Mathematical Modelling of Gravity Effects in Plant Cooling
Mathematics
Master by Research and MRes Stochastic Processes
Mathematical modelling of gravity effects in plant cooling
Supervisor: Dr L. Bridge
Email:
Recent mathematical modelling results suggest that plant shoot architecture could be indicative of a plant’s cooling strategy. So far, a 2-dimensional model has been studied, with acceleration due to gravity kept constant. This project will review recent results, extend the model to 3 dimensions, and consider the effect of varying gravitational parameters. A background in mathematical modelling and computational methods for partial differential equations is required.
Mathematical modelling of biased signalling in pharmacology
Supervisor: Dr L. Bridge
Email:
Mathematical pharmacology is an exciting, growing field. This project will use mathematical models based on systems of differential equations to investigate the concept of “biased signalling” – the ability of a drug molecule to activate multiple effects at the same receptor on a cell. A background in mathematical modelling and computational methods for ordinary differential equations is required.
Model reduction methods in mathematical pharmacology
Supervisor: Dr L. Bridge
Email:
Mathematical models of biochemical processes in pharmacology often comprise large systems of ordinary differential equations. These systems are usually hard to analyse, but algorithms which reduce the dimension of the systems may help us to understand the most important processes. This project will review model reduction methods and apply these methods to some pharmacological models. A background in mathematical modelling and computational methods for ordinary differential equations is required.
Mathematical chronopharmacokinetics
Supervisor: Dr L. Bridge
Email:
Pharmacokinetics (PK) is the study of drug distribution in the body, and is a topic which uses the results of differential equation based models. Classical PK analysis often assumes that key parameters controlling drug absorption and elimination are constant. Many of these parameters may vary with time of day, meal timings and other time-dependent factors. This project will review some classical PK analysis, then extend models to include time-dependent effects, finding analytical solutions wherever possible. A background in mathematical modelling and ordinary differential equations is required.
Singular limits of elliptic and parabolic system
Supervisor: DrE.Crooks
Email:
Singular limits of systems of nonlinear partial differential equations arise in conjunctionwith important biological and physical phenomena, such as spatial segregation of speciesinpopulation dynamics or phase separation, and are mathematically both interesting and useful.This researchproject is concerned with the study of the large-competition limit ofnonlinear ellipticand parabolic systems modelling populations thatcompete in some region. The mathematicaltechniques used will include,for example, maximum principles, variational methods, degreetheory,etc,and will provide excellent preparation for a student who intends to continue to studyfor a PhD in the analysis of nonlinear partial differential equations.
Invasion and spreading speeds in population models
Supervisor: DrE.Crooks
Email:
Invasions of biological species typically take place via propagation of populationwaves separating a zone where a species is absent from a zone where it ispresent, and it is crucial ecologically to determine the nature and spreading speed ofsuch invasions. Models of population dynamics include reaction-diffusionequations/systems, and also non-local variants involving non-local dispersalor/and non-local saturation effects.This research project is concerned with studying spreading speeds of biologicalinvasions and travelling waves in ecologically-realistic population-dynamics models.The mathematical approaches used will centre on techniques from the analysis ofpartial differential equations and will provide excellent preparation for a studentwho intends to continue to study for a PhD in either partial differential equationsor mathematical biology
Topology, the new language of data.
Supervisor:Dr P. Dlotko
Email:
In the recent years we are facing new data-related challenges. Understanding of those data may bring us understanding of the underlying processes, and therefore understanding of time. In this project we will focus on topological approach to data analysis(TDA). TDA takes into account a shape of data, and often is able to bring valuable conclusions by understanding the shape. Your task will be to study the existing TDA techniques and build new ones on top of them. Your research will be driven by real data taking from neuroscience and material science. You will introduce a new theory and at the same time, provide efficient implementations of your ideas so that they can be used in data analysis.
Knowledge of topology and some experience in programming is required in this project.
Functions of Relevance: Symbols of Random Processes.
Supervisor: Dr K Evans
Email:
The propagation of light, the random motion of a particle, or the price of an option in the stock market are modelled by rather complicated mathematical objects, e.g. partial differential operators and stochastic processes. In many cases however it is possible to reduce these objects to just one function which we call the symbol of this object. Investigating symbols is often possible with tools learnt in undergraduate studies, in particular those met in real analysis, complex analysis and Euclidean geometry. Constructing new, interesting (classes of) symbols will lead to new interesting objects, e.g. stochastic processes. In this project the research part is to construct and study new symbols giving rise to new classes of random processes (however no previous knowledge of random processes is needed). In a more theoretical part the student will learn more about the general framework, i.e. the relation between symbols and processes. The project is part of a larger project and will feed into the research of our group and hence the student will collaborate closely with all academics as well as with other research students in the team.
Asymptotic properties of multi-dimensional convolution of functions
Supervisor: Dr. D. Filkelshtein
Email:
Convolution of functions is a classical object which appears regularly in Analysis. In many problems, motivated, in particular, Probability Theory, Stochastic Processes, and PDE, one needs to know which asymptotic behaviour the convolution of two functions has, given the information about their asymptotic behaviour. In the one-dimensional case an important role play the so-called sub-exponential functions whose convolutions asymptotically similar to the functions themselves. The corresponding theory for the multi-dimensional setting is less studied. It is supposed to consider the topic for the case of radially symmetric multidimensional functions and their convolutions.
Travelling waves in reaction-diffusion equations with non-local terms
Supervisor: Dr. D. Filkelshtein
Email:
Travelling waves are special type of solutions to some classes of PDE's propagating in time with preservation of the space-shape. They usually have a minimal speed of propagation which corresponds to the wave whose behaviour for big times is close, in a certain sense, to the behaviour of solutions from a wide class. Nowadays, these topics which are well-developed for the classical reaction-diffusion equations are actively studied for the equations which contain non-local terms in diffusion and reaction. It is supposed to apply recently developed methods to the study of one or several particular non-local equations.
Exactly solvable stochastic dynamics on the cone of discrete measures
Supervisor: Dr. D. Filkelshtein
Email:
Discrete measures correspond to the marked point processes whose points may have accumulations however the marks are such that the measures of bounded sets are finite. Cone of discrete measures is an interesting object for modern Infinite-Dimensional Analysis. Stochastic dynamics of such measures have numerous applications in Life and Social Sciences. Exactly solvable dynamics have the property that the system of differential equations, describing the correlations of different orders within the dynamics, can be solved inductively. It is supposed to consider birth, death, and jump dynamics with this property.
Tropical projective duality
Supervisor: Dr J.Giansiracusa
Email:
In this project we will explore how recent discoveries in tropical geometry relate to the classical subject of projective duality.
Tropical geometry is an exciting developing area of geometry that lies at the confluence of algebraic geometry, commutative algebra and combinatorial geometry. The objects studied in tropical geometry have been viewed as polyhedral complexes, but hidden algebraic structure has now been uncovered through work of Giansiracusa (the supervisor for this project) and Maclagan-Rincon.
Classical projective duality is a bijection between d-dimensional objects and codimension d objects in projective space. This theory has an analogue for tropical objects at the polyhedral level. The goal of this project is to investigate how this interacts with the richer structure of tropical schemes.
This project is suitable for a student who has a solid foundation in abstract algebra (ring and module theory). Knowledge of basic algebraic geometry would be helpful but is not required. This project would be excellent preparation for a student who intends to continue on to a PhD in algebra or geometry.
Monte-Carlo evaluation of complex integrals with the LLR method
Supervisor: Prof. B. Lucini
Email:
Monte-Carlo methods are a powerful set of stochastic techniques for evaluating numerically integrals in multi-dimensional spaces. While these techniques work reliably for real functions, oscillating factors such as those appearing in Fourier transforms create large numerical cancellations; as a result, in the latter case often the numerical error dominates the final result. In this project, the student will explore a recently proposed novel method (the Logarithmic-Linear-Relaxation algorithm) that has been devised explicitly to deal with this problem. The project can be developed either in a numerical direction (optimisation of the algorithm for a set of relevant integrals) or in an analytical direction (evaluation of rigorous bounds for numerical results).
Laplace operators on completely random measures
Supervisor: Prof. E. Lytvynov
Email:
In this project, you will study random measures, i.e., measures which depend on a variable from a probability space. These measures can be finite (for example, random probability measures) or infinite. It appears that one can develop and study differential calculus on measures, including Laplace operators.
Umbral calculus in finite and infinitely many variables
Supervisors: Prof. E. Lytvynov,Dr. D. Filkelshtein
Email ,
Study of Sheffer polynomial sequences, a.k.a. umbral calculus, is an important topic in modern mathematics, which has numerous interconnections with Algebra, Analysis, Probability Theory, Mathematical Physics, Topology etc. as well as a wide range of applications in physics. The existing constructions of this theory for the polynomials of several variables are dimensional and basis dependent. It is suppose to modify an algebraic approach to the umbral calculus to construct a unified multi-dimensional version of it.You may also study umbral calculus of polynomials of a variable from a Hilbert (infinite-dimensional space). Relations to probability measure (orthogonal polynomials) may also be considered.
Mathematical theory of graphene
Supervisor: DrV. Moroz
Email:
Graphene is a recently discovered material which consists of exactly one atomic layer of carbon.Mathematically, this means that graphene could be considered as a two-dimensional object.Two dimensionality of graphene leads to a number of remarkable physical and mechanical properties which also give rise to new challenging mathematical problems.Mathematical theory of graphene is not yet fully understood.The aim of the project is to study from mathematical prospective the density functional theory models of electrons screening in a single layer graphene and their potential applications.
Diffusions, Constrained Hamiltonian Systems and Semiclassical mechanics. (MRes Stochastic Processes)
Supervisor: Dr A Neate
Email:
Classical mechanics governs the behaviour of the macroscopic and Quantum mechanics governs the behaviour of the microscopic. The classical world is entirely deterministic, whilst the quantum is inherently probabilistic. Between these two regimes lies the semiclassical, and it should be possible to mathematically move from the classical to the quantum using an appropriate semiclassical framework. In this project we will look at aspects of the semiclassical mechanics that can be associated to a given quantum state using diffusion processes and constrained Hamiltonian systems. This project will involve both a knowledge of mechanics and probability theory. The student will be required to study courses on Stochastic Processes, Ito Calculus and Analytical Dynamics.
Graphs, syzygies and multivariate splines
Supervisor: Dr N Villamizar
Email:
The space of piecewise polynomial functions, commonly called splines, on a polyhedral partition of a real domain can be viewed as the polynomial relations (or syzygies) between powers of linear polynomials, vanishing on the facets of the partitions, and its dual graph. Due to the wide range of applications of splines, their study has been approached by different methods and from several areas of mathematics such as approximation theory and applied geometry. However, in general, the description of the space of splines depends on the geometry of the partition and many questions remain open even for low degree polynomials. The idea of the project is to use algebraic techniques and explore certain polyhedral partitions that meet particularly good conditions so that the space of splines can be characterized in terms of dimension, and algebraic structure.
The requirements for the project are: linear and abstract algebra, basic combinatorics and geometry, and some knowledge of commutative algebra.
Discrete stochastic processes and dynamic data analysis
Supervisor: Prof J-L.Wu
Email:
The project will start with a systematical study of stochastic processes andstochasticdifferential/difference equations. Then, the topic is designed to analyse data samplings anddatamining, and furtherto identify certain dynamical structures like time serious with certain unknownparameters,aiming to establish reasonable stochastic models to predict the further developmentbased on the (big) data samples arising in financial and medical studies.
Population Dynamics
Supervisor: Prof. C.Yuan
Email:
A population is a group of individuals who live together in the same habitat. In this project, we shall establish stochastic population models to analysis the size of a population, we then study itspermanence, explosion and extinction by using Ito’s calculus.