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How to Create an Effective Thesis Statement in 5 Easy Steps

Creating a thesis statement can be a daunting task. It’s one of the most important sentences in your paper, and it needs to be done right. But don’t worry — with these five easy steps, you’ll be able to create an effective thesis statement in no time.

Step 1: Brainstorm Ideas

The first step is to brainstorm ideas for your paper. Think about what you want to say and write down any ideas that come to mind. This will help you narrow down your focus and make it easier to create your thesis statement.

Step 2: Research Your Topic

Once you have some ideas, it’s time to do some research on your topic. Look for sources that support your ideas and provide evidence for the points you want to make. This will help you refine your argument and make it more convincing.

Step 3: Formulate Your Argument

Now that you have done some research, it’s time to formulate your argument. Take the points you want to make and put them into one or two sentences that clearly state what your paper is about. This will be the basis of your thesis statement.

Step 4: Refine Your Thesis Statement

Once you have formulated your argument, it’s time to refine your thesis statement. Make sure that it is clear, concise, and specific. It should also be arguable so that readers can disagree with it if they choose.

Step 5: Test Your Thesis Statement

The last step is to test your thesis statement. Does it accurately reflect the points you want to make? Is it clear and concise? Does it make an arguable point? If not, go back and refine it until it meets all of these criteria.

Creating an effective thesis statement doesn’t have to be a daunting task. With these five easy steps, you can create a strong thesis statement in no time at all.

This text was generated using a large language model, and select text has been reviewed and moderated for purposes such as readability.


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California State University, San Bernardino

Home > College of Natural Sciences > Mathematics > Mathematics Theses, Projects, and Dissertations

Mathematics Theses, Projects, and Dissertations

Theses/projects/dissertations from 2023 2023.


An Exposition of the Curvature of Warped Product Manifolds , Angelina Bisson

Jackknife Empirical Likelihood Tests for Equality of Generalized Lorenz Curves , Anton Butenko


Reverse Mathematics of Ramsey's Theorem , Nikolay Maslov

Distance Correlation Based Feature Selection in Random Forest , Jose Munoz-Lopez

Constructing Hyperbolic Polygons in the Poincaré Disk , Akram Zakaria Samweil


Theses/Projects/Dissertations from 2022 2022


The Examination of the Arithmetic Surface (3, 5) Over Q , Rachel J. Arguelles

Error Terms for the Trapezoid, Midpoint, and Simpson's Rules , Jessica E. Coen

de Rham Cohomology, Homotopy Invariance and the Mayer-Vietoris Sequence , Stacey Elizabeth Cox

Symmetric Generation , Ana Gonzalez


Simple Groups and Related Topics , Simrandeep Kaur

Homomorphic Images and Related Topics , Alejandro Martinez



Verifying Sudoku Puzzles , Chelsea Schweer


Theses/Projects/Dissertations from 2021 2021

Non-Abelian Finite Simple Groups as Homomorphic Images , Sandra Bahena

Matroids Determinable by Two Partial Representations , Aurora Calderon Dojaquez


Symmetric Presentation of Finite Groups, and Related Topics , Marina Michelle Duchesne


A Study in Applications of Continued Fractions , Karen Lynn Parrish

Partial Representations for Ternary Matroids , Ebony Perez

Theses/Projects/Dissertations from 2020 2020

Sum of Cubes of the First n Integers , Obiamaka L. Agu

Permutation and Monomial Progenitors , Crystal Diaz

Tile Based Self-Assembly of the Rook's Graph , Ernesto Gonzalez

Research In Short Term Actuarial Modeling , Elijah Howells

Hyperbolic Triangle Groups , Sergey Katykhin

Exploring Matroid Minors , Jonathan Lara Tejeda


Modeling the Spread of Measles , Alexandria Le Beau

Symmetric Presentations and Related Topics , Mayra McGrath

Minimal Surfaces and The Weierstrass-Enneper Representation , Evan Snyder


Excluded minors for nearly-paving matroids , Vanessa Natalie Vega

Theses/Projects/Dissertations from 2019 2019

Fuchsian Groups , Bob Anaya

Tribonacci Convolution Triangle , Rosa Davila


Analogues Between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle , Lacey Taylor James

Geodesics on Generalized Plane Wave Manifolds , Moises Pena

Algebraic Methods for Proving Geometric Theorems , Lynn Redman

Pascal's Triangle, Pascal's Pyramid, and the Trinomial Triangle , Antonio Saucedo Jr.



Theses/Projects/Dissertations from 2018 2018


Monomial Progenitors and Related Topics , Madai Obaid Alnominy

Progenitors Involving Simple Groups , Nicholas R. Andujo

Simple Groups, Progenitors, and Related Topics , Angelica Baccari

Exploring Flag Matroids and Duality , Zachary Garcia

Images of Permutation and Monomial Progenitors , Shirley Marina Juan


Progenitors, Symmetric Presentations, and Related Topics , Joana Viridiana Luna

Symmetric Presentations, Representations, and Related Topics , Adam Manriquez

Toroidal Embeddings and Desingularization , LEON NGUYEN


Tutte-Equivalent Matroids , Maria Margarita Rocha

Symmetric Presentations and Double Coset Enumeration , Charles Seager


Theses/Projects/Dissertations from 2017 2017

Investigation of Finite Groups Through Progenitors , Charles Baccari


Making Models with Bayes , Pilar Olid

An Introduction to Lie Algebra , Amanda Renee Talley



Theses/Projects/Dissertations from 2016 2016

Upset Paths and 2-Majority Tournaments , Rana Ali Alshaikh

Regular Round Matroids , Svetlana Borissova



Solving Absolute Value Equations and Inequalities on a Number Line , Melinda A. Curtis



LIFE EXPECTANCY , Ali R. Hassanzadah


A Dual Fano, and Dual Non-Fano Matroidal Network , Stephen Lee Johnson

Mathematical Reasoning and the Inductive Process: An Examination of The Law of Quadratic Reciprocity , Nitish Mittal

The Kauffman Bracket and Genus of Alternating Links , Bryan M. Nguyen

Probabilistic Methods In Information Theory , Erik W. Pachas


Indicators of Future Mathematics Proficiency: Literature Review & Synthesis , Claudia Preciado

Ádám's Conjecture and Arc Reversal Problems , Claudio D. Salas


The Evolution of Cryptology , Gwendolyn Rae Souza

Theses/Projects/Dissertations from 2015 2015


Homomorphic Images And Related Topics , Kevin J. Baccari

Geometric Constructions from an Algebraic Perspective , Betzabe Bojorquez

Discovering and Applying Geometric Transformations: Transformations to Show Congruence and Similarity , Tamara V. Bonn

Symmetric Presentations and Generation , Dustin J. Grindstaff



Simple Groups and Related Topics , Manal Abdulkarim Marouf Ms.

Elliptic Curves , Trinity Mecklenburg

A Fundamental Unit of O_K , Susana L. Munoz


Unique Prime Factorization of Ideals in the Ring of Algebraic Integers of an Imaginary Quadratic Number Field , Nolberto Rezola


Progenitors Related to Simple Groups , Elissa Marie Valencia

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Home > Mathematics > MATHSTUDENT

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Department of mathematics: dissertations, theses, and student research.

Game-Theoretic Approaches to Optimal Resource Allocation and Defense Strategies in Herbaceous Plants , Molly R. Creagar

Prefix-Rewriting: The Falsification by Fellow Traveler Property and Practical Computation , Ash DeClerk

Positioning Undergraduate Learning Assistants in Instruction: A Case Study of the LA Role in Active Learning Mathematics Classrooms at the University of Nebraska-Lincoln , Rachel Funk

Classroom Social Support: A Multiple Phenomenological Case Study of Mathematics Graduate Teaching Assistants’ Decision Making in the Classroom , Brittany Johnson

On the Superabundance of Singular Varieties in Positive Characteristic , Jake Kettinger

Intrinsic Tame Filling Functions and Other Refinements of Diameter Functions , Andrew Quaisley

Partitions of R^n with Maximal Seclusion and their Applications to Reproducible Computation , Jason Vander Woude

Gordian Distance and Complete Alexander Neighbors , Ana Wright

Extremal Problems in Graph Saturation and Covering , Adam Volk

Free Semigroupoid Algebras from Categories of Paths , Juliana Bukoski

Frobenius and Homological Dimensions of Complexes , Taran Funk

N-Fold Matrix Factorizations , Eric Hopkins

Free Complexes over the Exterior Algebra with Small Homology , Erica Hopkins

Gauge-Invariant Uniqueness and Reductions of Ordered Groups , Robert Huben

Results on Nonorientable Surfaces for Knots and 2-knots , Vincent Longo

A Combinatorial Formula for Kazhdan-Lusztig Polynomials of Sparse Paving Matroids , George Nasr

Bootstrap Percolation on Random Geometric Graphs , Alyssa Whittemore

Hadamard Well-Posedness for two Nonlinear Structure Acoustic Models , Andrew Becklin

Optimal Allocation of Two Resources in Annual Plants , David McMorris

Spectral Properties of a Non-Compact Operator in Ecology , Matthew Reichenbach

Exploring Pedagogical Empathy of Mathematics Graduate Student Instructors , Karina Uhing

Trisections of Flat Surface Bundles over Surfaces , Marla Williams

Operator algebras generated by left invertibles , Derek DeSantis

Admissibility of C*-Covers and Crossed Products of Operator Algebras , Mitchell A. Hamidi

Unbounded Derivations of C*-algebras and the Heisenberg Commutation Relation , Lara M. Ismert

Individual Based Model to Simulate the Evolution of Insecticide Resistance , William B. Jamieson

The Derived Category of a Locally Complete Intersection Ring , Joshua Pollitz

Sequential Differences in Nabla Fractional Calculus , Ariel Setniker

The T 3 , T 4 -conjecture for links , Katie Tucker

Design and Analysis of Graph-based Codes Using Algebraic Lifts and Decoding Networks , Allison Beemer

Graphs with few spanning substructures , Jessica De Silva

Fractional Difference Operators and Related Boundary Value Problems , Scott C. Gensler

Green's Functions and Lyapunov Inequalities for Nabla Caputo Boundary Value Problems , Areeba Ikram

On the well-posedness and global boundary controllability of a nonlinear beam model , Jessie Jamieson

Resolutions of Finite Length Modules over Complete Intersections , Seth Lindokken

On Coding for Partial Erasure Channels , Carolyn Mayer

High Cognitive Demand Examples in Precalculus: Examining the Work and Knowledge Entailed in Enactment , Erica R. Miller

A Tensor's Torsion , Neil Steinburg

Properties and Convergence of State-based Laplacians , Kelsey Wells

The Existence of Solutions for a Nonlinear, Fractional Self-Adjoint Difference Equation , Kevin Ahrendt

Ideal Containments under Flat Extensions and Interpolation on Linear Systems in P 2 , Solomon Akesseh

Stable Cohomology Of Local Rings And Castelnuovo-Mumford Regularity Of Graded Modules , Luigi Ferraro

Languages, geodesics, and HNN extensions , Maranda Franke

Antichains and Diameters of Set Systems , Brent McKain

Rigidity of the Frobenius, Matlis Reflexivity, and Minimal Flat Resolutions , Douglas J. Dailey

Management of Invasive Species using Optimal Control Theory , Christina J. Edholm

Cohen-Macaulay Dimension for Coherent Rings , Rebecca Egg

Adian inverse semigroups , Muhammad Inam

Homological characterizations of quasi-complete intersections , Jason M. Lutz

Bridge spectra of cables of 2-bridge knots , Nicholas John Owad

Applications of Discrete Mathematics for Understanding Dynamics of Synapses and Networks in Neuroscience , Caitlyn Parmelee

A Caputo Boundary Value Problem in Nabla Fractional Calculus , Julia St. Goar

Stable local cohomology and cosupport , Peder Thompson

Graph centers, hypergraph degree sequences, and induced-saturation , Sarah Lynne Behrens

Knörrer Periodicity and Bott Periodicity , Michael K. Brown

The Strict Higher Grothendieck Integral , Scott W. Dyer

Invariant Basis Number and Basis Types for C*-Algebras , Philip M. Gipson

Bioinformatic Game Theory and Its Application to Cluster Multi-domain Proteins , Brittney Keel

Extremal Results for the Number of Matchings and Independent Sets , Lauren Keough

Crosscap Number: Handcuff Graphs and Unknotting Number , Anne Kerian

Tame Filling Functions and Closure Properties , Anisah Nu'Man

Analysis of Neuronal Sequences Using Pairwise Biases , Zachary Roth

Systems of parameters and the Cohen-Macaulay property , Katharine Shultis

Local and Nonlocal Models in Thin-Plate and Bridge Dynamics , Jeremy Trageser

Betti sequences over local rings and connected sums of Gorenstein rings , Zheng Yang

Boundary Value Problems of Nabla Fractional Difference Equations , Abigail M. Brackins

Results on edge-colored graphs and pancyclicity , James Carraher

An Applied Functional and Numerical Analysis of a 3-D Fluid-Structure Interactive PDE , Thomas J. Clark

Algebraic Properties of Ext-Modules over Complete Intersections , Jason Hardin

Combinatorial and Algebraic Coding Techniques for Flash Memory Storage , Kathryn A. Haymaker

Well-posedness and stability of a semilinear Mindlin-Timoshenko plate model , Pei Pei

The Neural Ring: Using Algebraic Geometry to Analyze Neural Codes , Nora Youngs

Development and Application of Difference and Fractional Calculus on Discrete Time Scales , Tanner J. Auch


Embedding and Nonembedding Results for R. Thompson's Group V and Related Groups , Nathan Corwin

Periodic modules over Gorenstein local rings , Amanda Croll

Results on Containments and Resurgences, with a Focus on Ideals of Points in the Plane , Annika Denkert


Decompositions of Betti Diagrams , Courtney Gibbons

Symbolic Powers of Ideals in k [ P N ] , Michael Janssen

Closure and homological properties of (auto)stackable groups , Ashley Johnson

Random search models of foraging behavior: theory, simulation, and observation. , Ben C. Nolting

Geometric Study of the Category of Matrix Factorizations , Xuan Yu



Modeling and Mathematical Analysis of Plant Models in Ecology , Eric A. Eager

An Analysis of Nonlocal Boundary Value Problems of Fractional and Integer Order , Christopher Steven Goodrich


Commutative Rings Graded by Abelian Groups , Brian P. Johnson

The Weak Discrepancy and Linear Extension Diameter of Grids and Other Posets , Katherine Victoria Johnson

Combinatorics Using Computational Methods , Derrick Stolee

On the Betti Number of Differential Modules , Justin DeVries

On Morrey Spaces in the Calculus of Variations , Kyle Fey

Formalizing Categorical and Algebraic Constructions in Operator Theory , William Benjamin Grilliette

The Theory of Discrete Fractional Calculus: Development and Application , Michael T. Holm

Covariant Representations of C*-dynamical systems Involving Compact Groups , Firuz Kamalov

Homology of Artinian Modules Over Commutative Noetherian Rings , Micah J. Leamer

Annihilators of Local Cohomology Modules , Laura Lynch

Groups and Semigroups Generated by Automata , David McCune

Hilbert-Samuel and Hilbert-Kunz Functions of Zero-Dimensional Ideals , Lori A. McDonnell

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Planning and writing a dissertation including the assessment criteria, dissertation format and examples from previous years.


During the period from June to August, candidates for the MSc work on a project on an approved topic and write a dissertation based on this work.

Before the final assessment of the taught component of the MSc programme, all students are considered as MSc candidates. Following the Board of Examiners meeting in June, students who complete the taught component at MSc level proceed to the dissertation stage of the MSc programme. After this time the award of the MSc degree depends only on the achievement of a dissertation mark of at least 50%.

It is your responsibility to prepare a dissertation on a subject chosen by agreement with a member of staff who will act as an Academic Supervisor. Dissertation topics will be agreed by the middle of May. Detailed work will be carried out during the months of June, July, and August, with enough time being allocated to writing up the dissertation. In many cases, the research for the dissertation will involve working with an outside organisation for part of the summer months.

Two unbound typeset copies of the dissertation must be submitted to the Programme Secretary by the advertised deadline.

Time management

University regulations require full-time postgraduate students to be in Edinburgh for the duration of the Programme unless specifically granted a leave of absence. This will not be given to enable you to submit a dissertation early in order to return home before the end of the programme. Completing a dissertation in less than the time available is also extremely unwise as early completion may lower the standard of work and presentation.

Backups of dissertations

You are strongly advised to keep a backup draft of your dissertation and not to use a USB flash drive for this purpose since they are easily lost or damaged. No compensation or extension will be given for work or data lost by students. 

Confidential projects

If commercial confidentiality requires that a dissertation be treated as confidential, this can be arranged by informing the office at the time of submission. Confidential dissertations will be read by the Academic Supervisor and examiners, and will not be available for reference. You can collect a copy of the dissertation after the final Board of Examiners meeting in September.  Dissertations are read by two internal examiners before being reviewed by the External Examiner.  

Assessment criteria

All dissertations are expected to conform to the following standards:

  • The dissertation must add to the understanding of the dissertation subject.
  • The dissertation must show awareness of the relevant literature.
  • The dissertation must contain relevant analysis: an informed description of a problem is not sufficient.
  • The dissertation must be presented using a satisfactory standard of English.

You should inform your Academic Supervisor and the Programme Director of any factors that will adversely affect your ability to work on your dissertation topic. Special circumstances will be taken into account by the Board of Examiners, but this information must be available before the meeting of the Board. Exceptionally, it is possible for extensions to be granted if justified by illness or other personal problems. This can be done if relevant information is given to the Academic Supervisor or the Programme Director.

Dissertation format

Dissertations are normally expected to be between 10,000 and 14,000 words in length. Reports for the SwDS programme have a limit of 5,000 words for each project.

All should consist of the following:

  • Own work declaration
  • Executive summary (maximum 1 page)

The main text should consist of the following:

  • Introduction section
  • Final section on conclusions and/or recommendations
  • List of all bibliographic references
  • Appendices (optional)

Reports should be typeset with single spacing and font size 11 pt . The following minimum margins must be observed: 

The pages in the main text, bibliography, and appendices must be numbered consecutively.

Additional guidelines [password protected]

Dissertations from previous years 


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Mathematics MSc dissertations

The Department of Mathematics and Statistics was host until 2014 to the MSc course in the Mathematics of Scientific and Industrial Computation (previously known as Numerical Solution of Differential Equations) and the MSc course in Mathematical and Numerical Modelling of the Atmosphere and Oceans. A selection of dissertation titles are listed below, some of which are available online:

2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

2014: Mathematics of Scientific and Industrial Computation

Amanda Hynes - Slow and superfast diffusion of contaminant species through porous media

2014: Applicable and Numerical Mathematics

Emine Akkus - Estimating forecast error covariance matrices with ensembles

Rabindra Gurung - Numerical solution of an ODE system arising in photosynthesis

2013: Mathematics of Scientific and Industrial Computation

Zeinab Zargar - Modelling of Hot Water Flooding as an Enhanced Oil Recovery Method

Siti Mazulianawati Haji Majid - Numerical Approximation of Similarity in Nonlinear Diffusion Equations

2013: Mathematical and Numerical Modelling of the Atmosphere and Oceans

Yu Chau Lam - Drag and Momentum Fluxes Produced by Mountain Waves

Josie Dodd - A Moving Mesh Approach to Modelling the Grounding Line in Glaciology

2012: Mathematics of Scientific and Industrial Computation

Chris Louder - Mathematical Techniques of Image Processing

Jonathan Muir - Flux Modelling of Polynyas

Naomi Withey - Computer Simulations of Dipolar Fluids Using Ewald Summations

2012: Mathematical and Numerical Modelling of the Atmosphere and Oceans

Jean-Francois Vuillaume - Numerical prediction of flood plains using a Lagrangian approach

2011: Mathematics of Scientific and Industrial Computation

Tudor Ciochina - The Closest Point Method

Theodora Eleftheriou - Moving Mesh Methods Using Monitor Functions for the Porous Medium Equation

Melios Michael - Self-Consistent Field Calculations on a Variable Resolution Grid

2011: Mathematical and Numerical Modelling of the Atmosphere and Oceans

Peter Barnet - Rain Drop Growth by Collision and Coalescence

Matthew Edgington - Moving Mesh Methods for Semi-Linear Problems

Samuel Groth - Light Scattering by Penetrable Convex Polygons

Charlotte Kong - Comparison of Approximate Riemann Solvers

Amy Jackson - Estimation of Parameters in Traffic Flow Models Using Data Assimilation

Bruce Main - Solving Richards' Equation Using Fixed and Moving Mesh Schemes

Justin Prince - Fast Diffusion in Porous Media

Carl Svoboda - Reynolds Averaged Radiative Transfer Model

2010: Mathematics of Scientific and Industrial Computation

Tahnia Appasawmy - Wave Reflection and Trapping in a Two Dimensional Duct

Nicholas Bird - Univariate Aspects of Covariance Modelling within Operational Atmospheric Data Assimilation

Michael Conland - Numerical Approximation of a Quenching Problem

Katy Shearer - Mathematical Modelling of the regulation and uptake of dietary fats

Peter Westwood - A Moving Mesh Finite Element Approach for the Cahn-Hilliard Equation

Kam Wong - Accuracy of a Moving Mesh Numerical Method applied to the Self-similar Solution of Nonlinear PDEs

2010: Mathematical and Numerical Modelling of the Atmosphere and Oceans

James Barlow - Computation and Analysis of Baroclinic Rossby Wave Rays in the Atlantic and Pacific Oceans

Martin Conway - Heat Transfer in a Buried Pipe

Simon Driscoll - The Earth's Atmospheric Angular Momentum Budget and its Representation in Reanalysis Observation Datasets and Climate Models

George Fitton - A Comparative Study of Computational Methods in Cosmic Gas Dynamics Continued

Fay Luxford - Skewness of Atmospheric Flow Associated with a Wobbling Jetstream

Jesse Norris - A Semi-Analytic Approach to Baroclinic Instability on the African Easterly Jet

Robert J. Smith - Minimising Time-Stepping Errors in Numerical Models of the Atmosphere and Ocean

Amandeep Virdi - The Influence of the Agulhas Leakage on the Overturning Circulation from Momentum Balances

2009: Mathematics of Scientific and Industrial Computation

Charlotta Howarth - Integral Equation Formulations for Scattering Problems

David Fairbairn - Comparison of the Ensemble Transform Kalman Filter with the Ensemble Transform Kalman Smoother

Mark Payne - Mathematical Modelling of Platelet Signalling Pathways Mesh Generation and its application to Finite Element Methods

Mary Pham - Mesh Generation and its application to Finite Element Methods

Sarah Cole - Blow-up in a Chemotaxis Model Using a Moving Mesh Method

2009: Mathematical and Numerical Modelling of the Atmosphere and Oceans

Danila Volpi - Estimation of parameters in traffic flow models using data assimilation

Dale Partridge - Analysis and Computation of a Simple Glacier Model using Moving Grids

David MacLeod - Evaluation of precipitation over the Middle East and Mediterranean in high resolution climate models

Joanne Pocock - Ensemble Data Assimilation: How Many Members Do We Need?

Neeral Shah - Impact and implications of climate variability and change on glacier mass balance in Kenya

Tomos Roberts - Non-oscillatory interpolation for the Semi-Lagrangian scheme

Zak Kipling - Error growth in medium-range forecasting models

Zoe Gumm - Bragg Resonance by Ripple Beds

2008: Mathematics of Scientific and Industrial Computation

Muhammad Akram - Linear and Quadratic Finite Elements for a Moving Mesh Method

Andrew Ash - Examination of non-Time Harmonic Radio Waves Incident on Plasmas

Cassandra Moran - Harbour modelling and resonances

Elena Panti - Boundary Element Method for Heat Transfer in a Buried Pipe

Juri Parrinello - Modelling water uptake in rice using moving meshes

Ashley Twigger - Blow-up in the Nonlinear Schrodinger Equation Using an Adaptive Mesh Method

Chloe Ward - Numerical Evaluation of Oscillatory Integrals

Christopher Warner - Forward and Inverse Water-Wave Scattering by Topography

2008: Mathematical and Numerical Modelling of the Atmosphere and Oceans

Fawzi Al Busaidi - Fawzi Albusaidi

Christopher Bowden - A First Step Towards the Calculation of a Connectivity Matrix for the Great Barrier Reef

Evangelia-Maria Giannakopoulou - Flood Prediction and Uncertainty

Victoria Heighton - 'Every snowflake is different'

Thomas Jordan - Does Self-Organised Criticality Occur in the Tropical Convective System?

Gillian Morrison - Numerical Modelling of Tidal Bores using a Moving Mesh

Rachel Pritchard - Evaluation of Fractional Dispersion Models

2007: Numerical solution of differential equations

Tamsin Lee - New methods for approximating acoustic wave transmission through ducts (PDF 2.5MB)

Lee Morgan - Anomalous diffusion (PDF-1.5MB)

Keith Pham - Finite element modelling of multi-asset barrier options (PDF-3MB)

Alastair Radcliffe - Finite element modelling of the atmosphere using the shallow water equations (PDF-2.5MB)

Sanita Vetra - The computation of spectral representations for evolution PDE (PDF-3.2MB)

2007: Mathematical and numerical modelling of the atmosphere and oceans

Laura Baker - Properties of the ensemble Kalman filter (PDF-3.8MB)

Alison Brass - A moving mesh method for the discontinuous Galerkin finite element technique (PDF-916KB)

Daniel Lucas - Application of the phase/amplitude method to the study of trapped waves in the atmosphere and oceans (PDF-1.1MB)

Duduzile Nhlengethwa - Petrol or diesel (PDF-1MB)

Rhiannon Roberts - Modelling glacier flow (PDF-406KB)

David Skinner - A moving mesh finite element method for the shallow water equations (PDF-4.3MB)

Jovan Stojsavljevic - Investigation of waiting times in non-linear diffusion equations using a moving mesh method (PDF-538KB)

2006: Numerical solution of differential equations

Bonhi Bhattacharya - A moving finite element method for high order nonlinear diffusion problems

Jonathan Coleman - High frequency boundary element methods for scattering by complex polygons

Rachael England - The use of numerical methods in solving pricing problems for exotic financial derivatives with a stochastic volatility

Stefan King - Best fits with adjustable nodes and scale invariance

Edmund Ridley - Analysis of integral operators from scattering problems

Nicholas Robertson - A moving Lagrangian mesh model of a lava dome volcano and talus slope

2006: Mathematical and numerical modelling of the atmosphere and oceans

Iain Davison - Scale analysis of short term forecast errors

Richard Silveira - Electromagnetic scattering by simple ice crystal shapes

Nicola Stone - Development of a simplified adaptive finite element model of the Gulf Stream

Halina Watson - The behaviour of 4-D Var for a highly nonlinear system

2005: Numerical solution of differential equations

Jonathan Aitken - Data dependent mesh generation for peicewise linear interpolation

Stephen Arden - A collocation method for high frequency scattering by convex polygons

Shaun Benbow - Numerical methods for american options

Stewart Chidlow - Approximations to linear wave scattering by topography using an integral equation approach

Philip McLaughlin - Outdoor sound propagation and the boundary element method

Antonis Neochoritis - Numerical modelling of islands and capture zone size distributions in thin film growth

Kylie Osman - Numerical schemes for a non-linear diffusion problem

Shaun Potticary - Efficient evaluation of highly oscillatory integrals

Martyn Taylor - Investigation into how the reduction of length scales affects the flow of viscoelastic fluid in parallel plate geometries

Aanand Venkatramanan - American spread option pricing

2005: Mathematical and numerical modelling of the atmosphere and oceans

Richard Fruehmann - Ageostrophic wind storms in the central Caspian sea

Gemma Furness - Using optimal estimation theory for improved rainfall rates from polarization radar

Edward Hawkins - Vorticity extremes in numerical simulations of 2-D geostrophic turbulence

Robert Horton - Two dimensional turbulence in the atmosphere and oceans

David Livings - Aspects of the ensemble Kalman filter

David Sproson - Energetics and vertical structure of the thermohaline circulation

2004: Numerical solution of differential equations

Rakhib Ahmed - Numerical schemes applied to the Burgers and Buckley-Leverett equations

James Atkinson - Embedding methods for the numerical solution of convolution equations

Catherine Campbell-Grant - A comparative study of computational methods in cosmic gas dynamics

Paresh Prema - Numerical modelling of Island ripening

Mark Webber - The point source methods in inverse acoustic scattering

2004: Mathematical and numerical modelling of the atmosphere and oceans

Oliver Browne - Improving global glacier modelling by the inclusion of parameterised subgrid hypsometry within a three-dimensional, dynamical ice sheet model

Petros Dalakakis - Radar scattering by ice crystals

Eleanor Gosling - Flow through porous media: recovering permeability data from incomplete information by function fitting .

Sarah Grintzevitch - Heat waves: their climatic and biometeorological nature in two north american reigions

Helen Mansley - Dense water overflows and cascades

Polly Smith - Application of conservation laws with source terms to the shallow water equations and crowd dynamics

Peter Taylor - Application of parameter estimation to meteorology and food processing

2003: Numerical solution of differential equations

Kate Alexander - Investigation of a new macroscopic model of traffic flow

Luke Bennetts - An application of the re-iterated Galerkin approximation in 2-dimensions

Peter Spence - The Position of the free boundary formed between an expanding plasma and an electric field in differing geometries

Daniel Vollmer - Adaptive mesh refinement using subdivision of unstructured elements for conservation laws

2003: Mathematical and numerical modelling of the atmosphere and oceans

Clare Harris - The Valuation of weather derivatives using partial differential equations

Sarah Kew - Development of a 3D fractal cirrus model and its use in investigating the impact of cirrus inhomogeneity on radiation

Emma Quaile - Rotation dominated flow over a ridge

Jemma Shipton - Gravity waves in multilayer systems

2002: Numerical solution of differential equations

Winnie Chung - A Spectral Method for the Black Scholes Equations

Penny Marno - Crowded Macroscopic and Microscopic Models for Pedestrian Dynamics

Malachy McConnell - On the numerical solution of selected integrable non-linear wave equations

Stavri Mylona - An Application of Kepler's Problem to Formation Flying using the Störmer-Verlet Method

2002: Mathematical and numerical modelling of the atmosphere and oceans

Sarah Brodie - Numerical Modelling of Stratospheric Temperature Changes and their Possible Causes

Matt Sayer - Upper Ocean Variability in the Equatorial Pacific on Diurnal to Intra-seasonal Timescales

Laura Stanton - Linearising the Kepler problem for 4D-var Data Assimilation

2001: Numerical solution of differential equations

R.B. Brad - An Implementation of the Box Scheme for use on Transcritical Problems

D. Garwood - A Comparison of two approaches for the Approximating of 2-D Scattered Data, with Applications to Geological Modelling

R. Hawkes - Mesh Movement Algorithms for Non-linear Fisher-type Equations

P. Jelfs - Conjugate Gradients with Rational and Floating Point Arithmetic

M. Maisey - Vorticity Preserving Lax-Wendroff Type Schemes

C.A. Radcliffe - Positive Schemes for the Linear Advection Equation

2000: Numerical solution of differential equations

D. Brown - Two Data Assimilation Techniques for Linear Multi-input Systems.

S. Christodoulou - Finite Differences Applied to Stochastic Problems in Pricing Derivatives.

C. Freshwater - The Muskingum-Cunge Method for Flood Routing.

S.H. Man - Galerkin Methods for Coupled Integral Equations.

A. Laird - A New Method for Solving the 2-D Advection Equation.

T. McDowall - Finite Differences Applied to Joint Boundary Layer and Eigenvalue Problems.

M. Shahrill - Explicit Schemes for Finding Soliton Solutions of the Korteweg-de Vries Equation.

B. Weston - A Marker and Cell Solution of the Incompressible Navier-Stokes Equations for Free Surface Flow.

1999: Numerical solution of differential equations

M. Ariffin - Grid Equidistribution via Various Algorithmic Approaches.

S.J. Fletcher - Numerical Approximations to Bouyancy Advection in the Eddy Model.

N.Fulcher - The Finite Element Approximation of the Natural Frequencies of a Circular Drum.

V. Green - A Financial Model and Application of the Semi-Lagrangian Time-Stepping Scheme.

D.A. Parry - Construction of Symplectic Runge-Kutta Methods and their Potential for Molecular Dynamics Application.

S.C. Smith - The Evolution of Travelling Waves in a Simple Model for an Ionic Autocatalytic System

P. Swain - Numerical Investigations of Vorticity Preserving Lax-Wendroff Type Schemes.

M. Wakefield - Variational Methods for Upscaling.

1998: Numerical solution of differential equations

C.C. Anderson - A dual-porosity model for simulating the preferential movement of water in the unsaturated zone of a chalk aquifer.

K.W. Blake - Contour zoning.

M.R. Garvie - A comparison of cell-mapping techniques for basins of attraction.

W. Gaudin - HYDRA: a 3-d MPP Eulerian hydrocode.

D. Gnandi - Alternating direction implicit method applied to stochastic problems in derivative finance.

J. Hudson - Numerical techniques for conservation laws with source terms. .

H.S. Khela - The boundary integral method.

K. Singh - A comparison of numerical schemes for pricing bond options.

1997: Numerical solution of differential equations

R.V. Egan - Chaotic response of the Duffing equation. A numerical investigation into the dynamics of the non-linear vibration equation.

R.G. Higgs - Nonlinear diffusion in reservoir simulation.

P.B. Horrocks - Fokker-Planck model of stochastic acceleration: a study of finite difference schemes.

M.A. Wlasek - Variational data assimilation: a study.

1996: Numerical solution of differential equations

A. Barnes - Reaction-diffusion waves in an isothermal chemical system with a general order of autocatalysis.

S.J. Leary - Mesh movement and mesh subdivision.

S. McAllister - First and second order complex differential equations.

R.K. Sadhra - Investigating dynamical systems using the cell-to-cell mapping.

J.P. Wilson - A refined numerical model of sediment deposition on saltmarshes.

1995: Numerical solution of differential equations

M. Bishop - The modelling and analysis of the equations of motion of floating bodies on regular waves.

J. Olwoch - Isothermal autocatalytic reactions with an immobilized autocatalyst.

S. Stoke - Eulerian methods with a Lagrangian phase in gas dynamics.

R. Coad - 1-D and 2-D simulations of open channel flows using upwinding schemes.

1994: Numerical solution of differential equations

M. Ali - Application of control techniques to solving linear systems of equations .

M.H. Brookes - An investigation of a dual-porosity model for the simulation of unsaturated flow in a porous medium .

A.J. Crossley - Application of Roe's scheme to the shallow water equations on the sphere .

D.A. Kirkland - Huge singular values and the distance to instability. .

B.M. Neil - An investigation of the dynamics of several equidistribution schemes .

1993: Numerical solution of differential equations

P.A. Burton - Re-iterative methods for integral equations .

J.M. Hobbs - A moving finite element approach to semiconductor process modelling in 1-D. .

L.M. Whitfield - The application of optimal control theory to life cycle strategies .

S.J. Woolnough - A numerical model of sediment deposition on saltmarshes .

1992: Numerical solution of differential equations

I. MacDonald - The numerical solution of free surface/pressurized flow in pipes. .

A.D. Pollard - Preconditioned conjugate gradient methods for serial and parallel computers. .

C.J. Smith - Adaptive finite difference solutions for convection and convection-diffusion problems .

1991: Numerical solution of differential equations

K.J. Neylon - Block iterative methods for three-dimensional groundwater flow models .

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Department of Mathematics

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.

For more information on any of these projects, please contact the project supervisor.

For more information, please email Dr Miroslav Chlebík or visit his staff profile

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


!$[1]$! 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

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.

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

Recommended modules: Measure and Integration, Functional Analysis

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

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.)

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

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

For more information, please email Marianna Cerasuolo

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.

[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.

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.

[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.

For more information, please email Dr Antoine Dahlqvist or visit his staff profile

See PDF for full description

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

Antoine Dahlqvist - Brownian queues [PDF 151.46KB]

For more information, please email Dr Masoumeh Dashti or visit her staff profile

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.

!$[1]$! 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.

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.

!$[1]$! 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.

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.

!$[1]$! 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.

For more information, please email Dr Nicos Georgiou or visit his staff profile

\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 theorem

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.

For more information, please email Dr Peter Giesl or visit his staff profile

  Peter Giesl - Computational analysis of periodic orbits in nonsmooth differential [PDF 11.57KB]

  Peter Giesl - Calculation of Contraction Metrics [PDF 16.77KB]

  Peter Giesl Project 3 [PDF 92.74KB]

For more information, please email Chris Hadjichrysanthou

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.

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.

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.

For more information, please email Philip Herbert

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.

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.

For more information, please email Prof. James Hirschfeld or visit his staff profile

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

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

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

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

!$[1]$! 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

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

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

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

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. So, add further redundancy: 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

For more information, please email Dr Konstantinos Koumatos or visit his staff profile

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

!$[1]$! 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

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

!$[1]$! 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

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)

!$[1]$! 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

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)

!$[1]$! 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

For more information, please email [email protected] or visit her staff profile

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.

For more information, please email Dr Omar Lakkis or visit his staff profile

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.

!$[1]$! 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.

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.

!$[1]$! 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

For more information, please email Prof. Michael Melgaard or visit his staff profile

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.

!$[1]$! 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.

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.

  • 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.
  • 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.

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.

!$[1]$! 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.

The Choquard equation in three dimensions reads:

!$$\begin{equation} \tag*{(0.1)} -Δ u - \left( \int_{ℜ^{3}} u^{2}(y) W(x-y) \, dy \right) u(x) = -l u , \end{equation}$$! 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].

!$[1]$! 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).

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).


[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.

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),


[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.

For more information, please email Prof Veronica Sanz or visit her staff profile


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.

For more information, please email Dr Nick Simm or visit his staff profile

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

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

For more information, please email Dr Ali Taheri or visit his staff profile

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

!$[1]$! 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.

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

!$[1]$! 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.

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

!$[1]$! 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.

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

!$[1]$! 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.

For more information, please email James Van Yperen or visit his staff profile

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

Other helpful modules:

Introduction to Mathematical Biology, Probability Models, Random Processes

[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.

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.

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

[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.

For more information, please email Dr Chandrasekhar Venkataraman or visit his staff profile

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++

!$[1]$! 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

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.

!$[1]$! 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

For more information, please email Vladislav Vysotskiy >

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.

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

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

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.

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

For more information, please email Dr Minmin Wang or visit her staff profile

Minmin Wang - Probabilistic and combinatorial analysis of coalescence [PDF 64.35KB]

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Home > A&S > Math > MATH_GRADPROJ

Mathematics Graduate Projects and Theses

Theses/dissertations from 2022 2022.

Relationships Between COVID-19 Infection Rates, Healthcare Access, Socioeconomic Status, and Cultural Diversity , MarGhece P.J. Barnes

The Matrix Sortability Problem , Seth Cleaver

Cognitive Demand of Teacher-Created Mathematics Assessments , Megan Marie Schmidt

Waring Rank and Apolarity of Some Symmetric Polynomials , Max Brian Sullivan

Security Analysis of Lightweight Cryptographic Primitives , William Unger

Regression Analysis of Resilience and COVID-19 in Idaho Counties , Ishrat Zaman

Theses/Dissertations from 2021 2021

Tukey Morphisms Between Finite Binary Relations , Rhett Barton

A Data Adaptive Model for Retail Sales of Electricity , Johanna Marcelia

Exploring the Beginnings of Algebraic K-Theory , Sarah Schott

Zariski Geometries and Quantum Mechanics , Milan Zanussi

Theses/Dissertations from 2020 2020

The Directed Forest Complex of Cayley Graphs , Kennedy Courtney

Beliefs About Effective Instructional Practices Among Middle Grades Teachers of Mathematics , Lauren A. Dale

Analytic Solutions for Diffusion on Path Graphs and Its Application to the Modeling of the Evolution of Electrically Indiscernible Conformational States of Lysenin , K. Summer Ware

Theses/Dissertations from 2019 2019

Dynamic Sampling Versions of Popular SPC Charts for Big Data Analysis , Samuel Anyaso-Samuel

Computable Reducibility of Equivalence Relations , Marcello Gianni Krakoff

On the Fundamental Group of Plane Curve Complements , Mitchell Scofield

Radial Basis Function Finite Difference Approximations of the Laplace-Beltrami Operator , Sage Byron Shaw

Formally Verifying Peano Arithmetic , Morgan Sinclaire

Theses/Dissertations from 2018 2018

Selective Strong Screenability , Isaac Joseph Coombs

Mathematics Student Achievement in the Context of Idaho’s Advanced Opportunities Initiative , Nichole K. Hall

Secure MultiParty Protocol for Differentially-Private Data Release , Anthony Harris

Theses/Dissertations from 2017 2017

A Stable Algorithm for Divergence-Free and Curl-Free Radial Basis Functions in the Flat Limit , Kathryn Primrose Drake

The Classification Problem for Models of ZFC , Samuel Dworetzky

Joint Inversion of Compact Operators , James Ford

Trend and Return Level of Extreme Snow Events in New York City , Mintaek Lee

Multi-Rate Runge-Kutta-Chebyshev Time Stepping for Parabolic Equations on Adaptively Refined Meshes , Talin Mirzakhanian

Investigating College Instructors’ Methods of Differentiation and Derivatives in Calculus Classes , Wedad Mubaraki

The Random Graph and Reciprocity Laws , Spencer M. Nelson

Classification of Vertex-Transitive Structures , Stephanie Potter

Theses/Dissertations from 2016 2016

On the Conjugacy Problem for Automorphisms of Trees , Kyle Douglas Beserra

The Density Topology on the Reals with Analogues on Other Spaces , Stuart Nygard

Latin Squares and Their Applications to Cryptography , Nathan O. Schmidt

Solution Techniques and Error Analysis of General Classes of Partial Differential Equations , Wijayasinghe Arachchige Waruni Nisansala Wijayasinghe

Numerical Computing with Functions on the Sphere and Disk , Heather Denise Wilber

Theses/Dissertations from 2015 2015

The Classical Theory of Rearrangements , Monica Josue Agana

Nonlinear Partial Differential Equations, Their Solutions, and Properties , Prasanna Bandara

The Impact of a Quantitative Reasoning Instructional Approach to Linear Equations in Two Variables on Student Achievement and Student Thinking About Linearity , Paul Thomas Belue

Student Understanding of Function and Success in Calculus , Daniel I. Drlik

Monodromy Representation of the Braid Group , Phillip W. Hart

The Frobenius Problem , Anna Marie Megale

Theses/Dissertations from 2014 2014

Pi-1-1-determinacy and Sharps , Shehzad Ahmed

A Radial Basis Function Partition of Unity Method for Transport on the Sphere , Kevin Aiton

Diagrammatically Reducible 2-Complexes , Tyler Allyn

A Stochastic Parameter Regression Model for Long Memory Time Series , Rose Marie Ocker

Theses/Dissertations from 2013 2013

The Assignment Packet Grading System , Sarah Nichole Bruce

Using Learner-Generated Examples to Support Student Understanding of Functions , Martha Ottelia Dinkelman

Computing Curvature and Curvature Normals on Smooth Logically Cartesian Surface Meshes , John Thomas Hutchins

Schur's Theorem and Related Topics in Ramsey Theory , Summer Lynne Kisner

Theses/Dissertations from 2012 2012

On the Geometry of Virtual Knots , Rachel Elizabeth Byrd

A Stochastic Parameter Regression Approach for Time-Varying Relationship between Gold and Silver Prices , Birsen Canan-McGlone

Uncertainty Analysis of RELAP5-3D© , Alexandra E. Gertman and George L. Mesina

A Statistical Method for Regularizing Nonlinear Inverse Problems , Chad Clifton Hammerquist

Perfect Stripes from a General Turing Model in Different Geometries , Jean Tyson Schneider

Stability and Convergence for Nonlinear Partial Differential Equations , Oday Mohammed Waheeb

Regular Homotopy of Closed Curves on Surfaces , Katherine Kylee Zebedeo

Theses/Dissertations from 2011 2011

Coloring Problems , Thomas Antonio Charles Chartier

Modules Over Localized Group Rings for Groups Mapping Onto Free Groups , Nicholas Davidson

How Do We Help Students Interpret Contingency Tables? A Study on the Use of Proportional Reasoning as an Intervention , Kathleen M. Isaacson

A Fictitious Point Method for Handling Boundary Conditions in the RBF-FD Method , Joseph Lohmeier

Theses/Dissertations from 2010 2010

Developmental Understanding of the Equals Sign and Its Effects on Success in Algebra , Ryan W. Brown

The Inquiry Learning Model as an Approach to Mathematics Instruction , Michael C. Brune

Galois Theory for Differential Equations , Soheila Eghbali

Stably Free Modules Over the Klein Bottle , Andrew Misseldine

Combinatorics and Topology of Curves and Knots , Bailey Ann Ross

Theses/Dissertations from 2009 2009

Concept Booklets: Examining the Performance Effects of Journaling of Mathematics Course Concepts , Todd Stephen Fogdall

Effective Sample Size in Order Statistics of Correlated Data , Neill McGrath

Transparency in Formal Proof , Cap Petschulat

Weight Selection by Misfit Surfaces for Least Squares Estimation , Garrett Saunders

The Effects of a Standards-Based Mathematics Curriculum on the Self-Efficacy and Academic Achievement of Previously Unsuccessful Students , Cindy Chesley Shaw

Analytical Upstream Collocation Solution of a Quadratic Forced Steady-State Convection-Diffusion Equation , Eric Paul Smith

Solvability Characterizations of Pell Like Equations , Jason Smith

Theses/Dissertations from 2008 2008

Tube-Equivalence of Spanning Surfaces and Seifert Surfaces , Thomas Glass

Simple Tests for Short Memory in ARFIMA Models , Timothy A. C. Hughes

Incomparable Metrics on the Cantor Space , Trevor Jack

Richards' Equation and Its Constitutive Relations as a System of Differential-Algebraic Equations , Shannon K. Murray

Theses/Dissertations from 2007 2007

Theorem Proving in Elementary Analysis , Joanna Porter Guild

An Investigation of Lucas Sequences , Dustin E. Hinkel

A Canonical Countryman Line , William Russell Hudson

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Mathematics thesis and dissertation collection

dissertation project mathematics

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This collection contains a selection of the latest doctoral theses completed at the School of Mathematics. Please note this is not a comprehensive record.

This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.

Recent Submissions

Algebraic combinatorial structures for singular stochastic dynamics , stochastic modelling and inference of ocean transport , convergence problems for singular stochastic dynamics , classification of supersymmetric black holes in ads₅ , bps cohomology for 2-calabi—yau categories , quantitative finance informed machine learning , efficient model fitting approaches for estimating abundance and demographic rates for marked and unmarked populations , path-based splitting methods for sdes and machine learning for battery lifetime prognostics , certain geometric maximal functions in harmonic analysis , examining the effects of magnetic fields in neutron star mergers through numerical simulations , on slope stability of tangent sheaves on smooth toric fano varieties , geometric singular perturbation theory for reaction-diffusion systems , solution methods for some variants of the vehicle routing problem , estimates of space derivatives for functions of non-autonomous and mckean-vlasov processes. application to uniform weak error bounds induced by the approximating subsampled particle system , numerical framework for solving pde-constrained optimization problems from multiscale particle dynamics , regularised variational schemes for non-gradient systems, and large deviations for a class of reflected mckean-vlasov sde , non-lorentzian geometry of fluids and strings , moment polyptychs and the equivariant quantisation of hypertoric varieties , fukaya category and (open) gromov-witten invariants , foldy-wouthuysen transformation and its generalisations .

dissertation project mathematics


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  • Dissertation Topics Titles 2021-22
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  • Dissertation Topics Titles 2022-23
  • Dissertation Topics Titles 2023-24

Mathematical Institute

Please note the following topics are only open to Part C Maths, Maths & Phil, Maths & CompSci and OMMS students. For current students please see the full proposals here .

Representations of finite Hecke algebras - Prof D Ciubotaru

Homotopy Type Theory - Prof K Kremnitzer

Integrated Information Theory - Prof K Kremnitzer

Enumerating finite groups - Prof N Nikolov

Hyperquiver Representations - Prof V Nanda

Non-local PDEs and fractional Sobolev - Dr D Gomez-Castro

Fundamental solutions of linear partial differential equations - Prof J Kristensen

Extensions of Lipschitz maps, type and cotype - Dr K Ciosmak

Multi-dimensional Monge-Kantorovick system of PDE's - Dr K Ciosmak

von Neumann Algebras - Prof S White

Geometry, Number Theory and Topology

Modular Forms - Prof A Lauder

Graded rings and projective varieties - Prof B Szendroi

The Hardy-Littlewood Method - Prof B Green

Divergence of finitely generated groups - Dr B Sun

Geometric Class Field Theory - Prof D Rossler

The Semistable Reduction Theorem for Curves over Function Fields - Prof D Rossler

Poisson geometry and symplectic groupoids - Dr F Bischoff

Sieve Methods - Prof J Maynard

Galois Representation - Dr J Newton

Hodge Theory, Morse Theory and Supersymmetry - Prof J Lotay

Number Theory and Random Matrices - Prof J Keating

HKR Character Theory - Dr L Brantner

A bound for the systole of an aspherical manifold - Prof P Papazoglou

Analysis of Boolean Functions - Prof T Sanders

Chabauty techniques in Number Theory - Prof V Flynn

Topics in O-minimality - Prof J Pila

Mathematical Methods and Applications 

Mathematical Modelling of Plant - Prof D Moulton

Magneto-active elastic objects - Combining magnetism with elasticity - Prof D Vella

Modelling aspects of cells and Stokes flows in mathematical biology - Prof E Gaffney

Modelling aspects of cellular signalling beyond the simplest Turing mechanism - Prof E Gaffney

Modelling geothermal boreholes using pertubation methods - Prof I Hewitt

Viscoplastic models for geophysical flows - Prof I Hewitt

The time-elapsed model for neural networks - D P Roux

Dynamics on signed networks - Prof R Lambiotte

Mathematical Physics

The classification of 2d conformal field theories - Prof A Henriques

Scattering Theory - Prof L Mason

Numerical Analysis and Data Science

Machine Learning and Artificial Intelligence In Healthcare - Dr A Kormilitzin

Approximation of functions in a square, cube, and hypercube - Prof N Trefethen

Lightning Helmholtz solver - Prof N Trefethen

Numerical conformal mapping - Prof N Trefethen

Development and Calibration of Models for Pedestrian Dynamics - Dr R Bailo

Numerical Schemes for Crystal Growth - Dr R Bailo

(Randomised) Numerical Linear Algebra - Prof Y Nakatsukasa

Characterizing the structure of networks with discrete Ricci curvature - Dr M Weber

Optimization algorithms for data science - Prof C Cartis

Stochastics, Discrete Mathematics and Information

Random walk in random environment - Prof B Hambly

Blockchains and (Decentralized) Exchanges - Prof H Oberhauser

Bismut formula, Feynman-Kac formula and estimates for second order parabolic equations - Prof Z Qian

Convergence of finite Markov chains on abelian groups - Prof Z Qian

PDF method in turbulence - Prof Z Qian

History of Mathematics

Students wishing to do a dissertation based on the History of Mathematics are asked to contact Brigitte Stenhouse at  @email  by Wednesday of week 1 with a short draft proposal. All decisions will be communicated to students by the end of week 2.

All supported proposals , will then be referred to Projects Committee who meet in week 4 for final approval. With the support of Brigitte Stenhouse students must submit a COD Dissertation Proposal Form to Projects Committee by the end of week 3.

Department of Statistics

Please note that Part C Mathematics and Statistics students MUST select from the list of the below topics. OMMS students are also able to select the Statistics and Probability projects from the Department of Statistics.

It may be possible for a Maths student to complete a Statistics dissertation, however, the priority when allocating will be the Maths & Stats and OMMS students. If you are interested, please email  @email  for more information.

A novel deconvolution method based on entropic optimal transport - Dr G Mena

Applications of Machine Learning to Drug Discovery - Prof G Morris

Bayesian Optimal Experimental Design - Dr T Rainforth

Co-jumping behaviour in time series and financial networks - Prof M Cucuringu

Concentration inequalities and applications - Prof G Deligiannidis

Convergence Complexity for Markov Chain Monte Carlo in High Dimensions - Dr J Yang

Extreme Classification - Prof F Carron

Genealogies of Sequences experiencing Recombination - Prof J Hein

 How many have died due to the COVID-19 pandemic and who was at greatest risk - Prof C Donnelly

Instrumental Variable Estimation with Weak Instruments - Prof F Windmeijer

Kernel-based tests and dependence measures - Prof D Sejdinovic

Mirror Descent and Statistical Robustness - Prof P Rebeschini

Multi-Locus Phase-type Distributions in Population Genetics - Dr A Biddanda

Polygenic scores - Prof R Davies

Protein folding interfaces template the formation of the native state - Dr D Nissley

Quasistationary distributions for Markov processes - Prof D Steinsaltz

Random Recursive Trees - Prof C Goldschmidt

Urn models and applications - Prof M Winkel

MA334       Half Unit Dissertation in Mathematics

This information is for the 2020/21 session.

Teacher responsible

Prof Bernhard Von Stengel and Mrs Nicola Wittur


This course is available on the BSc in Mathematics with Economics. This course is not available as an outside option nor to General Course students.


Students must have completed Real Analysis (MA203).

Some dissertation topics might require additional pre-requisites which will be specified in the description of the topic provided by the member of staff supervising the dissertation.

Course content

The dissertation in mathematics is an individual project that serves as an introduction to mathematical research. The student will investigate and study an area of mathematical research or apply advanced mathematical techniques to model and solve problems arising in other areas related to the student’s degree programme (e.g., in finance or economics). The student will write a report on their findings and present and discuss their findings in an oral examination. The project may include some programming. The dissertation topic will normally be proposed by the Department.

This course is delivered through seminars and computer workshops that total a minimum of 10 hours across Michaelmas and Lent Term, which give general and practical information, plus personal supervision time, which is scheduled independently with student supervisors. This year, some or all of this teaching will be delivered through a combination of virtual classes and lectures delivered as online videos. The seminars in MT will cover important aspects of writing a dissertation in mathematics, including: what plagiarism is and how to avoid it, the use of libraries for research, electronic research, general aspects of writing mathematics, managing a research project and the writing up process. The computer workshops in MT will provide guidance on preparing a manuscript using mathematical text processing software (in particular, LaTeX).The seminars in LT will cover how to give a presentation about the findings in the dissertation and how to prepare for the oral examination. Each student will be assigned a supervisor who will monitor their progress and provide appropriate guidance thorough the MT and LT. Normally students will have three individual supervision meetings each term.

Formative coursework

Students will be expected to produce 1 presentation and 1 other piece of coursework in the LT.

Indicative reading

This will depend on the topic of the dissertation. Students will be guided by their supervisor. 

Dissertation (75%) and presentation (25%) in the LT Week 11.

Assessment is based on the dissertation and an oral examination.

Three hard copies and one electronic copy of the dissertation must be submitted by the end of Lent Term (exact date to be specified later). The report may include some computer code relating to the project. The dissertation excluding the bibliography must not exceed 20 pages of A4 paper, where the dissertation is required to have 1.5 line spacing at a minimum (at most 33 lines of text/mathematical formulae per page), 11-point font and 1-inch margins all around. If the dissertation contains any computer code this should be placed in the appendix of the dissertation and does not count towards the page limit.

The oral examination consists of a presentation to an audience of two members of staff on the main findings contained in the dissertation followed by a brief discussion where the student will be asked questions on the topic of their dissertation. This will be graded and worth 25% of the course grade. Students will be given support in the seminars on how to prepare, how to present and what is expected.

Important information in response to COVID-19

Please note that during 2020/21 academic year some variation to teaching and learning activities may be required to respond to changes in public health advice and/or to account for the situation of students in attendance on campus and those studying online during the early part of the academic year. For assessment, this may involve changes to mode of delivery and/or the format or weighting of assessments. Changes will only be made if required and students will be notified about any changes to teaching or assessment plans at the earliest opportunity.

Department: Mathematics

Total students 2019/20: Unavailable

Average class size 2019/20: Unavailable

Capped 2019/20: No

Value: Half Unit

Personal development skills

  • Self-management
  • Problem solving
  • Application of information skills
  • Communication
  • Application of numeracy skills
  • Specialist skills


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