# Dissertation Topics

This page contains details for the topics available for final year dissertations for MMath students, and for projects for BSc students. For full information on the BSc and MMath Final Year Projects, please see this page.

These topics are also offered to students in MSc Mathematics.

Miroslav Chlebik

Project 1: Absolutely Minimizing Lipschitz Extensions and Infinity Laplacian (Dr M. Chlebík)

A continuous real-valued function !$u$! defined on a domain !$U\subseteq \mathbb{R}^n$! (!$n\geq 2$!) is called absolutely minimizing, if for any open set !$V\subset U$! and any Lipschitz function !$v$! on !$\overline{V}$! !$$v\bigm|_{\partial V}=u\bigm|_{\partial V} \qquad \implies \qquad \|\nabla u\|_{L^\infty(V)}\leq \|\nabla v\|_{L^\infty(V)}.$$! It is well-known that !$u$! is absolutely minimizing if and only if it is the solution of the infinity Laplacian, which is the (highly degenerate) Euler-Lagrange equation for the prototypical problem in the calculus of variations in !$L^\infty$!. The problem of regularity of these functions is widely open, at this time it is unknown whether they are differentiable everywhere if !$n>2$!. We examine various techniques to study pointwise behaviour of these functions.

Miroslav Chlebik Presentation [PDF 309.98KB]

Key words: Lipschitz mappings, optimal Lipschitz extension,degenerate elliptic PDEs, infinity harmonic functions.

Recommended modules: Functional Analysis, Partial Differential Equations

References:

!$[1]$! Aronsson, G., Crandall, M. G. and Juutinen, P., A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. 41(2004), no. 4, 439--505

!$[2]$! Crandall, M. G., Evans, L. C. and Gariepy, R. F., Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Diff. Equations 13(2001), no. 2, 123--139

Project 2: Metric Dimension (Dr M. Chlebík)

Hausdorff dimension is the principal notion of dimension in the context of fractal sets in !$\mathbb{R}^n$!, or even for general metric spaces. However, other definitions are in widespread use, for example, packing dimension, upper and lower box-counting dimension, upper and lower Minkowski dimension, ... We will examine some of these and their inter-relationship.

Miroslav Chlebik Presentation [PDF 309.98KB]

Key words: Hausdorff dimension, Lipschitz mappings, rectifiable sets, fractals

Recommended modules: Measure and Integration, Functional Analysis

References:

!$[1]$! Falconer, K., Fractal geometry: Mathematical Foundations and Applications, John Wiley & Sons Ltd., 1990

Project 3: Arcs with Increasing Chords (Dr M. Chlebík)

A curve !$C$! in the plane has the increasing chord property if !$\|x_2-x_3\|\leq \|x_1-x_4\|$! whenever !$x_1$!, !$x_2$!, !$x_3$! and !$x_4$! lie in that order on !$C$!. Larman & Mc Mullen showed that !$$L\leq 2\sqrt 3|a-b|,$$! where !$C$! is a plane curve with the increasing chord property with length !$L$! and endpoints !$a$! and !$b$!. We will examine how to improve the above constant "!$2\sqrt 3$!". (It is conjectured that !$L\leq \frac23\pi|a-b|$!, with equality if !$C$! consists of two sides of a Reuleaux triangle.)

Miroslav Chlebik Presentation [PDF 309.98KB]

Key words: curve length, Lipschitz curve, calculus of variations

Recommended modules: Functional Analysis, Partial Differential Equations

References:

!$[1]$! Larman, D. G. and McMullen P., Arcs with increasing chords, Proc. Cambridge Philos. Soc. 72(1972), 205--207

Marianna Cerasuolo

Project 1:Tumour modelling: Understanding your enemy is the first step towards victory!

This project will focus on understanding, through a strong mathematical approach, the dynamics of tumour cells. From Britton: “Biological processes such as cell proliferation are normally extremely tightly controlled through feedback processes that are mainly chemically mediated... There are cell populations that escape from such controls through mutations that allow them to manipulate their local environment. Some mutations that cancer cells undergo may be sufficient to allow the immune system to recognise them as foreign, and hence to mount a defence against them.” However, such defence is not always effective. The use of reaction diffusion equation models for the description of the dynamics of tumour growth and the processes involved is explored. The project will focus on the effect of environmental inhomogeneity on the mutations (evolution) of a tumour growing cell-population.

References:

[1] Burbanks, A., Cerasuolo, M., Ronca, R., & Turner, L. (2023). A hybrid spatiotemporal model of PCa dynamics and insights into optimal therapeutic strategies. Mathematical Biosciences, 355, 108940.

[2] Krause, A. L., Gaffney, E. A., & Walker, B. J. (2023). Concentration-Dependent Domain Evolution in Reaction–Diffusion Systems. Bulletin of Mathematical Biology, 85(2), 14.

Project 2:How can we measure and interpret entropy in chaotic data?

Physiological signals such as ECG and blood pressure contain a high degree of chaos. Often the absence of chaos and the consequent regularization of the signal implies a deterioration of the patient’s health status. In this project baseline and LPS data of blood pressure over 24 hours will be represented and interpreted using various mathematical techniques, for example:

• Heart rate variability (HRV) measure

• SPAR waveform variability measures

• Fractal dimension measures All different methods used to extract the deterioration information will be compared with the aim of finding the optimal technique for the data of interest.

References:

[1] Aston P.J., Nandi M., Christie M., & Huang Y., (2014),“Comparison of Attractor Reconstruction and HRV Methods for Analysing Blood Pressure Data”, Computing in Cardiology. Volume 41

[2] Steven H. Strogatz,(1994), “Nonlinear Dynamics and Chaos: With applications to physics Biology, Chemistry and Engineering”,Perseus Books Publishing.

Antoine Dahlqvist

Project 1: RANDOM MATRICES AND FREE PROBABILITY

See PDF for full description

Antoine Dahlqvist - Random matrices and Free Probability [PDF 345.10KB]

Project 2: FROM M/M/1 TO BROWNIAN QUEUES, SYMMETRIES OF THE BROWNIAN PATH.

See PDF for full description

Antoine Dahlqvist - Brownian queues [PDF 151.46KB]

Masoumeh Dashti

Project 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 problems

Recommended modules: Introduction to Probability, Measure and Integration.

References:

!$[1]$! Gibbs A. L. and Su F. E. (2002) On choosing and bounding probability metrics.

!$[2]$! Robert, C. P. and Casella, G. (2004) Monte Carlo statistical methods. Second edition. Springer Texts in Statistics. Springer-Verlag, New York.

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 regularization

Recommended modules: Partial differential equations, Functional analysis, Probability and statistics, Measure and Integration.

References:

!$[1]$! Engl H. W., Hanke M. and Neubauer A. (2000) Regularization of inverse problems, Kluwer Academic Publishers.

!$[2]$! Stuart A. (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 theory

Recommended modules: Partial differential equations, Functional analysis, Measure and Integration.

References:

!$[1]$! Chae L. and Lee J. (2001) Regularity criterion in terms of pressure for the NavierStokes equations, Nonlinear Analysis 46. 727-735

!$[2]$! Serrin J. (1962) On the interior regularity of weak solutions of the Navier-Stokes equations. Archive for Rational Mechanics and Analysis, 9, 187-195.

!$[3]$! Temam R. (2001) Navier-Stokes Equations: Theory and Numerical Analysis, American Mathematical Society.

Nicos Georgiou

Project 1: Corner growth processes;(Dr N. 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. For any rectangle !$[m]\times[n] =\{ 1,2,...,m \}\times\{1,2,...,n\}$! 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 !$n^{-1}L([nx],[ny])\rightarrow \Psi(x,y)$! a.s. and in !$L^1$!. Part of the project will be to prove the closed formula for !$\Psi(x,y)$! given by

$$\Psi(x,y) =\left\{ \begin{array}{lll} x, & \textrm {if } x < py \\ \displaystyle \frac{2\sqrt{pxy}-p(x+y)}{q}, & \textrm {if } p^{-1}y\geq x\geq py \\ y, &\textrm {if } y < px \end {array} \right.$$

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 theorem

Project 2: Traffic flow models via totally asymmetric simple exclusion processes;(Dr N. Georgiou)

The totally asymmetric simple exclusion process (TASEP) is a stochastic particle system in which particles move only in one direction, without being able to overtake each other. The model has been used a few times to model traffic flow in narrow highways, for which there are rigorous mathematical results, or even implemented to make predictions about traffic in city grids but in this case without the mathematical rigour.

The goal of this project is three-fold. First there is the theoretical component of understanding the mathematics behind the hydrodynamic limits of the particle system and find the limiting PDE. Second, we will use free traffic data and develop statistical tests to identify and estimate relevant parameters that appear in the hydrodynamic limit above. The third is to develop Monte Carlo algorithms that take the estimated parameters, build the stochastic model, and show us the traffic progress in a given road network.

Supervisor: Dr. Nicos Georgiou

Helpful mathematical background: Random processes, Monte Carlo Simulations, Statistical Inference.

Some Bibliography:

[1] N. Georgiou, R. Kumar and T. Seppäläinen TASEP with discontinuous jump rates https://arxiv.org/pdf/1003.3218.pdf

[2] H.J. Hilhorst and C. Appert-Rolland, A multi-lane TASEP model for crossing pedestrian traffic flows https://arxiv.org/pdf/1205.1653.pdf

[3] J.G. Brankov, N.C. Pesheva and N. Zh. Bunzarova, One-dimensional traffic flow models: Theory and computer simulations. Proceedings of the X Jubilee National Congress on Theoretical and Applied Mechanics, Varna, 13-16 September, 2005(1), 442–456.

Peter Giesl

Project 1: Computational analysis of periodic orbits in nonsmooth differential

See PDF for full description

Project 2: Calculation of Contraction Metrics

See PDF for full description

Project 3: Dimension of Attractors in Dynamical Systems (MMath and PGT only)

See PDF for full description

Project 1: Modelling the impact of prophylactic and therapeutic interventions to control viral infections

Novel models will be developed to describe the dynamical changes of different respiratory viruses, like SARS-CoV and influenza, at different levels, from the cellular level to the individual and population level. The models will be extended to incorporate the impact of a range of prophylactic and therapeutic interventions to control i) viral replication within an individual host, ii) transmission of viral infections between individuals. Following an analytical investigation of the models and the derivation of important quantities, such as the basic reproductive number, generation times and area under the viral load and epidemic curves, we will solve them numerically and fit them to real data from clinical and epidemiological studies. This will enable the improvement of the models based on a number of selection criteria and identifiability analysis techniques. Depending on the interests and skills, stochastic algorithms will be developed to simulate the stochastic processes and quantify uncertainty in the model outputs. Some of the questions that you will be able to answer by the end of the project are:

- What are the most appropriate mathematical models to describe within- and between-host viral infection dynamics given the infection, the available data and the quantities we want to consider?

- What is the time window for prophylactic and therapeutic interventions to prevent an infection, or the development of mild/severe symptoms? What should be the optimal treatment efficacy?

- What should be the optimal vaccination and/or treatment strategy to prevent an outbreak, or reduce severe cases/hospitalisations/deaths below a certain threshold?

- Who should be prioritised for vaccination/treatment in a highly heterogenous population? An old, isolated person or a highly connected child?

The various components of this project can be extended in different ways and could constitute individual projects. During the project, you may have the opportunity to meet with leading researchers in the area of infectious disease epidemiology, and attend meetings with pharmaceutical companies, so you see how theory is linked with practice and real-world problems.

Project 2: Epidemic and evolutionary processes in highly heterogeneous populations

We will describe complex evolutionary and/or epidemic processes in non-homogeneous populations, characterised by high heterogeneity in demographic factors and contacts between individuals. Starting from the master equations we will introduce approximations that can reduce the number of system’s states while maintaining the accuracy of the prediction of the stochastic process. Both deterministic and stochastic systems will be tested and compared on a range of real-world networks, using data from epidemiological studies. The importance of the properties of the contact structure in the evolution of different systems will be studied.

Project 3: Modelling Alzheimer’s disease

Alzheimer’s disease is a progressive neurodegenerative disease which is rapidly becoming one of the leading causes of disability and mortality. We aim to develop mathematical, statistical and computational tools that will generate insights into the development and progression of Alzheimer’s disease, address the therapeutic challenges and accelerate the development of much-needed treatments. Clinical data from thousands of individuals will be analysed to try to identify changes in biological and clinical markers that indicate disease progression. Statistical, mathematical and computational techniques will be employed to describe the long-term changes of potential biomarkers, and indicators of cognitive and functional abilities, using short-term data. The focus will be on the stage prior to the clinical presentation of the disease.

- What is the expected probability and time required for an individual to develop Alzheimer’s dementia given its demographic and genetic characteristics, as well as levels of certain biological and cognitive markers?

- How factors like education could affect the clinical progression of the disease?

- If hypothetical treatments that reduce the accumulation of certain proteins in the brain lead to the decrease of the rate of cognitive decline, what is the time window for intervention to delay the occurrence of Alzheimer’s clinical symptoms for x years, given a certain treatment efficacy?

The project requires advanced statistical analysis skills.

Philip Herbert

Project 1:PDEs on surfaces

Many processes may be modelled by partial differential equations (PDE), some of these may take place in a thin region. In the limit of the thinness tending to zero, one might justify modelling the process by a PDE on a surface. To begin with, this project would seek to describe surfaces and various quantities upon that surface, for example the normal vector. With geometric notions in mind, one may define a surface gradient and pose surface PDEs. Finally, one might be able to provide well-posedness for a simple surface PDE. Computational results would accompany this project well.

Project 2:Optimisation with (PDE) constraints

In this project, we wish to understand some of the mathematical background for optimisation under constraints. Frequently constraints will take the form of a partial differential equation (PDE), and the optimisation may be related to quantities of interest from that PDE. A prototype example is: where should I heat (or cool) the room in order to ensure that the room has a temperature profile which suits the task at hand. Here the quantity of interest is the deviation from the desired temperature profile and the PDE constraint is Laplace's equation. This project will investigate the applications of functional analytic theorems to show well-posedness of a variety of optimisation problems. Tjis project would be well complimented by computational results.

James Hirschfeld

Project 1: Algebraic Geometry (Professor J. W. P. 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.

James Hirschfeld Presentation 1 [PDF 36.89KB]

Key words: polynomial, algebraic geometry

Recommended modules: Coding Theory

References:

!$[1]$! Reid, M. Undergraduate Algebraic Geometry, University Press, 1988.

!$[2]$! Semple, J. G. and Roth, L. Introduction to Algebraic Geometry, Oxford University Press, 1949

Project 2: Cubic Curves (Professor J. W. P. Hirschfeld)

Cubic curves in the plane may have a singular point or be non-singular. The non-singular points on a cubic form an abelian group, which leads to many remarkable properties such as the theory of the nine associated points, from which many other results can be deduced. A non-singular (elliptic) cubic is one of the most beautiful structures in mathematics.

James Hirschfeld Presentation 2 [PDF 25.58KB]

Key words: algebraic curve, cubic, group

Recommended modules: Coding Theory

References:

!$[1]$! Seidenberg, A. Elements of the Theory of Algebraic Curves Addison-Welsley 1968

!$[2]$! Clemens, C.H. A scrapbook for Complex Curve Theory Plenum Press 1980

Project 3: Finite Geometry (Professor J. W. P. Hirschfeld)

In defining a vector space, the scalars belong to a field, which can also be finite, such as the integers modulo a prime. Many combinatorial problems reduce to the study of geometrical configurations, which in turn can be analysed in a geometry over a finite field.

James Hirschfeld Presentation 3 [PDF 26.96KB]

Key words: geometry, projective plane, finite field

Recommended modules: Coding Theory

References:

!$[1]$! Dembowski, P. Finite Geometries, Springer Verlag, 1968

!$[2]$! Hirschfeld, J.W.P. Projective Geometries over a Finite Field Oxford University Press, 1998.

Project 4: Coding Theory (Professor J. W. P. Hirschfeld)

Error correction codes are used to correct errors when messages are transmitted through a noisy communication channel.
Here is the basic idea.

To send just the two messages YES and NO, the following encoding suffices:
YES = 1, NO = 0:

If there is an error, say 1 is sent and 0 arrives, this will go undetected. So, add some redundancy:
YES = 11, NO = 00:

Now, if 11 is sent and 01 arrives, then an error has been detected, but not corrected, since the original messages 11 and 00 are equally plausible.
YES = 111, NO = 000:

Now, if 010 arrives, and it is supposed that there was at most one error, we know that 000 was sent: the original message was NO.
Most of the theory depends on vector spaces over a finnite field.

References
1. R. Hill, A First Course on Coding Theory, Oxford, 1986; QE 1302 Hil. The course
is mostly based on this book.
2. V.S. Pless, Introduction to the Theory of Error-Correcting Codes, Wiley, 1982, 1989;
QE 1302 Ple.
3. S. Ling and C.P. Xing, Coding Theory, a First Course, Cambridge, 2004; QE 1302
Lin.
4. https://www.maths.sussex.ac.uk/Staff/JWPH/TEACH/CODING21/index.html

Konstantinos Koumatos

Project 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.g. 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 problems

Recommended modules: Continuum Mechanics, Partial Differential Equations, Functional Analysis, Measure and Integration

References:

!$[1]$! J. M. Ball, Mathematical models of martensitic microstructure, Materials Science and Engineering A 378, 61--69, 2004

!$[2]$! J. M. Ball and R. D. 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. Chen, V. Srivastava, V. Dabade R. D. 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, 1999

Project 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 !$u(x)\in \mathbb{R}^n$!. The function !$W$! is the energy density associated to the material and physical requirements force one to assume that !$$W(F) \to \infty, \mbox{ as }\det F\to0^+ \mbox{ and } W(F) = \infty, \,\det F \leq 0. \tag{\ast}$$! 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 constraints

Recommended modules: Continuum Mechanics, Partial Differential Equations (essential), Functional Analysis (essential), Measure and Integration

References:

!$[1]$! J. M. 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, 2007

Project 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. In a series of papers in the 1970's, L. Tartar and F. 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.g. 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 + \partial_x f(u^\varepsilon) & = \varepsilon \partial_{xx} 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 limit

Recommended modules: Continuum Mechanics, Partial Differential Equations (essential), Functional Analysis (essential), Measure and Integration (essential)

References:

!$[1]$! C. M. Dafermos, Hyperbolic conservation laws in continuum physics, Springer, 2010

!$[1]$! L. C. 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, 1979

Project 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 !$$\frac{d}{dt}X(t) = b(X(t)),\quad X(0) = x\in \mathbb{R}^n. \tag{!\ast!}$$! 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. However, in many cases (e.g. 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.e. !$$u(X(t),t) = u(X(0),0) = u_0(X(0)) = u_0(x).$$!

Key words: ODEs with Sobolev coefficients, DiPerna-Lions, transport equation, renormalised solutions, continuity equation

Recommended 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. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae, 98, 511--517, 1989

!$[3]$! L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998

Yuliya Kyrychko

Project 1: Mathematical modelling of infectious diseases
This project is devoted to the development and analysis of the mathematical models for the spread of infectious diseases. The models developed in this project will be analysed analytically using methods of nonlinear dynamics and  numerically in MATLAB to compare the analytical findings on the disease progression with numerical observations. A number of various scenarios will be considered as the models are developed to account for the effects of vaccination, quarantine, delayed onset of symptoms and similar.
Keywords: mathematical modelling, epidemics, nonlinear dynamics.
Recommended modules: Introduction to Mathematical Biology, Dynamical Systems

Project 2: Time-delayed models of couples systems

This project aims to identify and analyse models of coupled elements, which are connected with time-delays. These types of systems arise in various different disciplines, such as engineering, physics, biology etc. The interesting feature where the current state of the system depends on the state of the system some time ago makes such models much more realistic and leads to various potential scenarios of dynamical behaviour. The models in this project will be analysed analytically to understand their stability properties and find critical time delays as well as numerically using MATLAB.

Keywords: mathematical modelling; delay differential equations; stability analysis
Recommended modules: Introduction to Mathematical Biology, Dynamical Systems
Omar Lakkis

Project 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. Extra references will be given in that case. One way of performing this extension would be to implement computer code simulating geometric motions and analysing the algorithms.

Omar Lakkis Presentations [PDF 358.53KB]

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 Analysis

Recommended modules: Finite Element Methods, Measure and Integration, Numerical Linear Algebra, Numerical Differential Equations, Intro to Math Bio, Applied Whatever Modelling.

References:

!$[1]$! Gurtin, Morton E., Thermomechanics of evolving phase boundaries in the plane. Oxford Mathematical Monographs. 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. Vol. III. Second edition. Publish or Perish, 1979. 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. Ser., 2, Amer. Math. Soc., Providence, RI, 1996.

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.

Omar Lakkis Presentations [PDF 358.53KB]

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.

References:

!$[1]$! L.C. Evans, An Introduction to stochastic differential equations. Lecture notes on authors website (google: Lawrence C Evans). University of California Berkley.

!$[2]$! C. W. Gardiner, Handbook of stochastic methods for physics, chemistry and the natural sciences. 3rd ed., Springer Series in Synergetics, vol. 13, Springer-Verlag, Berlin, 2004. ISBN 3-540-20882-8

!$[3]$! P.E. Kloeden; E. Platen; H. Schurz, Numerical solution of SDE through computer experiments. Universitext. Springer-Verlag, Berlin, 1994. xiv+292 pp. ISBN 3-540-57074-8

!$[4]$! A. Beskos and A. Stuart, MCMC methods for sampling function space, ICIAM2007 Invited Lectures (R. Jeltsch and G. Wanner, eds.), 2008.

!$[5]$! Joseph L. Doob, Classical potential theory and its probabilistic counterpart, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1984 edition. ISBN 3-540-41206-9

Michael Melgaard

Project 1: Spectral and scattering properties of Quantum Hamiltonians (Prof. M. 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.g., 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 Schrödinger operator (Kato's theorem and all that);
• The periodic Schrödinger operator (describing crystals);
• Scattering properties of Schrödinger 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.g., the Stability of Matter or Turbulence.

Key words: differential operators, spectral theory, scattering theory.

Recommended modules: Functional Analysis, Measure and Integration theory, Partial Differential Equations.

References:

!$[1]$! M. Melgaard, G. Rozenblum, Schrödinger operators with singular potentials, in: Stationary partial differential equations Vol. II, 407--517, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005.

!$[2]$! Reed, M., Simon, B., Methods of modern mathematical physics. Vol. I-IV. Academic Press, Inc., New York, 1975, 1978,1979,1980.

Project 2: Variational approach to Kohn-Sham models with magnetic fields (Prof. M. 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 Schrödinger 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.e., the ground state) to !$HΨ=EΨ$!, where !$H$! denotes the Hamiltonian of the molecular system, !$Ψ$! 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, !$$E= \inf \{ \, \mathcal{E}^{\rm QM}(Ψ) \, : \, Ψ \in \mathcal{H}, \:\: \| Ψ \|_{L^{2}} =1 \, \}, where \ \mathcal{E}^{\rm QM}(Ψ):= \langle Ψ, H Ψ \rangle_{L^{2}}$$! 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 !$ℜ^{3N}$! on which the wavefunctions are defined and the problem has to be approximated. Quantum Chemistry (QC), as pioneered by Fermi, Hartree, Löwdin, 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.e., excited states).

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 !$-eħm^{-1} {s} \cdot \mathcal{B}$! to the Hamiltonian, while the second, diamagnetic effect arises from the usual kinetic energy !$(2m)^{-1} | {\mathbf p} |^{2}$! being replaced by !$(2m)^{-1} | \mathbf {p} -(e/c) \mathcal{A}|^{2}$!. 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.

1. 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.e., 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.
2. 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:

!$[1]$! M. Melgaard, G. Rozenblum, Schrödinger operators with singular potentials, in: Stationary partial differential equations Vol. II, 407--517, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005.

!$[2]$! Reed, M., Simon, B., Methods of modern mathematical physics. Vol. I-IV. Academic Press, Inc., New York, 1975, 1978,1979,1980.

Project 3: Resonances for Schrödinger and Dirac operators (Prof. M. Melgaard)

Resonances play an important role in Chemistry and Molecular Physics. They appear in many dynamical processes, e.g. 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 Schrödinger 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 !$z_{k} =E_{k}-i Γ_{k}/2$!, !$Γ_{k}>0$!, which are called resonances of the scattering system. The probability density of the corresponding "eigenfunction" !$Ψ_{k}(x)$! decays in time like !$e^{-t Γ_{k}/ \hbar}$!, thus physically !$Ψ_{k}$! represents a metastable state with a decay rate !$Γ_{k}/ \hbar$! or, re-phrased, a lifetime !$\tau_{k}=\hbar / Γ_{k}$!. In the semi-classical limit !$\hbar \to 0$!, resonances !$z_{k}$! satisfying !$Γ_{k}=\mathcal{O}(\hbar)$! (equivalently, with lifetimes bounded away from zero) are called "long-lived".

Physically, the eigenfunction !$Ψ_{k}(x)$! 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 !$L^{2}$!-eigenfunctions !$\widetilde{Ψ}_{k}$!.

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 !$\widetilde{Ψ}_{k}$!, !$Ψ_{k}(x)$! 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), Sjöstrand (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.

References:

!$[1]$! J. Kungsman, M. Melgaard, Complex absorbing potential method for Dirac operators. Clusters of resonances, J. Ope. Th., to appear.

!$[2]$! P. Stefanov, Approximating resonances with the complex absorbing potential method, Comm. Part. Diff. Eq. 30 (2005), 1843--1862.

Project 4: Critical point approach to solutions of nonlinear, nonlocal elliptic equations arising in Astrophysics (Prof. M. Melgaard)

The Choquard equation in three dimensions reads:

!$$$$\tag*{(0.1)} -Δ u - \left( \int_{ℜ^{3}} u^{2}(y) W(x-y) \, dy \right) u(x) = -l u ,$$$$! where !$W$! is a positive function. It comes from the functional:

!$$\mathcal{E}^{\rm NR}(u) = \int_{ℜ^{3}} | \nabla u |^{2} \, dx -\int \int | u(x) |^{2} W(x-y) |u(y)|^{2} \, dx dy,$$!

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 !$E^{\rm NR}(\nu) = \inf \{ \, \mathcal{E}(u) \, : \, u \in \mathcal{H}^{1}(ℜ^{3}), \| u \|_{L^{2}} \leq \nu \, \}$!.

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 (0.1) possesses infinitely many solutions. For the constrained problem, seeking radially symmetric, normalized functions !$\| u \|_{L^{2}} =+1$!, or more generally, seeking solutions belonging to:

!$$\mathcal{C}_{N}= \{ \, φ \in \mathcal{H}_{\rm r}^{1} (ℜ^{3}) \, : \, \| φ \|_{L^{2}} =N \, \} ,$$! 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.e., the pseudodifferential operator !$\sqrt{ -δ +m^{2} } -m$!; 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 !$\sqrt{k^{2} +m^{2}} -m$!, 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). This system models a Boson star.

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.g., [1] and [2].

Key words: operator and spectral theory, semiclassical analysis, micro local analysis.

Recommended modules:Functional Analysis, Measure and Integration theory, Partial Differential Equations.

References:

!$[1]$! S. Cingolani, M. Clapp, S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Zeitschrift fr Angewandte Mathematik und Physik (ZAMP) , vol. 63 (2012), 233-248.

!$[2]$! M. Melgaard, F. D. Zongo, Multiple solutions of the quasi relativistic Choquard equation, J. Math. Phys. !${53}$!(2012), 033709 (12 pp).

Project 5: QUANTUM THREE-BODY (e.g., He) SYSTEMS. BOUND AND QUASI-BOUND STATES.(Prof. M. Melgaard)
Project 6: CHEMOTAXIS MODELS USED IN MEDICINE (Keller-Segel equations, pattern formation)(Prof. M. Melgaard)
Project 7: DOES THE BORN-OPPENHEIMER APPROXIMATION WORK? (Prof. M. Melgaard)
Project 8: FINITE-ELEMENT METHOD IN APPLIED SCIENCES (Prof. M. Melgaard)
Project 9: HYDROLOGY OF GROUNDWATER AND CONTAMINANT TRANSPORT (using Fractional Calculus). (Prof. M. Melgaard)
Project 10: OCEAN MODELS AND CLIMATE (Prof. M. Melgaard)
Project 11: NEURAL NETWORKS AND DEEP LEARNING(Prof. M. Melgaard)

The purpose is to study neural networks and deep learning, applied to a specific real-world problem. Projects include applications to Finance (deep hedging, calibration, option pricing, etc.), Quantum Physics/Chemistry (deep variational Monte Carlo simulations, solving (generalized) eigenvalue problem for the Schrödinger equation), and other applied topics where we need to solve Partial Differential Equations (PDEs).

Bibliogrpahy

[1] J. Berner, P. Grohs, G. Kutyniok, P. Petersen, The modern mathematics of deep learning. Mathematical aspects of deep learning, 1–111, Cambridge Univ. Press, Cambridge, 2023.

[2] M. López de Prado, Advances in Financial Machine Learning, J. Wiley and Sons, Ltd, 2018.

[3] E. O. Pyzer-Knapp, M. Benatan, Deep Learning for Physical Scientists: Accelerating Research with Machine Learning, John Wiley and Sons Ltd, 2021.

Project 12: TENSOR METHODS IN DEEP LEARNING, COMPUTER VISION, AND SCIENTIFIC COMPUTING(Prof. M. Melgaard)

Tensor methods are increasingly finding significant applications in deep learning, computer vision, and scientific computing. Possible projects include image classification, image reconstruction, noise filtering, sensor measurements, low memory optimization, solving PDEs, supervised/unsupervised learning, grid-search and/or DMRG-type algorithms, hidden Markov models, convolutional rectifier networks, neuroscience (neural data, medical images etc), biology (genomic signal processing, low-rank tensor model for gene–gene interactions) or multichannel EEG signals.

Quantum physics (grid-based electronic structure calculations using tensor decomposition approach etc.), Latent Variable Models (community detection through tensor methods, topic models (say, co-occurrence of words in a document), latent trees etc),

Bibliography

[1] W. Hackbusch, Tensor spaces and numerical tensor calculus. Springer, Cham, 2019.

[2] T. G. Kolda, B, W. Bader, Tensor decompositions and applications. SIAM Rev. 51 (2009), no. 3, 455--500.

[3] Y. Panagakis, J. Kossaifi et al, Tensor Methods in Computer Vision and Deep Learning, Proc. IEEE 109 (2021), no. 5, 863-890.

Project 13: PSEUDO DIFFERENTIAL OPERATORS, AND SEMICLASSICAL ANALYSIS WITH APPLICATIONS (e.g., in solid state physics)(Prof. M. Melgaard)
Project 14: MATHEMATICALLY MODELING OF SOLAR ENERGY HARVESTING AND OPTIMIZATION (Hamilton-Jacob-Bellman equations etc)(Prof. M. Melgaard)
Project 15: PERTURBATION OF (EMBEDDED) EIGENVALUES (near thresholds), LOW-ENERGY LIMIT, QUANTUM TIME EVOLUTION (Prof. M. Melgaard)
Project 16: COHERENT STATES IN QUANTUM PHYSICS(Prof. M. Melgaard)
Project 17: SPECTRAL AND SCATTERING THEORY IN ELECTROMAGNETIC THEORY, OR QUANTUM MECHANICS(Prof. M. Melgaard)
Veronica Sanz (This project only available to MSc Data Science students)

PLEASE NOTE THAT PROF SANZ IS NOT AVIALABLE FOR PROJECT SUPERVISION IN 19/20

Project 1: Searching for fundamental theories of Nature with unsupervised machine learning (Prof Veronica Sanz)

In High Energy Particle Physics we contrast data with new theories of Nature. Those theories are proposed to solve mysteries such as 1.) what is the Dark Universe made of, 2.) why there is so much more matter than antimatter in the Universe, and 3.) how can a light Higgs particle exist.

To answer these questions, we propose mathematical models and compare with observations. Sources of data are quite varied and include complex measurements from the Large Hadron Collider, underground Dark Matter detection experiments and satellite information on the Cosmic Microwave Background. We need to incorporate all this data in a framework which allows us to test hypotheses, and this is usually done via a statistical analysis, e.g. Bayesian, which provides a measure of how well a hypothesis can explain current observations. Alas, this approach has so far been unfruitful and is driving the field of Particle Physics to an impasse.

In this project, we will take a different and novel approach to search for new physics. We will assume that our inability to discover new physics stems from strong theoretical biases which have so far guided analyses. We will instead develop unsupervised searching techniques, mining on data for new phenomena, avoiding as much theoretical prejudices as possible. The project has a strong theoretical component, as the candidate will learn the mathematical/physical basis of new physics theories including Dark Matter, the Higgs particle and Inflation. The candidate will also learn about current unsupervised-learning techniques and the interpretation of data in High-Energy Physics.

The strategy adopted for this project holds the potential to open a new avenue of research in High Energy Physics. We are convinced that this departure from conventional statistical analyses mentioned above is the most effective way to discover new physics from the huge amount of data produced in the Large Hadron Collider and other experiments of similar scale.

Reaching the scientific goals outlined here would require modelling huge amounts of data at different levels of purity (raw measurements, pseudo-observables, re-interpreted data), and finding patterns which had not been detected due to a focus on smaller sets of information. Hence, we believe that research into unsupervised learning in this context will have far reaching applications beyond academic pursuits. As the world becomes increasingly data-orientated, so does our reliance on novel algorithms to make sense of the information we have in our possession. To give some examples, we can easily expect the development of unsupervised learning integrated into facial recognition software and assist in the discovery of new drugs, which provides a boost in the security and medical sector respectively.

Nick Simm

Project 1: Random matrix theory and the Riemann zeta function

See PDF for full description

Simm: Random matrix theory and the Riemann zeta function [PDF 156.59KB]

Project 2: Asymptotic analysis of integrals and applications

See PDF for full description

Simm: Asymptotic analysis of integrals and applications [PDF 124.34KB]

Ali Taheri

Project 1: Hardy Spaces !$H^p$! and Boundary Behaviour of Holomorphic Functions (Dr A. Taheri)

The study of boundary behaviour of holomorphic functions in the unit disc is a classical subject which has been revived and generalised to higher dimensions as well as other geometries due to recent developments in the theory of ellipic PDEs, e.g., one such development being the H1 and BMO duality.

The aim of this project is more modest and lies in understanding the interplay between holomorphic functions in the disc on the one hand and the Poisson integral of Borel measures on the boundary circle. The results here lead to surprising qualitative properties of holomorphic functions.

Key words: Poisson integrals, Nevanlinna class, Non-tangential convergence, M&F Riesz theorem

Recommended modules: Complex Analysis, Functional Analysis, Measure Theory

References:

!$[1]$! Real and Complex Analysis by Rudin

!$[2]$! Introduction to !$H^p$! spaces by Koosis

!$[3]$! Theory of !$H^p$! spaces by Duren

!$[4]$! Bounded Analytic Functions by Garnett.

Project 2: Fourier Series in !$L^p$! Spaces and Kolmogorov's Theorem (Dr A. Taheri)

Fourier analysis has been one of the major sources of interesting and fundamental problems in analysis. It alone plays one of the most significant roles in the development of mathematical analysis in the past 2 centuries.

The aim of this project is to study Fourier series, specifically in the context of: !$L^2$! -- the Hilbert space approach, continuous functions, and !$L^p$! with !$1 < p < ∞$!.

Particular emphasis goes towards the convergence/divergence properties using Functional analytic tools, Baire category arguments, singular integrals.

Key words: !$L^p$! spaces, Summability kernels, Baire category, Singular integrals, Hilbert transform

Recommended modules: Complex Analysis, Functional Analysis, Measure and Integration

References:

!$[1]$! Fourier Analysis, T.W. Koner, Cambridge University Press, 1986

!$[2]$! Real and Complex Analysis, W. Rudin, McGraw Hill, 1987

!$[3]$! Real Variable Methods in Harmonic Analysis, A. Torchinsky, Dover, 1986.

Project 3: Oscillation and Concentration Effects in Nonlinear PDEs (Dr A. Taheri)

In the theory of nonlinear partial differential equations the study of the oscillation and concentration phenomenon plays a key role in settling the question of the existence of solutions. Here the aim is to understand the basics of weak versus strong convergence for sequences of functions and to introduce a tool known as Young measures for detecting the mechanisms that could prevent strong convergence.

Key words: Young measures, Weak convergence, Div-Curl lemma

Recommended modules: Partial Differential Equations, Functional Analysis, Measure Theory

References:

!$[1]$! Parameterised Measures and Variational Principles, P.Pedregal, Birkhäuser, 1997.

!$[2]$! Partial Differential Equations, L.C. Evans, AMS, 2010.

!$[3]$! Weak Convergence Methods in PDEs, L.C. Evans, AMS, 1988.

Project 4: Singularities in Harmonic Maps Between Manifolds (Dr A. Taheri)

Harmonic maps between manifolds are extremals of the Dirichlet energy. It is well-known that depending on the topology and global geometry of the domain and target manifolds these harmonic maps can develop singularities in all forms and shapes. The aim of this project is to introduce the student to the theory and some of the basic ideas and important tools involved.

Key words: Harmonic maps, Dirichlet energy, Minimal connections, Singular cones.

Recommended modules: Partial Differential Equations, Introduction to Topology, Algebraic Topology, Functional Analysis

References:

!$[1]$! Infinite dimensional Morse theory by Chang

!$[2]$! Two reports on Harmonic maps by Eells and Lemaire

!$[3]$! Cartesian Currents in the Calculus of Variations by Giaquinta, Modica and Soucek.

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James Van Yperen

Project 1: Parameter estimation techniques for birth-death processes

Mathematical models found in nature are typically stochastic in nature, and thus difficult to calibrate and analyse. Birth-death processes (a model to describe population evolution) are no different, however they are Markovian – that is, they satisfy the Markov property. Under some assumptions about the size of the population, one can take expectation of the process and derive ODEs for the mean of the population over time. In this project, we will develop a parameter estimation framework to calibrate the ODE to some given data. Dependent on the student’s interest, we can look at nonlinear birth-death processes, parameter identifiability issues and analysis, or deriving an ODE for the variance and improving the calibration method.

Recommended Modules:

Advanced Numerical Analysis, Programming through Python, Statistical Inference

Introduction to Mathematical Biology, Probability Models, Random Processes

References:

[1] : Raol JR, Girija G, Singh J. Modelling and parameter estimation of dynamic systems. Iet; 2004 Aug 13.

[2] : Jones DS, Plank M, Sleeman BD. Differential equations and mathematical biology. CRC press; 2009 Nov 9.

[3] : Stortelder WJ. Parameter estimation in dynamic systems. Mathematics and Computers in Simulation. 1996 Oct 1;42(2-3):135-42.

[4] : Allen L. An introduction to mathematical biology. Prentice Hall, 2007.

Project 2: Numerical simulation of curve shortening flow

Curve shortening flow, mean curvature flow for curves, is a mathematical phenomenon where a curve is moving in a direction and velocity proportional to its own curvature. Formally, it is a type of geometric partial differential equation. By parametrising the curve, one derives a nonlinear first order in time and second order in arc-length partial differential equation for the coordinates of the curve as it moves over time. In this project we will be looking at the numerical simulation of curve shortening flow for different curves using the finite element method, which will involve the derivation programming of a finite element scheme. Depending on the student’s interest, we can look into using linear and quadratic elements, different types of parametrisations, or conduct some finite element analysis on a simpler parabolic PDE.

Recommended Modules:

Advanced Numerical Analysis, Numerical Solution to Partial Differential Equations, Partial Differential Equations, Programming through Python

References:

[1] Deckelnick K, Dziuk G, Elliott CM. Computation of geometric partial differential equations and mean curvature flow. Acta numerica. 2005 May;14:139-232.

[2] Brenner SC. The mathematical theory of finite element methods. Springer; 2008.

[3] Thomée V. Galerkin finite element methods for parabolic problems. Springer Science & Business Media; 2007 Jun 25.

[4] Barrett JW, Garcke H, Nürnberg R. Parametric finite element approximations of curvature-driven interface evolutions. InHandbook of numerical analysis 2020 Jan 1 (Vol. 21, pp. 275-423). Elsevier.

Chandrasekhar Venkataraman

Project 1: Modelling, analysis and simulation of biological pattern formation (Dr C Venkataraman)

The formation of structure or patterns from homogeneity is ubiquitous in biological systems such as the intricate markings on sea shells, pigment patterns on the wings of butterflies and the regular structures made by populations of cells. Their is a rich theory for mathematical modelling of these phenomena that typically involves systems of PDEs. In this project we will understand and analyse some classical models for pattern formation and then extend them to take into account phenomena such as non-local interactions or growth and curvature. Dependent on the interests of the student we will either focus on the approximation of the models or their analysis.

Recommended modules: Introduction to Mathematical Biology, Advanced Numerical Analysis, Numerical Solution of Partial Differential Equations, Partial Differential Equations, Programming in C++

References:

!$[1]$! Turing, A. M. (1952). The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B

!$[2]$! Murray JD (2013) Mathematical Biology II: Spatial Models and Biomedical Applications. Springer New York

!$[3]$! Kondo, S.,and Miura, T. (2010). Reaction-diffusion model as a framework for understanding biological pattern formation. Science.

!$[4]$! Plaza, R. G., Sanchez-Garduno, F., Padilla, P., Barrio, R. A., & Maini, P. K. (2004). The effect of growth and curvature on pattern formation. Journal of Dynamics and Differential Equations

Project 2: Mathematical problems from cell biology

Mathematical modelling, analysis and simulation can help us understand a number of cell biological questions such as, How do cells move? How do they interact with their environment and each other? How do cell scale interactions influence tissue level phenomena? In this project we will review and extend models for either cell migration, receptor-ligand interactions or cell signalling. The models typically involve geometric PDE with coupled systems of equations posed in different domains, cell interior, cell-surface, extracellular space. Dependent on the interests of the student we will either focus on the derivation, the approximation, or the analysis of the models.

Recommended modules: Introduction to Mathematical Biology, Advanced Numerical Analysis, Numerical Solution of Partial Differential Equations, Partial Differential Equations, Programming in C++

References:

!$[1]$! Elliott, C. M., Stinner, B., and Venkataraman, C. (2012). Modelling cell motility and chemotaxis with evolving surface finite elements. Journal of The Royal Society Interface

!$[2]$! Croft, W., Elliott, C. M., Ladds, G., Stinner, B., Venkataraman, C., and Weston, C. (2015). Parameter identification problems in the modelling of cell motility. Journal of mathematical biology

!$[3]$! Elliott, C. M., Ranner, T., and Venkataraman, C. (2017). Coupled bulk-surface free boundary problems arising from a mathematical model of receptor-ligand dynamics. SIAM Journal on Mathematical Analysis

!$[4]$! Ptashnyk, M., and Venkataraman, C. (2018). Multiscale analysis and simulation of a signalling process with surface diffusion. arXiv preprint

Project 1: Beta-expansions of real numbers

The topic of this project is at the intersection of probability, ergodic theory, number theory, and dynamical systems. It is well-known that any real number can be represented by its decimal, binary, ternary, etc. expansion. But what if we try to expand in a non-integer basis? Such expansions are known as beta-expansions. Do they have the same properties as the usual ones? For example, what can we say about frequencies of digits? Are all patterns of digits possible? Does every real number have a unique beta-expansion? The project aims to address questions of such type to study basic properties of beta-expansions.

Bibliography

[1] A. Renyi. Representations for real numbers and their ergodic properties (1957)

[2] W. Parry. Representations for real numbers (1964)

Project 2: Bernoulli convolutions and self-similar measures

The topic of this project is at the intersection of probability and measure theory. The Bernoulli convolution is the distribution of a power series in x whose coefficients are independent identically distributed Bernoulli(1/2) random variables. These distributions have surprising different properties depending on the value of x, e.g. they are singular for all 0<x<1/2 and have density for almost all (but not all!) 1/2<x

What are the properties that make certain values of x special? What happens at such x? Is there any connection with the famous Cantor function (aka Devil’s staircase)? The project aims to address questions of such type to study basic properties of Bernoulli convolutions.

Bibliography

[1] B. Solomyak. Notes on Bernoulli convolutions (2017)

Minmin Wang