associated time-dependent equation arises as an effective dynamical description for an !$N$!-body quantum system of relativistic bosons with two-body interaction given by Newtonian gravity, as recently shown by Elgart and Schlein (2007).

Several questions arise for the quasi-relativistic Choquard equation (existence, uniqueness, positive solutions etc) and the first part of the project would be to get acquainted with recent (related) results, e.

Key words: operator and spectral theory, semiclassical analysis, micro local analysis. Recommended modules:Functional Analysis, Measure and Integration theory, Partial Differential Equations.

Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Zeitschrift fr Angewandte Mathematik und Physik (ZAMP) , vol.

Zongo, Multiple solutions of the quasi relativistic Choquard equation, J. Mariapia PalombaroProject 1: The brachistochrone and the minimal surface of revolution problem (Dr M. Palombaro)A classical problem in the Calculus of Variations is to find minimizers of the problem !\inf u\in X \int a

We will study the problem of the existence and uniqueness of minimizers.

In particular we will focus on the well-known brachistochrone problem and the minimal surface of revolution problem. The first consists in finding the shape of a curve down which a material point of given mass accelerated by gravity moves between two fixed points in the least time.

The second consists in finding the curve connecting two given points !$(x 1,y 1)$!, !$(x 2,y 2)$! such that, when revolved around the !$x$!-axis, yields the surface with smallest area. Key words: minimization of integral functionals, Euler-Lagrange equation, brachistochrone, minimal surface of revolution.

, Measure theory and fine properties of functions. Palombaro)The length of a curve or the area of a hypersurface in !$For such "small" sets one needs the notion of "lower dimensional" measures. We will study their definition and basic properties and we will see that the !$n$!-dimensional Hausdorff measure !$\mathcal H

, Measure theory and fine properties of functions. Derek RobinsonFor more information, please email Dr Derek Robinson or visit his staff profile.

Project 1: Modelling House Prices (Dr on)Apparently similar houses can vary drastically in their asking price.

The project is to identify the degree of influence on price of both tangible factors such as number of bedrooms and condition, and intangible factors such as the desirability of the district. It will be necessary to obtain relevant data from estate agents (the data on the internet will probably not be sufficient by itself) and create a computer database.

You will then analyse the data, mainly using the methods described in the Linear Statistical Models module taught in the Autumn semester, in order to measure the effect on price of factors such as an extra bedroom or proximity to a good school. Project 1: The replication crisis in science (Prof. Scalas)In the last decade, several results published even by self-important journals such as Nature and Science had to be retracted because they could not be reproduced. In this project, we will not focus on deliberate scientific fraud, but rather on errors made by applied scientists because of the misuse and abuse of probability and statistics.

The so-called replication crisis virtually affects all the natural sciences 1 . The situation is so serious that, recently, 72 applied statisticians called for a revision of the significance level to be used for p-values in hypothesis testing 2 .

However, there are serious doubts that this will cure the problem 3 . In this dissertation, you will review the main factors leading to the current dire situation in science and you will work out possible solutions based on probabilistic and statistical techniques.

"1,500 scientists lift the lid on reproducibility". Available at: https:// /preprints/psyarxiv/mky9j.

Vanessa StylesProject 1: Finite Difference Approximation of the Allen-Cahn Equation with Forcing (Dr V. Styles)The Allen-Cahn equation is a parabolic partial differential equation that approximates motion by mean curvature of an evolving interface.

A forcing term can be added to the equation to give forced motion by mean curvature. In this project we derive a finite difference approximation of the Allen-Cahn equation with forcing which we then solve to produce some computational results.

Key words:Partial differential equations, numerical approximation, computer programming (MATLAB/C)Recommended modules: Advanced Numerical Analysis, Numerical Solution of Partial Differential Equations, Partial Differential Equations, Programming in C++References: !$1$! K. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numerica (2005) 130-232. Project 2: Numerical approximations for solving partial differential equations on evolving curves (Dr V. Styles)We consider two methods for numerically solving partial differential equations (PDEs) on evolving curves:!$(1)$! the Evolving Surface Finite Element method!$(2)$! the Arbitrary Eulerian Evolving Surface Finite Element method.

The curve is approximated by straight line segments joining nodes on the curve and the solution of the PDE is approximated at these nodes. We investigate how the approximation solution of the PDE is influenced by the approximation of the evolving curve.

Key words: Partial differential equations, numerical approximation, computer programming (MATLAB/C)Recommended modules: Advanced Numerical Analysis, Numerical Solution of Partial Differential Equations, Partial Differential Equations, Programming in C++References: !$1$! C. Styles, An Arbitrary Eulerian Evolving Surface Finite Element Method for solving PDEs on evolving surfaces, Milan Journal of Mathematics (2012) 1-33.

Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numerica (2005) 130-232. Project 3: Image segmentation using a numerical approximation of a modified Allen-Cahn equation (Dr V. Styles)In order to partition a grayscale image we consider a modified Allen-Cahn equation.

This gives rise to a parabolic partial differential equation. We derive a finite element approximation of this equation which we then solve to produce some computational examples.

Key words: Partial differential equations, numerical approximation, computer programming (MATLAB/C)Recommended modules: Advanced Numerical Analysis, Numerical Solution of Partial Differential Equations, Partial Differential Equations, Programming in C++References: !$1$! M. Mikula, Geometric image segmentation by the Allen-Cahn equation, Appl.

!$2$! An Arbitrary Eulerian Evolving Surface Finite Element Method for solving PDEs on evolving surfaces, Milan Journal of Mathematics (2012) 1-33. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numerica (2005) 130-232.

Ali TaheriProject 1: Homotopy Groups of Sphere and Freudenthal Suspension (Dr A.

Taheri)The aim of this project is to understand the theory and some of the main tools in evaluating the homotopy groups of spheres. There are many interesting open problems with potentially vast implications.

Key words: Hurewicz and Serre fibrations, Homotopy groups, Spectral sequences, SuspensionsRecommended modules: Algebraic Topology