applied mathematics research project ideas

Applied mathematics projects

• Research projects

By undertaking a project with us, you’ll have the chance to create change within a range of diverse areas.​

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• Summer/Winter/SCIE3250

Mathematical and computational modelling of biological tissues: mechanical properties and morphogenesis

Mathematical / computational neuroscience, loewner evolution, interactive demonstrations using mathematica, information and large deviations in statistical estimation, importance sampling for diffusion processes via path-dependent mapping, estimation and inference on random networks, root systems associated to graphs, empirical study of stochastic network games, machine learning for defined contribution superannuation, numerical methods for hamilton-jacobi-bellman equations in finance, graphical cryptography, risk and uncertainty quantification in environmental modelling, effective computing in model predictive control for urban road traffic networks, forecasting future global fisheries production under climate change using systems of differential equations, fishery-dependent monitoring of queensland’s fisheries, queensland state-wide estimation of recreational fish catches, modeling the dynamics of the faintest dwarf galaxies, advanced monte carlo methods, could geoengineering the atmosphere to minimise global warming compromise global fisheries, using systems of partial differential equations to understand changes in tuna fisheries in response to climate change, how could marine heatwaves impact marine biodiversity in the future, computational biology of actin filament networks, simulation of collisions in intra-cellular transport, tug-of-war in intracellular transport, modelling and simulation of tempo-spatial dynamics in australian petrol retail pricing, the value of information for saving the environment, modern statistical inference, sequential importance sampling with mixture models, modelling the growth and treatment of brain tumours, mathematical modelling of epidermal self-renewal and radiation induced cancer progression.

Applied Mathematics Research

In applied mathematics, we look for important connections with other disciplines that may inspire interesting and useful mathematics, and where innovative mathematical reasoning may lead to new insights and applications.

Applied Mathematics Fields

• Combinatorics
• Computational Biology
• Physical Applied Mathematics
• Computational Science & Numerical Analysis
• Theoretical Computer Science
• Mathematics of Data

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Research projects suitable for undergraduates

What follows is a sample, provided by members of the faculty, of mathematical research projects where undergraduate students in the honors program in mathematics could participate. Interested students should contact either the faculty members directly, one of the honors advisors: professors Al Novikoff or Steve Childress .

A joint research project of Helmut Hofer and Esteban Tabak studies the behavior of Hamiltonian flows on a prescribed energy surface. Computer experiments using symplectic integrators could give some new insight. Such a project would be ideal for a team of an undergraduate and a graduate student. Codes would be developed and experiments would be conducted, shedding new light on the intriguing dynamics of these flows.

Charles Newman has recently studied zero-temperature stochastic dynamics of Ising models with a quenched (i.e., random) initial configuration. When the Ising models are disordered (e.g., a spin glass), there are a host of open problems in statistical physics which could be profitably investigated via Monte Carlo simulations by students (graduate and undergraduate) without an extensive background in the field. For example, on a two-dimensional square lattice, in the +/- J spin glass model, it is known that some sites flip forever and some don't; what happens in dimension three?

Current experiments in the Applied Mathematics Laboratory (WetLab/VisLab) include one project on dynamics of friction, and another involving the interaction of fluid flow with deformable bodies. Gathering data, mathematical modeling, and data analysis all provide excellent opportunities for undergraduate research experiences. In the friction experiment of Steve Childress, for example, the formulation and numerical solution of simplified models of stick/slip dynamics gives exposure to modern concepts of dynamical systems, computer graphics and analysis, and the mathematics of numerical analysis.

Marco Avellaneda's current research in mathematical finance demands econometric data to establish a basis for mathematical modeling and computation. The collection and analysis of such data could be done by undergraduates. The idea is to get comprehensive historical price data from several sources and perform empirical analysis of the correlation matrices between different price shocks in the same economy. The goal of the project is to map the ``principal components'' of the major markets.

Joel Spencer is studying the enumeration of connected graphs with given numbers of vertices and edges. The approach turns asymptotically into certain questions about Brownian motion. Much of the asymptotic calculation is suitable for undergraduates, while the subtleties of going to the Brownian limit would need a more advanced student.

A joint project of David McLaughlin, Michael Shelley, and Robert Shapley (Professor, Center for Neural Science, NYU) is developing a computer model of the area V1 of the monkey's primary visual cortex. Simplifications of this complex network model can provide projects for advanced undergraduate students, giving excellent exposure to mathematical and computational modeling, as well as to biological experiment and observation.

Peter Lax has carried out many numerical experiments with dispersive systems, and with systems modeling shock waves. The basic theory of these equations is well within the grasp of interested undergraduates, and calculations can reveal new phenomena.

A joint research project of David Holland and Esteban Tabak investigates ocean circulation at regional, basinal and global scales. Their approach is based on a combination of numerical and analytical techniques. There is an opportunity within this framework for undergraduate and graduate students to work together to further develop the simplified analytical and numerical models so as to gain insight into various mechanisms underlying and controlling ocean circulation.

Aspects of Lai-Sang Young's work in dynamical systems, chaos, and fractal geometry are suitable for undergraduate research projects. Simple analytic tools for iterations are accessible to students. Research in this area brings together material the undergraduate student has just learned from his or her classes. With proper guidance, this can be a meaningful scientific experience with the possibility of new discoveries.

David McLaughlin and Jalal Shatah's work on dynamical systems provides opportunities for undergraduate research experiences. For instance, the study of normal forms and resonances can be simplified to require only calculus and linear algebra. Thus undergraduate students can study analytically what is resonant in a given physical system, as well as its concrete consequences on qualitative behavior.

Leslie Greengard and Marsha Berger's work on adaptive computational methods plays an increasingly critical role in scientific computing and simulation. There are a number of opportunities for undergraduate involvement in this research. These range from designing algorithms for parallel computing to using large-scale simulation for the investigation of basic questions in fluid mechanics and materials science.

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Department of Mathematics

Research projects

Find a postgraduate research project in your area of interest by exploring the research projects we offer in the Department of Mathematics.

Programme directors

If you are not sure which supervisors are the best match for your interests, contact the postgraduate programme directors:

• Sean Holman  (applied mathematics and numerical analysis)
• Olatunji Johnson (probability, statistics and financial mathematics)
• Marcus Tressl  (pure mathematics)

You can also get in touch with the postgraduate research leads through our  research themes  page.

Opportunities within the department are advertised by supervisors as either:

• Specific, well-defined individual projects : which you can apply for directly after contacting the named supervisor
• Research fields with suggestions for possible projects : where you can discuss a range of potential projects available in a specific area with the supervisor.

Choosing the right PhD project depends on matching your interests to those of your supervisor.

Our  research themes  page gives an overview of the research taking place in the Department and contacts for each area. Potential supervisors can also be contacted directly through the academic staff list . They will be able to tell you more about the type of projects they offer and/or you can suggest a research project yourself.

Please note that all PhD projects are eligible for funding via a variety of scholarships from the Department, the Faculty of Science and Engineering and/or the University; see our  funding page  for further details. All scholarships are awarded competitively by the relevant postgraduate funding committees.

Academics regularly apply for research grants and may therefore be able to offer funding for specific projects without requiring approval from these committees. Some specific funded projects are listed below, but many of our students instead arrive at a project through discussion with potential supervisors.

Specific, individual projects

Browse all of our specific, individual projects listed on FindAPhD:

Research field projects

In addition to individual projects listed on FindAPhD, we are also looking for postgraduate researchers for potential projects within a number of other research fields.

Browse these fields below and get in contact with the named supervisor to find out more.

Applied Mathematics and Numerical Analysis

Adaptive finite element approximation strategies.

Theme: Numerical analytics and scientific computing | Supervisor: David Silvester

I would be happy to supervise projects in the general area of efficient solution of elliptic and parabolic partial differential equations using finite elements. PhD projects would involve a mix of theoretical analysis and the development of proof-of-concept software written in MATLAB or Python. The design of robust and efficient error estimators is an open problem in computational fluid dynamics. Recent papers on this topic include Alex Bespalov, Leonardo Rocchi and David Silvester, T--IFISS: a toolbox for adaptive FEM computation, Computers and Mathematics with Applications, 81: 373--390, 2021. https://doi.org/10.1016/j.camwa.2020.03.005 Arbaz Khan, Catherine Powell and David Silvester, Robust a posteriori error estimators for mixed approximation of nearly incompressible elasticity, International Journal for Numerical Methods in Engineering, 119: 1--20, 2019. https://doi.org/10.1002/nme.6040 John Pearson, Jen Pestana and David Silvester, Refined saddle-point preconditioners for discretized Stokes problems, Numerische Mathematik, 138: 331--363, 2018. https://doi.org/10.1007/s00211-017-0908-4

Bayesian and machine learning methods for statistical inverse problems

Theme: Statistics, inverse problems, uncertainty quantification and data science | Supervisor: Simon Cotter

A range of projects are available on the topic of statistical inverse problems, in particular with application to problems in applied mathematics. Our aim is to construct new methods for the solution of statistical inverse problems, and to apply them to real problems from science, biology, engineering, etc. These may be more traditional Markov chain Monte Carlo (MCMC) methods, Piecewise-deterministic Markov processes (PDMPs), gradient flows (e.g. Stein gradient descent), or entirely new families of methods. Where possible the methods will be flexible and widely applicable, which will enable us to also apply them to real problems and datasets. Some recent applications involve cell matching in biology, and characterisation of physical properties of materials, for example the thermal properties of a manmade material, or the Young's modulus of a tendon or artery. The project will require the candidate to be proficient in a modern programming language (e.g. Python).

Complex deformations of biological soft tissues

Theme: Continuum mechanics | Supervisor: Tom Shearer , Andrew Hazel

The answers to many open questions in medicine depend on understanding the mechanical behaviour of biological soft tissues. For example, which tendon is most appropriate to replace the anterior cruciate ligament in reconstruction surgery? what causes the onset of aneurysms in the aorta? and how does the mechanics of the bladder wall affect afferent nerve firing? Current work at The University of Manchester seeks to understand how the microstructure of a biological soft tissue affects its macroscale mechanical properties. We have previously focused on developing non-linear elastic models of tendons and are now seeking to incorporate more complex physics such as viscoelasticity, and to consider other biological soft tissues, using our “in house” finite element software oomph-lib. The work will require development and implementation of novel constitutive equations as well as formulation of non-standard problems in solid mechanics. The project is likely to appeal to students with an interest in continuum mechanics, computational mathematics and interdisciplinary science.

Efficient solution for PDEs with random data

I would be happy to supervise projects in the general area of efficient solution of elliptic and parabolic partial differential equations with random data. PhD projects would involve a mix of theoretical analysis and the development of proof-of-concept software written in MATLAB or Python. The design of robust and efficient error estimators for stochastic collocation approximation methods is an active area of research within the uncertainty quantification community. Recent papers on this topic include Alex Bespalov, David Silvester and Feng Xu. Error estimation and adaptivity for stochastic collocation finite elements Part I: single-level approximation, SIAM J. Scientific Computing, 44: A3393--A3412, 2022. {\tt https://doi.org/10.1137/21M1446745} Arbaz Khan, Alex Bespalov, Catherine Powell and David Silvester, Robust a posteriori error estimators for stochastic Galerkin formulations of parameter-dependent linear elasticity equations, Mathematics of Computation, 90: 613--636, 2021. https://doi.org/10.1090/mcom/3572 Jens Lang, Rob Scheichl and David Silvester, A fully adaptive multilevel collocation strategy for solving elliptic PDEs with random data, J. Computational Physics, 419, 109692, 2020. https://doi.org/10.1016/j.jcp.2020.109692

Fluid flow, interfaces, bifurcations, continuation and control

Theme: Continuum mechanics | Supervisor: Alice Thompson

My research interests are in fluid dynamical systems with deformable interfaces, for example bubbles in very viscous fluid, or inkjet printed droplets. The deformability of the interface can lead to complex nonlinear behaviour, and often occurs in configurations where full numerical simulation of the three-dimensional system is computationally impossible. This computational difficulty leaves an important role for mathematical modelling, in using asymptotic or physical arguments to devise simpler models which can help us understand underlying physical mechanisms, make testable predictions, and to directly access control problems for active (feedback) or passive control mechanisms. Most recently I am interested in how different modelling methodologies affect whether models are robust in the forward or control problems. I am also interested in how control-based continuation methods can be used in continuum mechanics to directly observe unstable dynamical behaviour in experiments, even without access to a physical model. This research combines fluid dynamics, mathematical modelling, computational methods (e.g. with the finite-element library oomph-lib), experiments conducted in the Manchester Centre for Nonlinear Dynamics, control theory and nonlinear dynamics. I would not expect any student to have experience in all these areas and there is scope to shape any project to your interests.

Granular materials in industry and nature

Theme: Continuum mechanics | Supervisor: Chris Johnson

The field of granular materials encompasses a vast range of materials and processes, from the formation of sand dunes on a beach and snow avalanches in the mountains, to the roasting of coffee beans and the manufacture of pharmaceutical tablets. The science of granular materials is still in its relative infancy, and many aspects of flowing grains cannot yet be predicted with a continuum rheology. Insights into granular material behaviour come from a range of methods, and my research therefore combines mathematical modelling, computation, and laboratory experiments, undertaken at the Manchester Centre for Nonlinear Dynamics laboratories. Some example areas of work suitable for a PhD project include: - Debris flows and their deposits Debris flows are rapid avalanches of rock and water, which are triggered on mountainsides when erodible sediment is destabilised by heavy rainfall or snowmelt. These flows cause loss of life and infrastructure across the world, but many of the physical mechanisms underlying their motion remain poorly understood. Because it is difficult to predict where and when a debris flow will occur, scientific observations are rarely made on an active flow. More often, all we have to work from is the deposit left behind, and some detective work is required to infer properties of the flow (such as its speed and composition) from this deposit. This project focuses on developing theoretical models for debris flows that predict both a debris flow and its deposit -- in particular the way in which grains of different sizes are distributed throughout the deposit. The aim is then to invert such models, allowing observations of a deposit, when combined with model simulations, to constrain what must have happened during the flow. - Modelling polydispersity Much of the current theory of granular materials has been formulated with the assumption of a single type of grain. When grains vary in size, shape or density, it opens up the possibility that such grains with different properties separate from one another, a process called segregation. A fundamental question in this area is predicting the rate of segregation from a description of a granular material, such as the distribution of particle sizes. Thanks to some recent developments, we are approaching a point where this can be done for very simple granular materials (in particular those containing only two, similar, sizes sizes of grain), but many practical granular materials are much more complex. For example, it is common for mixtures of grains used in industry to vary in diameter by a factor of more than 100, and the complex segregation that can occur in these mixtures is poorly understood. This project will make measurements of the segregation behaviour of such mixtures and use these to put together a theoretical framework for describing segregation in complex granular materials.

Machine learning with partial differential equations

Theme: Statistics, inverse problems, uncertainty quantification and data science | Supervisor: Jonas Latz

Machine learning and artificial intelligence play a major part in our everyday life. Self-driving cars, automatic diagnoses from medical images, face recognition, or fraud detection, all profit especially from the universal applicability of deep neural networks. Their use in safety critical applications, however, is problematic: no interpretability, missing mathematical guarantees for network or learning process, and no quantification of the uncertainties in the neural network output. Recently, models that are based on partial differential equations (PDEs) have gained popularity in machine learning. In a classification problem, for instance, a PDE is constructed whose solution correctly classifies the training data and gives a suitable model to classify unlabelled feature vectors. In practice, feature vectors tend to be high dimensional and the natural space on which they live tends to have a complicated geometry. Therefore, partial differential equations on graphs are particularly suitable and popular. The resulting models are interpretable, mathematically well-understood, and uncertainty quantification is possible. In addition, they can be employed in a semi-supervised fashion, making them highly applicable in small data settings. I am interested in various mathematical, statistical, and computational aspects of PDE-based machine learning. Many of those aspects translate easily into PhD projects; examples are - Efficient algorithms for p-Laplacian-based regression and clustering - Bayesian identification of graphs from flow data - PDEs on random graphs - Deeply learned PDEs in data science Depending on the project, applicants should be familiar with at least one of: (a) numerical analysis and numerical linear algebra; (b) probability theory and statistics; (c) machine learning and deep learning.

Mathematical modelling of nano-reinforced foams

Theme: Maths: Continuum mechanics | Supervisor: William Parnell

Complex materials are important in almost every aspect of our lives, whether that is using a cell phone, insulating a house, ensuring that transport is environmentally friendly or that packaging is sustainable. An important facet of this is to ensure that materials are optimal in some sense. This could be an optimal stiffness for a given weight or an optimal conductivity for a given stiffness. Foams are an important class of material that are lightweight but also have the potential for unprecedented mechanical properties by adding nano-reinforcements (graphene flakes or carbon nanotubes) into the background or matrix material from which the foam is fabricated. When coupled with experimentation such as imaging and mechanical testing, mathematical models allow us to understand how to improve the design and properties of such foams. A number of projects are available in this broad area and interested parties can discuss these by making contact with the supervisor.

Multiscale modelling of structure-function relationships in biological tissues

Theme: Maths: Mathematics in the life sciences | Supervisor: Oliver Jensen

Biological tissues have an intrinsically multiscale structure. They contain components that range in size from individual molecules to the scale of whole organs. The organisation of individual components of a tissue, which often has a stochastic component, is intimately connected to biological function. Examples include exchange organs such as the lung and placenta, and developing multicellular tissues where mechanical forces play an crucial role in growth. To describe such materials mathematically, new multiscale approaches are needed that retain essential elements of tissue organisation at small scales, while providing tractable descriptions of function at larger scales. Projects are available in these areas that offer opportunities to collaborate with life scientists while developing original mathematical models relating tissue structure to its biological function.

Pure Mathematics and Logic

Algebraic differential equations and model theory.

Theme: Algebra, logic and number theory | Supervisor: Omar Leon sanchez

Differential rings and algebraic differential equations have been a crucial source of examples for model theory (more specifically, geometric stability theory), and have had numerous application in number theory, algebraic geometry, and combinatorics (to name a few). In this project we propose to establish and analyse deep structural results on the model theory of (partial) differential fields. In particular, in the setup of differentially large fields. There are interesting questions around inverse problems in differential Galois theory that can be address as part of this project. On the other hand, there are (still open) questions related to the different notions of rank in differentially closed fields; for instance: are there infinite dimensional types that are also strongly minimal? This is somewhat related to the understanding of regular types, which interestingly are quite far from being fully classified. A weak version of Zilber's dichotomy have been established for such types, but is the full dichotomy true?

Algebraic invariants of abelian varieties

Theme: Algebra, logic and number theory | Supervisor: Martin Orr

Project: Algebraic invariants of abelian varieties Abelian varieties are higher-dimensional generalisations of elliptic curves, objects of algebraic geometry which are of great interest to number theorists. There are various open questions about how properties of abelian varieties vary across a families of abelian varieties. The aim of this project is to study the variation of algebraic objects attached to abelian varieties, such as endomorphism algebras, Mumford-Tate groups or isogenies. These algebraic objects control much of the behaviour of the abelian variety. We aim to bound their complexity in terms of the equations defining the abelian variety. Potential specific projects include: (1) Constructing "relations between periods" from the Mumford-Tate group. This involves concrete calculations of polynomials, similar in style to classical invariant theory of reductive groups. (2) Understanding the interactions between isogenies and polarisations of abelian varieties. This involves calculations with fundamental sets for arithmetic group actions, generalising reduction theory for quadratic forms. A key tool is the theory of reductive groups and their finite-dimensional representations (roots and weights).

Algebraic Model Theory of Fields with Operators

Model theory is a branch of Mathematical Logic that has had several remarkable applications with other areas of mathematics, such as Combinatorics, Algebraic Geometry, Number Theory, Arithmetic Geometry, Complex and Real Analysis, Functional Analysis, and Algebra (to name a few). Some of these applications have come from the study of model-theoretic properties of fields equipped with a family of operators. For instance, this includes differential/difference fields. In this project, we look at the model theory of fields equipped with general classes of operators and also within certain natural classes of arithmetic fields (such as large fields). Several foundational questions remain open around what is called "model-companion", "elimination of imaginaries", and the "trichotomy", this is a small sample of the problems that can be tackled.

Homeomorphism groups from a geometric viewpoint

Theme: Algebra, logic and number theory | Supervisor: Richard Webb

A powerful technique for studying groups is to use their actions by isometries on metric spaces. Properties of the action can be translated into algebraic properties of the group, and vice versa. This is called geometric group theory, and has played a key role in different fields of mathematics e.g. random groups, mapping class groups of surfaces, fundamental groups of 3-manifolds, the Cremona group. In this project we will study the homeomorphisms of a surface by using geometric group theoretic techniques recently introduced by Bowden, Hensel, and myself. This is a new research initiative at the frontier between dynamics, topology, and geometric group theory, and there are many questions waiting to be explored using these tools. These range from new questions on the relationship between the topology/dynamics of homeomorphisms and their action on metric spaces, to older questions regarding the algebraic structure of the homeomorphism group.

The Existential Closedness problem for exponential and automorphic functions

Theme: Maths: Algebra, logic and number theory | Supervisor: Vahagn Aslanyan

The Existential Closedness problem asks when systems of equations involving field operations and certain classical functions of a complex variable, such as exponential and modular functions, have solutions in the complex numbers. There are conjectures predicting when such systems should have solutions. The general philosophy is that when a system is not "overdetermined" (e.g. more equations than variables) then it should have a solution. The notion of an overdetermined system of equations is related to Schanuel's conjecture and its analogues and is captured by some purely algebraic conditions. The aim of this project is to make progress towards the Existential Closedness conjectures (EC for short) for exponential and automorphic functions (and the derivatives of automorphic functions). These include the usual complex exponential function, as well as the exponential functions of semi-abelian varieties, and modular functions such as the j-invariant. Significant progress has been made towards EC in recent years, but the full conjectures are open. There are many special cases which are within reach and could be tackled as part of a PhD project. Methods used to approach EC come from complex analysis and geometry, differential algebra, model theory (including o-minimality), tropical geometry. Potential specific projects are: (1) proving EC in low dimensions (e.g. for 2 or 3 variables), (2) proving EC for certain relations defined in terms of the function under consideration, e.g. establishing new EC results for "blurred" exponential and/or modular functions, (3) proving EC under additional geometric assumptions on the system of equations, (4) using EC to study the model-theoretic properties of exponential and automorphic functions.

Statistics and Probability

Distributional approximation by stein's method theme.

Theme: Probability, financial mathematics and actuarial science | Supervisor: Robert Gaunt

Stein's method is a powerful (and elegant) technique for deriving bounds on the distance between two probability distributions with respect to a probability metric. Such bounds are of interest, for example, in statistical inference when samples sizes are small; indeed, obtaining bounds on the rate of convergence of the central limit theorem was one of the most important problems in probability theory in the first half of the 20th century. The method is based on differential or difference equations that in a sense characterise the limit distribution and coupling techniques that allow one to derive approximations whilst retaining the probabilistic intuition. There is an active area of research concerning the development of Stein's method as a probabilistic tool and its application in areas as diverse as random graph theory, statistical mechanics and queuing theory. There is an excellent survey of Stein's method (see below) and, given a strong background in probability, the basic method can be learnt quite quickly, so it would be possible for the interested student to make progress on new problems relatively shortly into their PhD. Possible directions for research (although not limited) include: extend Stein's method to new limit distributions; generalisations of the central limit theorem; investigate `faster than would be expected' convergence rates and establish necessary and sufficient conditions under which they occur; applications of Stein's method to problems from, for example, statistical inference. Literature: Ross, N. Fundamentals of Stein's method. Probability Surveys 8 (2011), pp. 210-293.

Long-term behaviour of Markov Chains

Theme: Probability, financial mathematics and actuarial science | Supervisor: Malwina Luczak

Several projects are available, studying idealised Markovian models of epidemic, population and network processes. The emphasis will mostly be on theoretical aspects of the models, involving advanced probability theory. For instance, there are a number of stochastic models of epidemics where the course of the epidemic is known to follow the solution of a differential equation over short time intervals, but where little or nothing has been proved about the long-term behaviour of the stochastic process. Techniques have been developed for studying such problems, and a project might involve adapting these methods to new settings. Depending on the preference of the candidate, a project might involve a substantial computational component, gaining insights into the behaviour of a model, via simulations, ahead of proving rigorous theoretical results.

Mathematical Epidemiology

Theme: Mathematics in the life sciences | Supervisor: Thomas House

Understanding patterns of disease at the population level - Epidemiology - is inherently a quantitative problem, and increasingly involves sophisticated research-level mathematics and statistics in both infectious and chronic diseases. The details of which diseases and mathematics offer the best PhD directions are likely to vary over time, but this broad area is available for PhD research.

Spatial and temporal modelling for crime

Theme: Maths: Statistics, inverse problems, uncertainty quantification and data science | Supervisor: Ines Henriques-Cadby , Olatunji Johnson

A range of projects are available on the topic of statistical spatial and temporal modelling for crime. These projects will focus on developing novel methods for modelling crime related events in space and time, and applying these to real world datasets, mostly within the UK, but with the possibility to use international datasets. Some examples of recent applications include spatio-temporal modelling of drug overdoses and related crime. These projects will aim to use statistical spatio-temporalpoint processes methods, Bayesian methods, and machine learning methods. The project will require the candidate to be proficient in a modern programming language (e.g., R or Python). Applicants should have achieved a first-class degree in Statistics or Mathematics, with a significant component of Statistics, and be proficient in a statistical programming language (e.g., R, Python, Stata, S). We strongly recommend that you contact the supervisor(s) for this project before you apply. Please send your CV and a brief cover letter to [email protected] before you apply. At Manchester we offer a range of scholarships, studentships and awards at university, faculty and department level, to support both UK and overseas postgraduate researchers. For more information, visit our funding page or search our funding database for specific scholarships, studentships and awards you may be eligible for.

Applied Mathematics

Faculty and students interested in the applications of mathematics are an integral part of the department of mathematics; there is no formal separation between pure and applied mathematics, and the department takes pride in the many ways in which they enrich each other. we also benefit tremendously from close collaborations with faculty and students in other departments at uc berkeley as well as scientists at  lawrence berkeley national laboratory  and visitors to the  mathematical sciences research institute.

The Department regularly offers courses in ordinary and partial differential equations and their numerical solution, discrete applied mathematics, the methods of mathematical physics, mathematical biology, the mathematical aspects of fluid and solid mechanics, approximation theory, scientific computing, numerical linear algebra, and mathematical aspects of computer science. Courses in probability theory, stochastic processes, data analysis and bioinformatics are offered by the Department of Statistics, while courses in combinatorial and convex optimization are offered by the Department of Industrial Engineering and Operations Research. Our students are encouraged to take courses of mathematical interest in these and many other departments. Topics explored intensively by our faculty and students in recent years include scientific computation and the mathematical aspects of quantum theory, computational genomics, image processing and medical imaging, inverse problems, combinatorial optimization, control, robotics, shape optimization, turbulence, hurricanes, microchip failure, MEMS, biodemography, population genetics, phylogenetics, and computational approaches to historical linguistics. Within the department we also have a  Laboratory for Mathematical and Computational Biology .

Chair:   Per-Olof Persson

Applied Mathematics Faculty, Courses, Dissertations

Senate faculty, graduate students, visiting faculty, meet our faculty, mina aganagic, david aldous, robert m. anderson, sunčica čanić, jennifer chayes, alexandre j. chorin, paul concus, james w. demmel, l. craig evans, steven n. evans, f. alberto grünbaum, venkatesan guruswami, ole h. hald, william m. kahan, richard karp, michael j. klass, hendrik w. lenstra, jr., lin lin (林霖), michael j. lindsey, c. keith miller, john c. neu, beresford n. parlett.

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Possible MPhil/PhD Projects in Applied Mathematics

Suggested projects for postgraduate research in applied maths.

• Applied Maths Postgraduate Research Projects
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Areas of expertise

Applied mathematics research in the School of Mathematics, Statistics and Physics is concentrated in the following areas:

• cosmology and quantum gravity
• astrophysical and geophysical MHD
• quantum matter
• mathematical biology and archaeology

If you're applying for a MPhil/PhD project in one of these areas, please provide the titles of up to three projects from the list below, in order of preference. Applicants are invited to apply online .

For further information, please contact the PG tutor/selector in applied mathematics:  Dr Gerasimos Rigopoulos

Cosmology and quantum gravity

Neutron stars are extremely dense cinders remaining after stellar explosions. They often have strong magnetic fields and rotate rapidly, and this combination often results in their appearing to pulsate with extreme regularity. We call these objects "pulsars", and their measurable rotation provides an opportunity to take precision measurements in some of the most extreme astrophysical environments accessible to observation.

This project will use existing observations and request and carry out new observations of pulsar systems. These observations will strongly constrain theoretical models of how matter falls onto neutron stars, and in fact probe the details of how gravity works - does it behave as Einstein predicted?

An understanding of the basics of astronomical observation and data analysis is required, as is an interest in understanding models of these phenomena and how to test them.

Supervisor: Dr Anne Archibald

The prediction by Stephen Hawking that black holes radiate particles from the quantum vacuum has had a profound impact on the development of quantum gravity. One aspect of this has been the realisation that quantum gravity theories have non-gravitational equivalents in one less dimension - the holographic principle. This PhD project is concerned with theoretical modelling of one such holographic dual in two dimensions-called the SYK model-as realised in graphene. The project will attack the theory from two sides-numerical modelling electron wave functions on graphene wafers and semi-analytic modelling using the Dirac equation from elementary particle theory. The work will support a possible experiment.

Supervisor:  Professor Ian Moss

The 21st century has transformed cosmology from a speculative field into a precision science driven by theoretical methods, numerical simulations and galaxy survey data. Measurements of galaxy clustering and weak gravitational lensing will map the 3d matter distribution over the past 10 billion years and answer fundamental questions in physics: What are the properties of the early universe? What is the nature of dark energy? What are the characteristics of dark matter?

  A slice through the Euclid Flagship mock catalog of 2.6 billion galaxies displaying the growth of structure over 10 billion years from early (green/right) to late times (red/left).

Unravelling these mysteries is difficult because the information is hidden in the galaxy distribution that has been shaped by nonlinear clustering and is characterised by complex non-Gaussian statistics. As a PhD student working on this project, you will develop and apply state-of-the art statistical, analytical and computational techniques to extract the maximal information on fundamental physics from the late-time matter distribution. You will be part of a local research team in Newcastle and have the option to join the Euclid Consortium to contribute to the ESA space mission Euclid which will launch a dedicated satellite in 2022 to map the dark universe across one third of the sky.

Supervisor:  Dr Cora Uhlemann

The cosmic large-scale structure is the skeleton of matter on the largest scales in the Universe. Galaxies trace this large-scale skeleton of dark matter and form in large gravitationally bound dark matter structures. With major upcoming galaxy surveys like Euclid and LSST, we will be able to track the growth of structure through time across large volumes. This will provide a cosmic laboratory for testing cosmology, fundamental physics and astrophysics with the large-scale structure. To extract the maximum amount of information from galaxy surveys, we need a) accurate models for the gravitational dynamics of the dominant dark matter component, and b) powerful statistics that capture key aspects of gravitational clustering.

This PhD project will tackle these two intertwined challenges. First, we will use novel techniques to describe gravitational dark matter dynamics, for example using the quantum-classical correspondence. The goal is to develop new analytical and computational tools to solve for the time-evolution of dark matter and hunt for signatures of particular dark matter candidates. Second, we will develop clustering statistics that capture non-Gaussian properties of the late-time matter distribution. The idea is to use a sweet spot of simple statistics that are easy to measure, and can be accurately predicted into the nonlinear regime. With this, we will seek to improve the standard analysis relying on two-point statistics to obtain unique insights into cosmology, fundamental physics and astrophysics.

Astrophysical and geophysical fluids

Key Research Questions: Anthropogenic effects are significantly compromising the oceans’ ability to function, the most notable example being the rise in plastic waste dumped into the ocean. We know that the amount of plastic entering our ocean is several orders of magnitude larger than the estimates of floating plastic on the surface of the ocean. Where is this missing plastic? Through the deterministic modelling of microplastic transport in the ocean we will establish realistic models of plastic transport in the oceans.

Project Description: Plastic waste entering the oceans is a mounting global crisis, which has adverse effects ocean ecology, and existential health and economic impacts for humanity. It is essential to understand the full impact of plastic pollution in the ocean and implement effective plans for its removal, therefore, gaining a better insight into how plastic waste is distributed throughout the ocean is of great importance. Studies on plastic particle distributions have highlighted a significant lack of smaller particles (< 5 mm) at the ocean’s surface and the fate of these smaller particles below the surface remains largely unknown. The process of “biofouling”, where the particles density changes due to the growth of organisms on the surface, has been proposed as one explanation. A one dimensional model has shown that such a process can generate vertical oscillations of plastic particulates within the vertical water column, however, this model is limited to a stagnant ocean. In reality, turbulent motions exist throughout the ocean, the intensity of which are strongest in the wind and buoyancy driven upper mixed layer, the pycnocline (where nonlinear internal waves break) and near topography. Upper ocean turbulence is expected to impact on biofouled microplastic trajectories. In this project we propose the development of a more realistic biofouling model in two- and three-dimensional fluid flow to investigate plastic particle distribution within the ocean. Mathematical and computational models will be devised to study the distributions of particles with time dependent density properties in flow fields of increasing complexity.

Prerequisites: Applicants should have strong mathematical and/or computational skills, and an enthusiasm to tackle this environmental challenge.

Supervisor:  Professor Andrew Willmott

Spectacular cyclones have recently been discovered in the polar regions of Jupiter’s atmosphere by the NASA Juno spacecraft. These cyclones are huge (about 2000km wide), persistent and are grouped in a cluster of five to eight circulating around the poles. In contrast, at lower latitudes, the dynamics of the atmosphere is dominated by a well-known banded structure, associated with strong eastward and westward alternating wind jets. In this project, we will study the origin of the polar cyclones and the preference for cyclones rather than jets at high latitudes.

The dynamics of the gas in Jupiter’s deep atmosphere is driven by convection and is strongly influenced by the rotation of the planet. We will model this system using numerical simulations, which will be performed with existing codes.

No prior knowledge of planetary physics is required, but a good understanding of fluid dynamics is essential for this project.

Supervisors:  Dr Céline Guervilly and  Dr Paul Bushby

Motions of liquid iron in the Earth’s outer core are driven by thermal and compositional convection due to the cooling of the planet. An on-going debate in the geophysics community is whether the whole outer core is convecting or whether a region near the core-mantle boundary is stably stratified. In this project, we propose to study:

• the mechanisms by which a stratified layer could form at the top of Earth’s core
• the nature of the motions in this layer
• the layer thickness

The project is based on numerical simulations, which will be performed starting from existing numerical codes. A good understanding of fluid dynamics is essential. No prior knowledge of geophysics or computational modelling is required — the necessary training in these areas will be provided during the early stages of the project.

Supervisors:  Dr Celine Guervilly  and  Dr Graeme Sarson

We describe Galaxies as islands in the Universe, each containing billions of stars.

Interstellar gas fills space between the stars. This gas is a complex hydrodynamical system involved in intense turbulent motions. It exhibits an exceedingly wide range of temperatures (from a few degrees on the Kelvin scale to several million degrees) and densities. It is permeated by magnetic fields and mixed with relativistic particles: cosmic rays.

The interstellar medium is especially rich in complexity in spiral galaxies, whose notable feature is rapid rotation. The interstellar medium feeds the formation of new stars. This largely controls the optical appearance of the host galaxy, and maintains magnetic fields. Despite the violently turbulent nature of the interstellar gas, these magnetic fields exhibit order at very large scales comparable to the size of the parent galaxy.

This project will focus on the origin of the large-scale magnetic fields in spiral galaxies. Recent developments in hydromagnetic dynamo theory have opened an opportunity to construct models that can be directly compared to astronomical observations, refined and perfected. Such a theoretical development is urgently required to plan and interpret observations with a new generation of powerful radio telescopes.

The work on the project will involve:

development of the theory of galactic magnetic fields, based on numerical and analytical studies of the dynamo equations

collection of relevant astronomical data, as required to adapt theoretical models to specific galaxies

comparison of the theoretical predictions with radio astronomical observations

The work on the project will involve regular international contacts with both theoreticians and observational astronomers. It will involve a modest amount of numerical calculations (eg with Matlab or Fortran).

Supervisor: Professor Anvar Shukurov

The evolution of stars and their ultimate demise is affected by hydrodynamic processes occurring within their interiors throughout their lifetime.

Dynamical processes such as convection, rotation, waves and magnetism all greatly impact how these stars explode, chemically enrich the galactic environment and the properties of the stellar remnant.

This project will involve using multi-dimensional hydrodynamic processes to understand these dynamical processes and how they contribute to stellar evolution. Using this understanding, combined with observational constraints, we will develop one-dimensional prescriptions for use in standard stellar evolution models.

Supervisor:  Professor Tamara Rogers

Neutron stars are extremely dense and rapidly rotating objects. They have the strongest magnetic fields in the Universe. Regular stars are powered by nuclear reactions. Neutron stars are powered by their vast reservoirs of rotational and magnetic energy.

A neutron star has a solid outer crust surrounding a superfluid core. Within this core the rotation and magnetic field are "quantised" into thin filaments called vortices and fluxtubes.

This project will develop a model for the dynamics of the vortices in the star's core, and their interaction with the strong magnetic field. We will base our model on suitably modified fluid equations that take account of the superfluid nature of the core.

Basic knowledge of fluid dynamics is required, as well as interest in developing computational skills.

Supervisors:  Dr Toby Wood and  Professor Carlo Barenghi

The surface of the Sun is characterised by a broad range of complex magnetic structures. The most prominent of these are sunspots, which form within (so-called) active regions. The distribution of active regions waxes and wanes, following a cyclic pattern with a period of approximately 22 years. It is believed that such regions are the surface manifestations of an underlying large-scale magnetic field that is buried deep within the solar interior, probably localised around the solar tachocline, where the magnetic field is subject to strong shearing motions. 

To understand the formation of active regions, we need to understand the evolution of the magnetic field in the solar tachocline. In particular, we need to understand the various competing magnetohydrodynamic instabilities that may be playing a role in this evolution. It is generally accepted that magnetic buoyancy plays a crucial role in this regard; it has been suggested more recently that the magnetorotational instability (which is driven by shearing motions) could also be an important factor, particularly at high latitudes.

This project will combine analytical theory with high-resolution numerical simulations, to determine how these instabilities shape the evolution of the solar interior.

We will assume no prior knowledge of solar physics, but a good understanding of fluid dynamics is essential.

Supervisors:  Dr Paul Bushby and  Dr Toby Wood

Quantum matter

Superfluid helium is an intimate mixture of two compenetrating fluids: the normal fluid , akin to an ordinary (classical) viscous fluid, and the superfluid , an inviscid fluid characterised by the presence of interacting vortex filaments. The dynamical evolution of superfluid helium is governed by the interplay of these two liquid phases which, as regularly observed in experiments, is capable of generating a doubly turbulent flow : turbulence indeed characterises the flow of both fluids, making the system richer with respect to ordinary turbulence in classical fluids. Besides an intrinsic and stimulating scientific interest per se , research on superfluid helium flows is also motivated by the use of helium in refrigerating superconductors employed in particle accelerators and nuclear fusion facilities. 

This project focuses on the theoretical and numerical modelling of superfluid helium flows. In particular, as a PhD student working in this project, you will develop analytical and computational techniques to model ongoing experiments whose recent results require further interpretation. You will benefit from a novel, cutting-edge algorithm developed in the School of Mathematics, Statistics and Physics in collaboration with Dr. Krstulovic in Nice, which has already proven to be successful in reproducing past experimental measurements. Building on existing numerical algorithms, you will develop advanced computational tools capable of taking advantage of the high performance of Graphics Processing Units (GPUs). Given the direct connection to ongoing experiments, as a PhD student you will benefit from existing collaborations with leading theoretical and experimental groups in the UK and beyond.

Supervisors: Dr Luca Galantucci and  Professor Carlo Barenghi

We know well what happens when two classical systems interact: they can mix (eg milk and water), or phase-separate (eg oil and water). What happens then when two quantum fluids overlap? This depends crucially on their interaction strength, with the quantum nature of the many-body system setting new rules for their coupling – critically also depending on whether the atomic system is composed of bosonic, or fermionic, particles.

Motivated by experiments with a plethora of different mixtures of ultracold quantum gases, at temperatures below micro-Kelvin, the aim of this project is to study the static and dynamic properties of such multi-component systems.

Questions to be studied include:

• How do such quantum mixtures emerge from their classical systems across the phase transition?
• What difference does the bosonic, or fermionic, nature of the individual components play, and how does a double superfluid (ie a fluid in which both bosonic and fermionic components of a Bose-Fermi mixture are superfluid) differ from other mixtures?
• In particular, how does rotation influence the dynamics of quantum mixtures? (a question of indirect relevance to the cores of neutron stars)
• How does the presence of external (electromagnetic) coupling between different components influence the system’s properties?

Such questions will be addressed in close collaboration with European experimental groups, where such experiments are underway.

Supervisors:  Professor Nick Proukakis  and  Professor Carlo Barenghi

Landmark experiments with atomic quantum gases since 2017 have demonstrated a new form of quantum matter - a quantum droplet. These droplets have several unusual properties:

• they are superfluid - meaning that they are free from viscosity and can support eternal flow
• they are self-supporting - like stars under their own gravity
• they also have such high particle density that quantum mechanical fluctuations and correlations, normally negligible in the gas phase, become significant

These unique features bring the droplets to the fore for studying exotic physics, such as laboratory analogs of neutron stars and highly-correlated quantum systems, and developing new technologies, such as ultra-precise sensors.

This timely project will develop computational and/or analytical models of the droplets to explore these state-of-the-art opportunities.

Supervisors:  Dr Nick Parker  and  Dr Tom Billam

Understanding the behaviour of matter often requires the use of stochastic methods. We add random noise to numerical equations in a controlled way. This mimics the physical response of a system.

This arises across all aspects of modelling, from biological, chemical to physical systems. An obvious example is the random jitter of particles. Here, a random displacing noise is added to the otherwise stationary particle evolution. In the physical setting, the noise usually arises from the interaction of the object with a so-called “heat bath”. The object can exchange energy and particle number with the "heat bath".

Major advances in the last decades have led to a system of appropriate equations to model multi-particle quantum systems confined in appropriate geometries. These have been linked to many recent Nobel Prizes. Beyond a curiosity, such systems are accessible in the lab and promise to revolutionise our future quantum technologies.

The aims of this project are to:

• become familiar with the mathematical background, physical origin, and numerical implementation of such stochastic approaches
• use this knowledge to model cutting-edge experiments in at least two different physical systems which exhibit quantum effects on a macroscopic scale

Supervisor:  Professor Nick Proukakis

Mathematicians, physicists and engineers have studied turbulence for more than a century. Almost all investigations into this fundamental problem of the natural sciences have concentrated the attention on two aspects:

• the geometry
• the dynamics of turbulence

Little attention has been paid, in comparison, to a third equally important aspect: the topology.

This is despite the fact that 19th-century pioneers of fluid dynamics such as Kelvin and Helmholtz were already aware of the possibility of vortices becoming twisted, linked and knotted. Unfortunately, until recently, the only vortex structures which could be created in the laboratory were either very simple (such as vortex rings) or utterly complex (such as turbulence): the 'hydrogen atom' of topological complexity was missing. This situation suddenly changed in 2013, when Kleckner and Irvine, at the University of Chicago, showed that it is possible to create trefoil vortex knots under controlled and reproducible laboratory conditions. This breakthrough is now driving a great interest in the study of the topology of vortices and turbulence.

The project aims to:

• perform a numerical investigation of turbulent flows by solving the governing Euler or Navier-Stokes equations
• look for evidence of knotted structures

The objectives are to:

• define and quantify the topological complexity of turbulence
• to explore the possibility of scaling laws

You should have an interest in fluid dynamics and methods of computational mathematics. You should be willing to learn tools from other relevant disciplines such as knot theory.

Supervisors:  Professor Carlo Barenghi  and  Dr Andrew Baggaley

The progress of cosmology is limited as it cannot be studied in the laboratory. This remark motivates attempts to find physical systems which model some of the fundamental properties of the universe but are also experimentally accessible. One of such systems is the formation of topological defects at phase transitions. Below the temperature of approximately a milliKelvin, superfluid helium 3 exhibits a phase transition to a superfluid state characterised by the breaking of various symmetries that are good analogues to those broken after the Big Bang.

The experimental set up is the following. Superfluid helium 3 is locally heated by neutrons, creating a hot spot of normal liquid which expands and quickly cools back down to the superfluid state. Coherent superfluid regions of the liquid grow simultaneously, and, when they come in contact, the mismatch of the order parameter creates linear topological defects called vortex lines. It is thought that this process, called the Kibble-Zurek mechanism, in the context of cosmology, may be responsible for the formation of inhomogeneous large-scale structures, such as super clusters of galaxies. In liquid helium, the Kibble-Zurek mechanism has been observed experimentally at Helsinki and Grenoble.

To make a better connection with theory, it is necessary to understand how the small region of vortex lines, a turbulent spot, diffuses in space and time: this is what we plan to do numerically. In particular, we want to find how strongly nonlinear processes such as vortex reconnections, which occur when two vortex lines collide with each other, affect the diffusion. Preliminary numerical experiments suggest that vortex rings evaporate away from the turbulent spot.

Supervisor:  Professor Carlo Barenghi

Experimental advances over the last decade are beginning to usher in an age of quantum devices, such as sensors and interferometers. They have the potential to surpass their classical predecessors in terms of:

• sensitivity
• reliability
• miniaturization

One paradigm for constructing such devices involves using ultracold atoms as the quantum element. This has led to the creation of "atomtronic" analogs of electronic (and optical) devices in which the electrons (or light) are replaced with a superfluid current of ultracold atoms. This atomic superfluid can be made to flow without viscosity through carefully shaped channels in a similar way to electricity flowing through circuits, and light travelling through photonic media such as optical fibres or nonlinear crystals. Owing to many-body quantum effects inherent in the atomic superfluid, atomtronics has possible applications in making ultra-sensitive, quantum-enhanced sensors and interferometers.

This project will develop computational and analytical models of novel atomtronic quantum interference devices. In particular, we will connect with recent developments in quantum electronics to generate and develop new proposals for atomtronic devices that exhibit quantum-enhanced performance.

Supervisors:  Dr Tom Billam  and Dr Clive Emary

Current experiments with atomic condensates are concerned with the motion of vortices confined in a small geometry (typically discs or spheres). The small size of these systems means that this motion is affected by the boundaries.

The aim of this project is to gain more understanding of the behaviour of vortices and vortex clusters and possibly make connections with experiments. In particular, we shall use the point vortex method of classical Euler fluid dynamics and the nonlinear Gross-Pitaevksii equation to determine vortex trajectories and to study the chaotic properties of these vortex systems.

Mathematical biology and archaeology

Key Research Questions: 1) How do environmental cues (light, gravity, fluid flow) bias the swimming of algae in snow? 2) How does the spatio-temporal distribution of algae change the optical and thermodynamic properties of the snow?

Project Description: Microalgae are photosynthetic microorganisms critical to life on Earth and to global Climate as key players in biogeochemical cycles. They occupy a wide variety of habitats, including snowfields, where they form patches (>100 m 2 ) on or below the snow surface, and are known to be important terrestial carbon sinks [Gray et al. 2020]. However, key questions, such what environmental conditions lead to the formation of microalgal patches in snow, and how these are affected by climate warming, remain unanswered. In particular, several important species of snow algae are known to swim, but the biophysics of their swimming [Haw & Croze 2012] has not been used to understand microalgal movements in snow. In this PhD research project, the biophysics of microalgal migration in snow will be studied through a combination mathematical modelling, laboratory and field experiments. The PhD student will develop an experimental setup to microscopically and macroscopically image the movements of swimming microalgae in a slab of snow (artificial and field-sampled), in collaboration with snow physicist Dr M Sandells of Northumbria University (co-supervisor), algal biologist Dr M Davey of the Scottish Association for Marine Science (collaborating partner) and biotech company Xanthella (non-CASE collaborative partner). The student will measure how migrations, and the resulting optical properties of the snow, are affected by light, gravity and flow, as a function of warming temperatures. The student will also adapt existing agent based models (ABM) of swimming algae to predict the distribution of microalgae in snow, comparing these with experiment. The student will gain valuable skills in biophysical imaging of microbial populations (tracking and Differential Dynamic Microscopy). Together with continuum-modelling skills and a grounding in practical microalgal biology, this will provide a broad skill set and cross-disciplinary training.

Prerequisites: Candidates should hold a first class or 2:1 degree in physics, applied mathematics, engineering or a related subject. Enthusiasm for research, ability to think and work independently, excellent analytical skills, and strong verbal and written communication skills are essential. Experience in modelling, experimentation and knowledge of biology is desirable

Supervisor:  Dr Otti Croze.

Key Research Questions: Climate change increases the risk to our native trees of devastating disease outbreaks caused by alien pests and diseases. Strategic forest planning and management can help build resilience to these outbreaks and contain them when they occur. Here we will address the question: What are effective and economical strategies to build climate-resilient treescapes? This topical and open problem will be tackled through scenario-testing with sophisticated spatio-temporal models of tree disease spread (see image).

Project Description: Native trees are under constant threat from alien pests and diseases, as exemplified by recent outbreaks affecting ash and sweet chestnut trees. Such outbreaks have massive social and economic impacts and motivate the need to sustain their existence through suitable planning and management. Climate change exacerbates this threat by promoting the migration, survival and growth of alien pathogens. The Department for Environmental, Food and Rural Affairs (Defra) has highlighted the importance of modelling in developing robust plans and management policies for minimising the impacts of these threats. The project aims to identify practical strategies to build resilience to tree disease outbreaks in the face of climate change. We will develop spatio-temporal models of disease spread, based upon national tree maps (see image), field data, and projected changes in pathogen transmission over the next decade. This approach will be applied to existing pests/diseases and hypothetical future pathogens and will incorporate effects such as seasonal weather patterns, year-on-year climate trends (following the latest projection scenarios from the IPCC) and regional variability. Importantly, we will use the model to test the effectiveness of a variety of scenarios designed to build resilience against outbreaks of tree disease. The project will take place within the Mathematical Biology and Archaeology Research group, and make use of Newcastle University’s state-of-the-art high-performance computing facilities. It will be performed in collaboration with Defra, giving the student direct experience of working with government policymakers, and will support Defra’s national strategies for mitigating the effects of alien tree pathogens.

Supervisor: Dr Nick Parker

Microbes, such as bacteria and microalgae, inhabit almost every habitat on Earth, from oceans to snow fields. As agents of biochemical transformation, they play critical roles in global biogeochemical cycles. For example, microalgae fix roughly half the planet’s atmospheric carbon, helping climate regulation and coupling to climate change. Microbes are also critical to green biotechnologies, where they can be used to treat waste or produce bioproducts in an environmentally friendly way.

Many microbes swim, and bias their swimming in response to environmental cues, such as light, gravity, chemical gradients and fluid flow. The mathematical study of swimming and its bias at the individual level, and the wonderful patterns arising at the collective level, is a topic of great topical interest in mathematical biology and biophysics. Current models do reasonably well in predicting the patterns that swimming activity and bias swimming cause in biological fluids in the laboratory. However, these predictions are often only qualitative and models have not been adapted to industrial or agricultural conditions outside the lab.

In this project, current models of swimmers will be developed, and if necessary substantially reformulated, so that they can be tested for their usefulness in industrial and agricultural settings. Specifically, interested students will be able to study one of the following research topics:

• Biofluid dynamics of swimming algae in photobioreactors and harvesting systems
• Migration of algae in snow, and coupling to its albedo and thermodynamics
• Response of swimming algae to toxic chemicals produced by other microbes in the ocean
• Degradation of pollutants by swimming soil bacteria
• Movement of soil bacteria near plant roots

All of these projects will involve interaction with our collaborators in biology, physics, engineering and industry. For the photobioreactor project, students with an interest in carrying out experimental work will be able to do experiments with our photobioreactors located in the labs of our collaborator Dr Gary Caldwell in School of Natural and Environmental Sciences.

Supervisors: Dr Otti Croze and  Dr Andrew Baggaley

Large aggregates of living entities, from biological cells to animals, can exhibit rich and complex collective behaviour. This behaviour often arises from:

• simple actions of the individual members
• their interaction with their immediate neighbours and environment

Striking examples of this in nature are bird flocks (most spectacularly the aerial display of huge numbers of starlings at dusk) and fish schools.

Collective behaviour also plays a key role on the microscopic scale of biological cells. In particular, in-vitro stem cells undergo complex dynamics as they evolve to form colonies and tissues. This process underlies future medical applications of stem cells for the controlled regeneration of biological tissue.

This project will develop a model for such emergent collective behaviour. You may look at the macroscopic domain of birds (Dr Baggaley / Dr Gillespie) or the microscopic realm of stem cells (Dr Parker / Prof Shukurov). Comparison to experimental observations (for birds, this will be through live imaging taken as part of the PhD project; for stem cells, this will be obtained through a collaboration with state-of-the-art experiments at the Institute of Genetic Medicine) will help deduce the biological, physical and geometrical processes which govern these dynamics, and can be expected to shed new light on collective behaviour in these systems.

Supervisors:  Dr Andrew Baggaley , Dr Colin Gillespie ,  Dr Nick Parker  and  Professor Anvar Shukurov

Advances in medical imaging enable clinicians to probe the body with remarkable precision. Clinicians can gather extensive images and datasets. Such advances demand sophisticated mathematical techniques to extract clinically-relevant information. Our researchers are working with clinicians to contribute to these challenges. We work in a range of settings and use a range of analytical and computational methods.

One example is our work in monitoring the recovery of the cornea to stem cell therapies (Prof Shukurov). Following major trauma, the natural cellular structure of the cornea is destroyed. Stem cell therapies can lead to a recovery of this structure. Working with clinicians and microscope images of the eye, we are developing advanced methods to quantitatively characterise the cell structure. This means we can assess levels of damage and monitor the recovery process.

Another example is our work in studying medical ultrasound imaging within the body, in collaboration with clinical medical physics (Dr Parker). The refraction of the ultrasound (eg at tissue boundaries) leads to a geometrical distortion of the images which is not accounted for. This mathematical modelling may lead to the development of corrective strategies which may be translated into clinical devices.

Supervisor:  Dr Andrew Baggaley ,  Dr Nick Parker  and  Professor Anvar Shukurov

Population dynamics is a well-established field of applied mathematics. It has a wide range of applications to biological and social systems. It has been especially successful in applications to prehistory where the fundamental features of the evolution of human populations were free from the unmanageable complications of politics, long-distance travel, etc. One of the most fascinating ages in the human prehistory was the Neolithic, the last period of the Stone Age. The defining feature of the Neolithic was the birth of agriculture and food production (as opposed to food-gathering and hunting). This resulted in:

• a more sedentary lifestyle
• the emergence of urbanism
• human societies as we now know them

The Neolithic first appeared in the Near East and China (perhaps apart from other relatively minor sources) about 12-10 thousand years ago. It then spread across Europe and Asia. There are well-developed mathematical models of this process, but they suffer from several shortcomings:

virtually all of them focus on a limited geographical region (eg Western Europe) and neglect any connections, spatial and temporal, with other regions it remains unclear how such environmental factors as topography, climate, soil quality, etc. affected the spread of the agriculturalists and their technologies. This project aims to develop comprehensive mathematical models of the spread (and subsequent development) of the Neolithic in Eurasia. It will allow for the environmental effects, and take account of the multiple centres where agriculture was independently introduced.

Mathematical modelling, mostly based on numerical simulations, will be constrained by the archaeological and other evidence available, which we will interpret using statistical tools. The project will involve close contacts not only with other mathematicians but also with archaeologists. It may include participation in archaeological field trips and excavations, if desired.

Supervisors:  Professor Anvar Shukurov  and  Dr Graeme Sarson

Experimental techniques such as DNA sequencing and genome editing enable elegant studies into the inner workings of ‘simple’ single-cell organisms. Designer bugs that can eat plastics, digest toxic chemicals and produce bio-fuels are being extensively researched. In parallel with such lab based work there is also a growing body of mathematical and computational research directed at furthering our understanding of how micro-organisms organize their world, and how we might use this knowledge to better our own. This PhD project aligns with the mathematical/computational approach, and seeks to develop hi-fidelity models that accurately describe the collective behaviour and emergent properties of colonies of micro-organisms.

Many species of bacteria have evolved the ability to manufacture and secrete a sort of ‘glue’, commonly referred to as extra-polymeric substance (EPS). This substance serves a variety of purposes, not least it enables bacteria to ‘stick together’ and form colonies. These may adhere to surfaces as bio-films, or be suspended in fluids as bio-flocs. The EPS forms a protective matrix in which the cells can grow and divide. It also acts as a medium through which nutrients and cell metabolites can be transported, and by which cells may exchange chemical signals. In building mathematical models for the growth and behaviour of microbial colonies it is therefore important to take into account the role played by this EPS matrix.

A widely used modelling approach is that of agent-based descriptions; the individual cells in a colony are represented as discrete entities (agents) that grow, divide and interact with each other (and the EPS) through imposed biochemical and mechanical rules. Conceptually simple, and allowing detailed interactions to be prescribed relatively easily, this approach is designed for computer simulation. It has proved very effective for simulating the behaviour of colony formation and growth at small spatial scales, but as colony size increases the large number of cells is computationally prohibitive. (A 1mm square patch of biofilm will contain millions of individual cells). At larger scales, therefore, alternative modelling strategies are needed.

The main focus of this project will be on the development and application of population-based continuum models. Continuum models for bio-films, whilst not new, are perhaps less well developed and studied than agent-based models. In particular, the inclusion of microscale mechanical properties of EPS within continuum descriptions is an area where there is considerable modelling work to be done. Established modelling techniques developed in the context of multicomponent and granular continua will be adapted and applied to the type of biological media central to this project. Analysis of resulting models will involve both theoretical and numerical methods, and will require the development of some research codes.

As with all modelling, a key aspect of the work will be calibration and validation. This will draw on recent and on-going experimental and simulation studies: Experiments into the micro- and macro-scale viscoelastic properties of bio-films are in progress within the School of Engineering at Newcastle University. Data from these studies will inform parameter selection within proposed models. In addition, bio-film growth in channel flows is also being investigated experimentally, providing data that can be used to assess model predictions. The project will also have access to a recently developed, and mechanically detailed, agent-based simulation code, offering further reference data against which new continuum type models can be assessed.

The project would suit a mathematics or physics graduate with some background in continuum (fluid and/or solid) mechanics. Numerical work will require the development and use of computer codes, and programming experience/interest is necessary.

Supervisor:  Dr David Swailes

Fluid flows often transport material in the form of small solid particles, liquid droplets or gas bubbles. Sometimes all three at once; sand grains, oil drops and air bubbles in water for example. The particulate material may be present by design (spray atomization is an integral part of many engineering processes), or be unwelcome (micro-plastics in water systems, volcanic emissions etc.). In many of these multi-phase systems the underlying flow is turbulent, and the way in which the dispersed particulates interact with this flow is crucial to the overall transport process. Aerosols, for example, tend to cluster in high-strain/low-vorticity regions, which influences the rate at which these droplets coalesce.

One way to study particle dynamics in turbulent flows is through computer simulations: Based on an underlying particle equation of motion we can simulate the trajectories of many hundreds of thousands of individual particles, and thereby build up a statistical picture of the collective behaviour of the disperse phase, described in terms particle concentrations, average velocities, kinetic energies etc. These will inevitably depend on (and perhaps influence) the statistical properties of the turbulence.

A second approach is to develop models that govern directly how these statistical measures evolve in both space and time. A model that allows us to compute directly the statistical distribution of particles obviates the need to perform time-consuming particle tracking simulations (other than to test that the models are correct!). It is this second approach that forms the basis for the research in this project.

By treating particle equations of motion as stochastic ordinary differential equations (SDEs) we can formulate transport equations for probability density functions (pdfs) that describe the resulting, ensemble-based distributions of particle properties such as position and velocity. The SDEs are non-standard in that they incorporate stochastic processes and fields that are correlated both in time and in space. This feature reflects correlation structures inherent in turbulent flow, and has a profound effect on the form of the resulting pdf transport equations; previous mathematical analysis has identified a number of subtle challenges associated with both theoretical and numerical treatment of these pdf models. In this work, we will consider how some of these issues may be addressed. Extended phase-space models (generalized Langevin equations) have been proposed. These eliminate non-Markovian features in the pdf models, but at the expense of higher-dimensionalities. Moreover, these extended models, as they now stand, are not capable of reproducing some key features of particle-phase transport associated with preferential sampling and drift. We will aim to develop and assess strategies that address these important issues.

The project would suit a mathematics or physics graduate with a background in fluid dynamics modelling and/or stochastic analysis. Numerical work will require the development and use of computer codes, and programming experience/interest is highly desirable.

We live in an era of marked global change from climate change, deforestation and urbanisation. This has major implications for natural ecological systems such as plants and trees, insects, animals and coral reefs.

Drs Baggaley and Parker are working with government to understand the spread of tree disease from invasive species. Topical examples of this are the dieback fungus and borer beetle affected UK ash trees. Our work focusses on understanding the key factors governing the spread of the disease and how the damage might be mitigated.

Meanwhile Dr Emary is studying ecological networks – abstract representations of the interactions between species in an ecosystem. Key questions include the response of such networks to species loss and environmental change. This work is performed in collaboration with field ecologists with applications in agriculture and climate-change mitigation.

Moreover, engineering ecological systems, such as bacterial colonies, may help us meet future challenges in energy provision and waste management. We are developing mathematical models for the bacterial colonies, so as to highlight conditions and strategies to optimise the efficiency of these processes.

Supervisors:  Dr Andrew Baggaley , Dr Clive Emary and  Dr Nick Parker

Find out about the applied maths research group

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Undergraduate Research Projects

Integrated research projects.

The department periodically applies for research grants that involve faculty, graduate students, and undergraduate students. This year the Department was awarded a grant for the MCTP: Colorado Advantage proposal to the National Science Foundation (NSF). For more information on this grant, please see  NSF-MCTP .

Oral Assessments

What are Orals and Why Participate? Mathematics Education research proclaims the importance of mathematical discourse in helping students to master difficult concepts. Teachers often report that they never “really” understood some concepts until they taught them. Perhaps what they are saying is that they really understood the concepts when they were able to explain them to someone else. We find that oral assessments provide students that opportunity to explain concepts to their fellow students. For more information on oral assessments please see  Oral Assessments .

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Applied mathematics articles from across Nature Portfolio

Applied mathematics is the application of mathematical techniques to describe real-world systems and solve technologically relevant problems. This can include the mechanics of a moving body, the statistics governing the atoms in a gas or developing more efficient algorithms for computational analysis. These ideas are closely linked with those of theoretical physics.

Long ties across networks accelerate the spread of social contagions

Long ties that bridge socially separate regions of networks are critical for the spread of contagions, such as innovations or adoptions of new norms. Contrary to previous thinking, long ties have now been found to accelerate social contagions, even for behaviours that involve the social reinforcement of adoption by network neighbours.

Latest Research and Reviews

Evaluation and impact factors of international competitiveness of China’s cobalt industry from the perspective of trade networks

• Zhengfang Zhong

A python based algorithmic approach to optimize sulfonamide drugs via mathematical modeling

• Wakeel Ahmed
• Fekadu Tesgera Agama

Reconstructing rodent brain signals during euthanasia with eigensystem realization algorithm (ERA)

• Khitam Aqel
• Pedro D. Maia

Quantum computing for several AGV scheduling models

Key node identification for a network topology using hierarchical comprehensive importance coefficients

• Fanshuo Qiu

Advanced stability analysis of a fractional delay differential system with stochastic phenomena using spectral collocation method

• Sami Ullah Khan
• Salman A. AlQahtani

News and Comment

The curious case of the test set AUROC

The area under the receiver operating characteristic curve (AUROC) of the test set is used throughout machine learning (ML) for assessing a model’s performance. However, when concordance is not the only ambition, this gives only a partial insight into performance, masking distribution shifts of model outputs and model instability.

• Michael Roberts
• Carola-Bibiane Schönlieb

The role of computational science in digital twins

Digital twins hold immense promise in accelerating scientific discovery, but the publicity currently outweighs the evidence base of success. We summarize key research opportunities in the computational sciences to enable digital twin technologies, as identified by a recent National Academies of Sciences, Engineering, and Medicine consensus study report.

• Karen Willcox
• Brittany Segundo

Distilling data into code

One of the greatest limitations of deep neural networks is the difficulty of interpreting what they learn from the data. Deep distilling addresses this problem by extracting human-comprehensible and executable code from observations.

• Joseph Bakarji

Why even specialists struggle with black hole proofs

Mathematical proofs of black hole physics are becoming too complex even for specialists.

• Alejandro Penuela Diaz

Packing finite numbers of spheres efficiently

A paper in Nature Communications reports experiments and simulations of spherical particles that help show how finite numbers of spheres pack in practice.

• Zoe Budrikis

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Department of Mathematics

• Math Research
• Senior Project

Senior Project Marketplace

Senior Project (MATH 401) is designed to be a capstone experience for Math undergraduates, where the students engage in research activities guided by a faculty member. To enroll in MATH 401, you need to choose a project topic and get in touch with a faculty member. You can come up with your own project idea, or choose one from the list below. The list is by no means exhaustive but will give you an idea of what kind of research each researcher is working on.  These are examples of projects supervised by math department faculty.

Senior project ideas

Joe champion - math education, statistical modeling of math education achievement.

Learn about large-scale educational achievement data and techniques for predicting students’ math achievement. Involves data wrangling, intermediate coding in R or Python (mostly adapting existing code), and a focus on data visualization. Background in mathematics education and/or statistics preferred.

High School Mathematics Curriculum Development

For mathematics education students – modify and create Desmos Teacher activities to align with high school mathematics standards. Focus on data, modeling, and technology-assisted representations.

Middle School Mathematics Curriculum Development

Adapt activities from the Algebra through Visual Patterns curriculum for delivery in the Desmos Teacher Activities platform. Involves some testing with students and collaboration with math education researchers.

Other project ideas

Dig into the history of a K-12 math topic and write a paper / make a poster, create an original math video.

Contact info

Joe Champion Faculty Profile

Samuel Coskey - Set theory, logic, and combinatorics

Combinatorics and graph theory.

Learn something new in one of these areas that wasn’t covered in Math 189/287/387. Use a new book, book chapter, online notes, or published article as a resource. Present the motivation, examples, and results with a poster.

Algebra or analysis or geometry

Learn something new in one of these areas that wasn’t covered in Math 305/311/314/405/414. Use a new book, book chapter, online notes, or published article as a resource. Present the motivation, examples, and results with a poster.

Math education

Choose a topic in college-level mathematics to present at the middle or high school level. Create a detailed lesson plan.

I am open to exploring anything in pure mathematics (and applied mathematics if you can take the lead). The important thing is to find resources at the right level for you.

Contact Info

https://scoskey.org

Jens Harlander - Topology and Algebra (Group Theory)

Topics in graph theory, topics concerning the topology of surfaces, topics in linear algebra over rings.

Jens Harlander Faculty Profile Page

Uwe Kaiser - Geometric and Algebraic Topology, Quantum Computing

Quantum computing algorithms.

This project asks you to have some programming experience. You work on a specific quantum algorithm. You study programming in Microsoft’s quantum computing kit, see https://www.microsoft.com/en-us/quantum/development-kit, try out examples, and study properties of the algorithm. Basic linear algebra skills are necessary in order to understand how algorithms are implemented using circuits. The minimal expectation is a poster to be presented in the senior showcase.

Tangles and Electrical Networks

In 1993 Goldman and Kauffman interpreted the continued fraction of a tangle as a conductance of a corresponding network. This projects asks to search for further relations between electrical networks and invariants of links and tangles. The starting point is a paper of mine on band-operations on links leading to a formula similar to the one by Goldman/Kauffman. Basic linear algebra and some familiarity with discrete mathematics like graph theory are helpful, also knowledge of basic notions concerning electrical networks. The minimal expectation is a poster to be presented in the senior showcase.

Robotics and Topology

Topology and robotics are related through notion of configuration space. For example the configuration space of a robot arm in 3-space is a product of 2-spheres. Restrictions on motions lead to more interesting topology of configuration spaces. The goal is to study the complexity of basic examples through well-known theory developed by Farber. Knowledge of basic topology (for example MATH 411) is necessary for an understanding of the theory. The minimal expectation is a poster to be presented in the senior showcase.

Analysis Situs

In the years 1899 to 1904, Poincare published a paper with the title Analysis Situs and five supplements introducing basic ideas of topology. This project aims to study the way he introduced a particular concept, like e.g. the orientation of a manifold, and to research pre-Poincare origins of this concept (what was a building on?) and how the concept developed into modern times. The prerequisite for this project is the maturity of a senior, the willingness to read old mathematics literature (English translations are available). The expectation is a poster to be presented in the senior showcase.

[email protected]

Michal Kopera - Computational Math, Ocean Modeling

Broncorank - a new university ranking.

In this project, you will explore the idea of using a PageRank algorithm, which Google is using to rank websites in their search engine, for creating a university ranking which does not depend on some editorial board decision but emerges from each university peer institution lists. You will get a chance to work at an intersection of mathematics, programming, data science, and contribute to creating a more fair tool to rank universities across the U.S.

To be successful in this project, you need some background in programming and ideally have enjoyed your MATH 265 and/or 365 courses. Knowledge of basic linear algebra (matrices) is a plus. The minimum expectation is a poster presentation at Senior Showcase.

Ice/ocean interactions

The modeling of the interface between ice and the ocean is of utmost importance for climate science. You will experiment with models of ice/ocean boundary developed in my group and evaluate whether they produce physical results. No ocean science background is required, but you should be comfortable with writing simple code. The minimum expectation is a poster presentation at Senior Showcase.

Computational modeling using ODEs and PDEs

The bulk of my work is using computational methods to simulate phenomena described by ordinary or partial differential equations. I am open to your ideas on what you would like to model, and we can create a project based on your input.

You will likely need to be able to program in MATLAB, Python, Julia, or other languages.  Knowing something about ODEs and/or PDEs is welcome. I am also open to problems that yield themselves to Machine Learning. The minimum expectation is a poster presentation at Senior Showcase.

Game of Life, Fractals and self-similarity

You will explore some of the concepts outlined above and write code to implement them. The minimum expectation is a poster presentation at Senior Showcase.

Mathematical Art

You will work with genetic (or other nature-inspired) algorithms that try to generate art. You can either focus on optimization algorithms that try to reproduce existing images or aim to generate original art and try to measure its esthetics. The minimum expectation is a poster presentation at Senior Showcase.

Michal Kopera Faculty Profile Page

Zach Teitler - Algebra and Algebraic Geometry

My specialty is algebra. I can work with you on projects in algebra, graph theory, combinatorics, number theory, any other area of pure math, or any subject that you’re interested in within pure math, applied math, statistics, or math education.

I am available to work with students on undergraduate senior thesis projects. You can email me if you’re looking for a senior thesis advisor, but first, read about what you can expect if we work together and what project ideas we can work on together.

https://zteitler.github.io/advising/senior_thesis/

Barbara Zubik-Kowal - Applied mathematics

Difference equations and applications.

Difference equations arise naturally in real-world applications involving discrete sets or populations, or as approximations to continuum models in science and engineering. Mathematically, difference equations can be described as mathematical equalities involving the values of a function of a discrete variable. A recurrence relation such as the logistic map, relevant to population dynamics, or the sequence of Fibonacci numbers, are simple examples. Many difference equations can be solved analogously to how one solves ordinary differential equations. However, it is well-known that most difference equations depicting real-life phenomena cannot be solved in closed form and other methods are necessary to obtain qualitative or quantitative information about the desired solutions, including their stability properties. This senior project can go in a number of directions depending upon the interests of the student. The project may involve theoretical aspects, including theoretical derivations and proof-writing, or computations, including writing new codes or modifying existing ones.

Integro-differential equations and applications

Integro-differential equations are central to modelling numerous natural and industrial phenomena across physics, biology, medicine, engineering, and other fields. As an example in the field of epidemiology, integro-differential equations are frequently used in the mathematical modelling of epidemics, such as when the age-structure of the population is important in determining the dynamics of an epidemic. Integro-differential equations involve both integrals and derivatives of a function. As very few systems of integro-differential equations have a closed-form solution, a range of mathematical methods are often used to obtain qualitative information about the solutions of classes of problems involving integro-differential equations, and approximation techniques are often used to obtain quantitative information about the corresponding solutions given some initial data. In contrast to ordinary and partial differential equations, initial data for integro-differential equations is frequently provided on a whole interval, rather than a single initial point in time. This means more initial data is used to supplement systems of integro-differential equations. This senior project can go in a number of directions depending upon the interests of the student. The project may involve theoretical aspects, including theoretical derivations and proof-writing, or computations, including writing new codes or modifying existing ones.

Differential inequalities and applications

Mathematical models for a range of biological, physical or industrial phenomena may be grouped into general classes of systems of differential equations. Even if the underlying mathematical models may involve complexities that make it hard or impossible to solve by hand, it is frequently possible to extract useful qualitative information about its solutions. Such qualitative information frequently suffices to answer key questions about a solution’s behaviour. Examples are its long-term behavior, existence and uniqueness, convergence properties, and its upper and lower bounds, such as maximal and minimal solutions. These properties, in turn, help us derive information about not only one, but a whole family of mathematical models constituting a given class of differential equations. This senior project can go in a number of directions depending upon the interests of the student. The project may involve theoretical aspects, including theoretical derivations and proof-writing, or computations, including writing new codes or modifying existing ones.

Principles of approximation and applications

Smooth functions arise frequently in the mathematical modeling of numerous real-world phenomena in the sciences and engineering, including both natural and industrial processes. An example is the solution to a SIR model of susceptible, infectious, or recovered individuals in epidemiology, or solutions to mathematical models of tumor growth. It is well known, however, that solutions to most mathematical models depicting real-world phenomena cannot, in general, be expressed in closed form. It is, however, possible to make progress by making appropriate approximations to obtain an estimate of the desired solution. Such approximations involve discretizing the domain from a continuous interval to a finite subset of grid points, solving the discrete systems of equations, computing continuous extensions, or interpolations, and performing error analysis. There are many ways of doing this, but it is important to understand how to do it in a way that preserves certain desired properties, in order to ensure that the resulting approximate solutions that we are getting are indeed approximate solutions to the problem we started out with, rather than spurious output. This senior project can go in a number of directions depending upon the interests of the student. The project may involve theoretical aspects, including theoretical derivations and proof-writing, or computations, including writing new codes or modifying existing ones.

Barbara Zubik-Kowal Faculty Profile Page

Research areas

Research in Applied Mathematics has changed dramatically over the past 30 years, with revolutionary developments in traditional areas, together with the emergence of exciting new areas. These changes have been triggered by the development of more powerful computers allowing researchers to address previously intractable problems, and developments in other fields which have led to new mathematical problems.

The department has strong research programs in:   

• Control and Dynamical Systems  (including differential equations)
• Fluid Mechanics
• Mathematical Medicine and Biology
• Mathematical Physics
• Mathematics of Data Science and Machine Learning
• Scientific Computing

Researchers in our department are at the forefront of a number of exciting research areas. Here are some examples:

• Math and Water
• Carbon Nanotubes
• Mathematics and medicine are powerful partners
• Saving the whales with mathematics
• Quantum sounds could reveal the shape of the universe

Using social media to help prevent the spread of disease

The following links give further examples of research conducted in the department:

• Recent PhD Theses in the Applied Mathematics Department
• Recent Master's Theses in the Applied Mathematics Department
• Conference Posters on Research Conducted in the Applied Mathematics Department

Applied Mathematics

Research areas.

Mathematical climate research seeks to develop models of the climate system, including interactions between the atmosphere, ocean, and ecosystems, and methods for interpretation of complex environmental data. This field is inherently interdisciplinary and leverages mathematical research in disciplines such as scientific computing, dynamical systems, and probability in collaboration with environmental scientists. Problems range from fundamental questions about the physics of the climate system to applied questions such as the climate impacts in a given region or the deployment of a particular climate solution. 

• Mara Freilich : network analysis, stochastic models, fluid dynamics, oceanography, ecology

Dynamical Systems and Partial Differential Equations

Research in this area focuses on nonlinear differential equations and dynamical systems that arise in the physical, social, and life sciences.  Among the equations considered are finite-dimensional dynamical systems, reaction-diffusion systems, hyperbolic conservation laws, max-plus operators and differential delay equations.  Questions that are addressed for these systems include the existence and stability of nonlinear waves and patterns, kinetic theory, phase transitions, domain coarsening, and statistical theories of turbulence, to name but a few.  Even though the techniques can vary widely from case to case, a unifying philosophy is the combination of applications and theory that is in the great Brown tradition in this area of mathematics, which is being fostered by close collaboration among the members of the group.

• Hongjie Dong : Linear and nonlinear elliptic and parabolic PDEs, fluid equations
• Yan Guo :  Partial differential equations, kinetic theory and fluids
• Govind Menon : Kinetics of phase transitions and models of domain coarsening, integrable systems, random matrix theory, statistical theories of turbulence
• Bjorn Sandstede : Applied dynamical systems, nonlinear waves and patterns, data science, computational biology

Dynamical Systems and Partial Differential Equations

New England Dynamics Seminars

Pattern Theory, Statistics, and Computational Molecular Biology

Research in pattern theory seeks to develop models of complex systems and statistical methods and algorithms for the interpretation of high-dimensional data.  Pattern theory research is also typically interdisciplinary; it includes collaborations with computer scientists, engineers, cognitive and neural scientists, and molecular biologists.  Most of pattern theory research relies on tools from mathematical analysis, probability theory, applied and theoretical statistics, and stochastic processes.  Recent applications include the development of models for computation and representation in primate visual pathways, as well as the development of statistical methods for Bayesian non-parametrics, network analysis, the interpretation of multi-electrode neurophysiological recordings, image processing and image analysis, and the analyses of genome-wide expression data and cellular regulatory pathways.

• Elie Bienenstock : theoretical neuroscience, computational vision, computational linguistics
• Stuart Geman : compositionality, statistical analysis of neurophysiological data, neural representation and neural modeling, statistical analysis of natural images, timing and rare events in the markets
• Basilis Gidas : Bayesian statistics, computer vision, speech recognition, computational molecular biology
• Matthew Harrison : statistics, machine learning, applications in neuroscience, neural engineering, ecology

Pattern Theory

Probability and Stochastic Processes

The Division has long been a leader in stochastic systems theory and its applications, as well as at the forefront of current developments in probability theory, random processes and related computational methods.  Research in probability theory and stochastic processes include stochastic partial differential equations, nonlinear filtering, measure-valued processes, deterministic and stochastic control theory, probabilistic approaches to partial differential equations, stability and the qualitative theory of stochastic dynamical systems, the theory of large deviations.  Our research endeavors also include Monte Carlo simulation, Gibbs measures and phase transitions, as well as stochastic networks.  There also exists a major program in numerical methods for a variety of stochastic dynamical systems, including Markov chain approximations and spectral methods. 

• Hongjie Dong : stochastic processes, stochastic control theory, probabilistic approaches of PDEs
• Paul Dupuis : applied probability, control theory, large deviation, numerical methods, Monte Carlo
• Oanh Nguyen : probability theory, stochastic processes, stochastic networks, phase transition, random polynomials
• Kavita Ramanan : probability theory and stochastic processes, reflected diffusions, Gibbs measures and phase transitions, large deviations, measure-valued processes, stochastic networks
• Hui Wang : stochastic optimization, large deviations, stochastic networks, fast simulation

Scientific Computation and Numerical Analysis

This research area is inherently multidisciplinary. It has undergone phenomenal growth in response to the successes of modern computational methods in increasing the understanding of fundamental problems in science and engineering. The Division’s program in scientific computation and numerical analysis has kept pace with these developments and relates to most of the other research activities in the Division. Special emphasis has been given to newly developed, high-order techniques for the solution of the linear and nonlinear partial differential equations that arise in control theory and fluid dynamics. Numerical methods for the discontinuous problems that arise in shock wave propagation and for stochastic PDEs and uncertainty modeling are being studied. Emphasis is also being placed on the solution of large-scale linear systems and on the use of parallel processors in linear and nonlinear problems.

• Mark Ainsworth : Finite element methods: adaptivity, a-posteriori error control, Implementation using Bernstein- Bézier techniques, dissipative and dispersive properties.
• Jerome Darbon : Efficient algorithms for variational/Bayesian estimations and connections with Hamilton-Jacobi PDEs, combinatorial optimization (especially network flows and graph-based algorithms), stochastic sampling algorithms (especially perfect samplers), algorithm/architecture co-design including low level implementation, applications to denoising, geomorphology, remote sensing, biological, medical, historical, radar and inverse problems in imaging sciences
• Johnny Guzmán : Discontinuous Galerkin methods, Mixed methods/compatible discretizations, local error analysis 
• George Em Karniadakis:  Stochastic PDEs and stochastic multi-scale modeling, fractional PDEs, spectral element methods, parallel computing
• Brendan Keith : PDE-Constrained optimization, stochastic and fractional PDEs, scientific machine learning, uncertainty quantification, finite element analysis
• Chi-Wang Shu : High order methods for hyperbolic and convection dominated PDEs, computational fluid dynamics

Research Groups

Crunch group.

Research conducted by the CRUNCH Group focuses on the development of stochastic multiscale methods for physical and biological applications, specifically numerical algorithms, visualization methods and parallel software for continuum and atomistic simulations in biophysics, fluid and solid mechanics, biomedical modeling and related applications. The main approach to numerical discretization is based on spectral/hp element methods, on multi-element polynomial chaos, and on stochastic molecular dynamics (DPD). The group is directed by Prof. George Em Karniadakis . We invite you to visit both our  DPD Club  and  Crunch FPDE Club  websites. 

Visit Crunch Group

Visit Fractional Partial Differential Equations ARO MURI Projects

Statistical and Molecular Biology Group

The Statistical Molecular Biology Group at Brown University is led by Chip Lawrence, Professor Emeritus of Applied Mathematics.  The group's research energies are focused on statistical inference in molecular biology, genomics, and paleo-climatology, most specifically on several different high-dimensional (High-D) discrete inference problems in sequence data.

Visit Statistical and Molecular Biology Group

URA Research Project Ideas

What follows is a list of some of the project topics that faculty members in the department of mathematics have suggested as suitable for undergraduate research projects. Students who wish to participate can register and receive credit for an independent study or may be able to obtain URA funding to get paid to work on these projects.

Details of the project requirements will be worked out between the faculty supervisor and the student. Some of these projects require little background and are suitable for freshmen or sophomores, while others require knowledge of linear algebra, ordinary differential equations, or group theory. This list is by no means exclusive: any student with a particular interest in some area of research is encouraged to seek out a faculty supervisor. Students are encouraged to contact the URA Program Coordinator for help finding a suitable faculty research mentor.

Looking for examples of undergraduate research? The Honors College maintains a repository of past honors thesis submissions ; use the Advanced Filters to search by discipline (Mathematics or Statistics & Data Science). It may also be helpful to look at past MathFest Student Papers , as well as SUnMaRC abstracts .

Students participating in undergraduate research for credit must submit a proposal form through the math academic office . Stop by the window at Math 108 once you have lined up your project advisor and topic.

Project ideas list is not exhaustive - there are additional faculty who are interested in working with undergraduates that have not provided information to us.

If a project has not been updated in a long time, check the professor's homepage to see what they have been working on most recently. Research areas tend not to change drastically.

*Honors Thesis MATH 498H credit available to students in the Honors College .

**Restrictions may apply. Ask the individual faculty member for details.

• Senior Thesis

A thesis is a more ambitious undertaking than a project. Most thesis writers within Applied Mathematics spend two semesters on their thesis work, beginning in the fall of senior year.  Students typically enroll in Applied Mathematics 91r or 99r (or Economics 985, if appropriate) during each semester of their senior year.  AM 99r is graded on a satisfactory/unsatisfactory basis.  Some concentrators will have completed their programs of study before beginning a thesis; in situations where this is necessary, students may take AM 91r for letter-graded credit, for inclusion in Breadth section (v) of the plan of study.  In the spring semester, the thesis itself may serve as the substantial paper on which the letter grade is based.  Econ 985 is also letter-graded, and may be included in the Breadth section of the plan of study in place of AM 91r.

Another, somewhat uncommon option, is that a project that meets the honors modeling requirement (either through Applied Mathematics 115 or 91r) can be extended to a thesis with about one semester of work.  Obviously the more time that is spent on the thesis, the more substantial the outcome, but students are encouraged to write a thesis in whatever time they have. It is an invaluable academic experience.

The thesis should make substantive use of mathematical, statistical or computational modeling,  though the level of sophistication will vary as appropriate to the particular problem context.  It is expected that conscientious attention will be paid to the explanatory power of mathematical modeling of the phenomena under study, going beyond data analysis to work to elucidate questions of mechanism and causation rather than mere correlation. Models should be designed to yield both understanding and testable predictions. A thesis with a suitable modeling component will automatically satisfy the English honors modeling requirement; however a thesis won't satisfy modeling Breadth section (v) unless the student also takes AM 91r or Econ 985.

Economics 985 thesis seminars are reserved for students who are writing on an economics topic. These seminars are full courses for letter-graded credit which involve additional activities beyond preparation of a thesis. They are open to Applied Mathematics concentrators with suitable background and interests.

Students wishing to enroll in AM 99r or 91r should follow the application instructions on my.harvard.

Thesis Timeline

The timeline below is for students graduating in May. The thesis deadline for May 2024 graduates is Monday, April 1 at 2:00PM. For off-cycle students, a similar timeline applies, offset by one semester. The thesis due date for March 2025 graduates is Friday, November 22, 2024. Late theses are not accepted.

Mid to late August:

Students often find a thesis supervisor by this time, and work with their supervisor to identify a thesis problem. Students may enroll in Econ 985 (strongly recommended when relevant), AM 91r, or AM 99r to block out space in their schedule for the thesis.

Early December:

All fourth year concentrators are contacted by the Office of Academic Programs. Those planning to submit a senior thesis are requested to supply certain information. This is the first formal interaction with the concentration about the thesis.

Mid-January:

A tentative thesis title approved by the thesis supervisor is required by the concentration.

Early February:

The student should provide the name and contact information for a recommended second reader, together with assurance that this individual has agreed to serve. Thesis readers are expected to be teaching faculty members of the Faculty of Arts and Sciences or SEAS. Exceptions to this requirement must be first approved by the Directors, Associate Director, or Assistant Director of Undergraduate Studies. For AM/Economics students writing a thesis on a mathematical economics topic for the March thesis deadline, the second reader will be chosen by the Economics Department. For AM/Economics students writing for the November deadline, the student should recommend the second reader.

On the thesis due date:

Thesis due at 2pm. Late theses are not accepted. Electronic copies in PDF format should be delivered by the student to the two readers and to [email protected] (which will forward to the Directors of Undergraduate Studies, Associate and Assistant Director of Undergraduate Studies) on or before that date and time. An electronic copy should also be submitted via the SEAS  online submission tool  on or before that date. SEAS will keep this electronic copy as a non-circulating backup and will use it to print a physical copy of the thesis to be deposited in the Harvard University Archives. During this online submission process, the student will also have the option to make the electronic copy publicly available via DASH, Harvard’s open-access repository for scholarly work.

Contemporaneously, the two readers will receive a rating sheet to be returned to the Office of Academic Programs before the beginning of the Reading Period, together with their copy of the thesis and any remarks to be transmitted to the student.

The Office of Academic Programs will send readers' comments to the student in late May, after the degree meeting to decide honors recommendations.

Thesis Readers

The thesis is evaluated by two readers, whose roles are further delineated below.  The first reader is the thesis adviser.  The second and reader is recommended by the student and adviser, who should secure the agreement of the individual concerned to serve in this capacity.  The reader must be approved by the Directors, Associate Director, or Assistant Director of Undergraduate Studies.  The second reader is normally are teaching members of the Faculty of Arts and Sciences, but other faculty members or comparable professionals will usually be approved, after being apprised of the responsibilities they are assuming.   For theses in mathematical economics, the choice of the second reader is made in cooperation with the Economics department.  The student and thesis adviser will be notified of the designated second reader by mid-March.

The roles of the thesis adviser and of the outside reader are somewhat different.  Ideally, the adviser is a collaborator and the outside reader is an informed critics.  It is customary for the adviser's report to comment not only on the document itself but also on the background and context of the entire effort, elucidating the overall accomplishments of the student.  The supervisor may choose to comment on a draft of the thesis before the final document is submitted, time permitting.  The outside reader is being asked to evaluate the thesis actually produced, as a prospective scientific contribution — both as to content and presentation.  The reader may choose to discuss their evaluation with the student, after the fact, should that prove to be mutually convenient.

The thesis should contain an informative abstract separate from the body of the thesis.  At the degree meeting, the Committee on Undergraduate Studies in Applied Mathematics will review the thesis, the reports from the two readers and the student’s academic record. The readers (and student) are told to assume that the Committee consists of technical professionals who are not necessarily conversant with the subject matter of the thesis so their reports should reflect this audience.

The length of the thesis should be as long as it needs to be to make the arguments made, but no longer!

Thesis Examples

The most recent thesis examples across all of SEAS can be found on the Harvard DASH (Digital Access to Scholarship at Harvard) repository . Search the FAS Theses and Dissertations collection for "applied mathematics" to find dozens of examples.

Note: Additional samples of old theses can be found in McKay Library. Theses awarded Hoopes' Prizes can be found in Lamont Library.

Recent thesis titles

Theses submitted in 2021, theses submitted in 2020, theses submitted in 2019, theses submitted in 2018 , senior thesis submission information for a.b. programs.

Senior A.B. theses are submitted to SEAS and made accessible via the Harvard University Archives and optionally via  DASH  (Digital Access to Scholarship at Harvard), Harvard's open-access repository for scholarly work.

In addition to submitting to the department and thesis advisors & readers, each SEAS senior thesis writer will use an online submission system to submit an electronic copy of their senior thesis to SEAS; this electronic copy will be kept at SEAS as a non-circulating backup. Please note that the thesis won't be published until close to or after the degree date. During this submission process, the student will also have the option to make the electronic copy publicly available via DASH.  Basic document information (e.g., author name, thesis title, degree date, abstract) will also be collected via the submission system; this document information will be available in  HOLLIS , the Harvard Library catalog, and DASH (though the thesis itself will be available in DASH only if the student opts to allow this). Students can also make code or data for senior thesis work available. They can do this by posting the data to the Harvard  Dataverse  or including the code as a supplementary file in the DASH repository when submitting their thesis in the SEAS online submission system.

Whether or not a student opts to make the thesis available through DASH, SEAS will provide an electronic record copy of the thesis to the Harvard University Archives. The Archives may make this record copy of the thesis accessible to researchers in the Archives reading room via a secure workstation or by providing a paper copy for use only in the reading room.  Per University policy , for a period of five years after the acceptance of a thesis, the Archives will require an author’s written permission before permitting researchers to create or request a copy of any thesis in whole or in part. Students who wish to place additional restrictions on the record copy in the Archives must contact the Archives  directly, independent of the online submission system. 

Students interested in commercializing ideas in their theses may wish to consult Dr. Fawwaz Habbal , Senior Lecturer on Applied Physics, about patent protection. See Harvard's policy for information about ownership of software written as part of academic work.

In Applied Mathematics

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Course info, instructors.

• Dr. Peter Kempthorne
• Dr. Choongbum Lee
• Dr. Vasily Strela
• Dr. Jake Xia

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• Mathematics

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• Applied Mathematics
• Probability and Statistics

Learning Resource Types

Topics in mathematics with applications in finance, sample topics for the final paper.

Twenty-five percent of the course grade is based upon a final paper on a math finance topic of the student’s choice. Below are some sample topics. Students may propose other topics as well.

Portfolio Management

Based on what you learned in class, research further and come up with your own views in portfolio risk management.

Regime-Shift Modeling

Detail one or more approaches to regime-shift modeling, addressing the statistical modeling methodology and its use in a specific, real-world application.

Low-Volatility Investing

Critically review the rationales of low-volatility investing strategies in the U.S. equity market and their connection to the portfolio theory covered in class; evaluate the performance of such strategies as implemented in exchange-traded funds and / or mutual funds.

Modeling Financial Bubbles

Detail one or more approaches to modeling asset bubbles; e.g., the work of Didier Sornette.

Relationship between Black-Scholes and Heat Equations

• Go through the change of variables to get from Black-Scholes PDE to Heat Equation.
• Go through calculations verifying that a European call option price for a lognormaly distributed stock is in fact a discounted expected value of the pay-off under risk neutral measure.
• Explore possible numerical methods for the solution with various boundary conditions.
• Go through computations showing that Black-Scholes price of a digital option is a partial derivative of the call option price with respect to strike.

Hybrid products

• Price zero coupon bonds in USD and EUR in this jump–diffusion model.
• Determine the dynamic hedging strategy. There are two sources of risk, so need at least 2 hedging instruments. FX forwards are a great candidate.

HJM vs Short-Rate Interest Rate Models. 

• Start from the equation for forward rates df tT = μ tT dt + σ tT dB t and derive the no-arbitrage condition for drift μ tT . 
• Derive drift at for the short rate Ho-Lee Model dr t = a t dt + σdB t . Next, show that the Ho-Lee model can be written in the HJM form. Remember that r t = f tt .
• Add a mean reversion to the Ho-Lee model dr t = (a t - κr t *)dt + σdB* t and write it in the HJM form.

Ross Recovery

• Try to offer financial intuition for the Perron Forbenius theorem for positive matrices.
• Try to extend Ross recovery to a countable state space for a Markov chain.

A Few Topics Chosen by Students Last Year

• Transformation of Black-Scholes PDE to Heat Equation
• From Black-Scholes-Merton model to heat equation: Derivations and numerical solutions
• Solving Black-Scholes equation with Initial conditions by change of variables
• Derive HJM no arbitrage condition
• HJM model and Ho-Lee model
• Pricing zero-coupon USD and EUR bonds in the FX jump diffusion model
• Pricing Asian options
• On the Minimal Entropy Martingale Measure in Finite Probability Financial Market Model
• Principal Component Analysis on Oil, Gas, Power and Currency Swap Curves before and after the 2008 Financial Crisis
• A review of finite grid summation method and Monte-Carlo method for a three-legged spread option integration
• Monte-Carlo option pricing using the heston model for stochastic volatility

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251+ Math Research Topics [2024 Updated]

Mathematics, often dubbed as the language of the universe, holds immense significance in shaping our understanding of the world around us. It’s not just about crunching numbers or solving equations; it’s about unraveling mysteries, making predictions, and creating innovative solutions to complex problems. In this blog, we embark on a journey into the realm of math research topics, exploring various branches of mathematics and their real-world applications.

How Do You Write A Math Research Topic?

Writing a math research topic involves several steps to ensure clarity, relevance, and feasibility. Here’s a guide to help you craft a compelling math research topic:

• Identify Your Interests: Start by exploring areas of mathematics that interest you. Whether it’s pure mathematics, applied mathematics, or interdisciplinary topics, choose a field that aligns with your passion and expertise.
• Narrow Down Your Focus: Mathematics is a broad field, so it’s essential to narrow down your focus to a specific area or problem. Consider the scope of your research and choose a topic that is manageable within your resources and time frame.
• Review Existing Literature: Conduct a thorough literature review to understand the current state of research in your chosen area. Identify gaps, controversies, or unanswered questions that could form the basis of your research topic.
• Formulate a Research Question: Based on your exploration and literature review, formulate a clear and concise research question. Your research question should be specific, measurable, achievable, relevant, and time-bound (SMART).
• Consider Feasibility: Assess the feasibility of your research topic in terms of available resources, data availability, and research methodologies. Ensure that your topic is realistic and achievable within the constraints of your project.
• Consult with Experts: Seek feedback from mentors, advisors, or experts in the field to validate your research topic and refine your ideas. Their insights can help you identify potential challenges and opportunities for improvement.
• Refine and Iterate: Refine your research topic based on feedback and further reflection. Iterate on your ideas to ensure clarity, coherence, and relevance to the broader context of mathematics research.
• Craft a Title: Once you have finalized your research topic, craft a compelling title that succinctly summarizes the essence of your research. Your title should be descriptive, engaging, and reflective of the key themes of your study.
• Write a Research Proposal: Develop a comprehensive research proposal outlining the background, objectives, methodology, and expected outcomes of your research. Your research proposal should provide a clear roadmap for your study and justify the significance of your research topic.

By following these steps, you can effectively write a math research topic that is well-defined, relevant, and poised to make a meaningful contribution to the field of mathematics.

251+ Math Research Topics: Beginners To Advanced

• Prime Number Distribution in Arithmetic Progressions
• Diophantine Equations and their Solutions
• Applications of Modular Arithmetic in Cryptography
• The Riemann Hypothesis and its Implications
• Graph Theory: Exploring Connectivity and Coloring Problems
• Knot Theory: Unraveling the Mathematics of Knots and Links
• Fractal Geometry: Understanding Self-Similarity and Dimensionality
• Differential Equations: Modeling Physical Phenomena and Dynamical Systems
• Chaos Theory: Investigating Deterministic Chaos and Strange Attractors
• Combinatorial Optimization: Algorithms for Solving Optimization Problems
• Computational Complexity: Analyzing the Complexity of Algorithms
• Game Theory: Mathematical Models of Strategic Interactions
• Number Theory: Exploring Properties of Integers and Primes
• Algebraic Topology: Studying Topological Invariants and Homotopy Theory
• Analytic Number Theory: Investigating Properties of Prime Numbers
• Algebraic Geometry: Geometry Arising from Algebraic Equations
• Galois Theory: Understanding Field Extensions and Solvability of Equations
• Representation Theory: Studying Symmetry in Linear Spaces
• Harmonic Analysis: Analyzing Functions on Groups and Manifolds
• Mathematical Logic: Foundations of Mathematics and Formal Systems
• Set Theory: Exploring Infinite Sets and Cardinal Numbers
• Real Analysis: Rigorous Study of Real Numbers and Functions
• Complex Analysis: Analytic Functions and Complex Integration
• Measure Theory: Foundations of Lebesgue Integration and Probability
• Topological Groups: Investigating Topological Structures on Groups
• Lie Groups and Lie Algebras: Geometry of Continuous Symmetry
• Differential Geometry: Curvature and Topology of Smooth Manifolds
• Algebraic Combinatorics: Enumerative and Algebraic Aspects of Combinatorics
• Ramsey Theory: Investigating Structure in Large Discrete Structures
• Analytic Geometry: Studying Geometry Using Analytic Methods
• Hyperbolic Geometry: Non-Euclidean Geometry of Curved Spaces
• Nonlinear Dynamics: Chaos, Bifurcations, and Strange Attractors
• Homological Algebra: Studying Homology and Cohomology of Algebraic Structures
• Topological Vector Spaces: Vector Spaces with Topological Structure
• Representation Theory of Finite Groups: Decomposition of Group Representations
• Category Theory: Abstract Structures and Universal Properties
• Operator Theory: Spectral Theory and Functional Analysis of Operators
• Algebraic Number Theory: Study of Algebraic Structures in Number Fields
• Cryptanalysis: Breaking Cryptographic Systems Using Mathematical Methods
• Discrete Mathematics: Combinatorics, Graph Theory, and Number Theory
• Mathematical Biology: Modeling Biological Systems Using Mathematical Tools
• Population Dynamics: Mathematical Models of Population Growth and Interaction
• Epidemiology: Mathematical Modeling of Disease Spread and Control
• Mathematical Ecology: Dynamics of Ecological Systems and Food Webs
• Evolutionary Game Theory: Evolutionary Dynamics and Strategic Behavior
• Mathematical Neuroscience: Modeling Brain Dynamics and Neural Networks
• Mathematical Physics: Mathematical Models in Physical Sciences
• Quantum Mechanics: Foundations and Applications of Quantum Theory
• Statistical Mechanics: Statistical Methods in Physics and Thermodynamics
• Fluid Dynamics: Modeling Flow of Fluids Using Partial Differential Equations
• Mathematical Finance: Stochastic Models in Finance and Risk Management
• Option Pricing Models: Black-Scholes Model and Beyond
• Portfolio Optimization: Maximizing Returns and Minimizing Risk
• Stochastic Calculus: Calculus of Stochastic Processes and Itô Calculus
• Financial Time Series Analysis: Modeling and Forecasting Financial Data
• Operations Research: Optimization of Decision-Making Processes
• Linear Programming: Optimization Problems with Linear Constraints
• Integer Programming: Optimization Problems with Integer Solutions
• Network Flow Optimization: Modeling and Solving Flow Network Problems
• Combinatorial Game Theory: Analysis of Games with Perfect Information
• Algorithmic Game Theory: Computational Aspects of Game-Theoretic Problems
• Fair Division: Methods for Fairly Allocating Resources Among Parties
• Auction Theory: Modeling Auction Mechanisms and Bidding Strategies
• Voting Theory: Mathematical Models of Voting Systems and Social Choice
• Social Network Analysis: Mathematical Analysis of Social Networks
• Algorithm Analysis: Complexity Analysis of Algorithms and Data Structures
• Machine Learning: Statistical Learning Algorithms and Data Mining
• Deep Learning: Neural Network Models with Multiple Layers
• Reinforcement Learning: Learning by Interaction and Feedback
• Natural Language Processing: Statistical and Computational Analysis of Language
• Computer Vision: Mathematical Models for Image Analysis and Recognition
• Computational Geometry: Algorithms for Geometric Problems
• Symbolic Computation: Manipulation of Mathematical Expressions
• Numerical Analysis: Algorithms for Solving Numerical Problems
• Finite Element Method: Numerical Solution of Partial Differential Equations
• Monte Carlo Methods: Statistical Simulation Techniques
• High-Performance Computing: Parallel and Distributed Computing Techniques
• Quantum Computing: Quantum Algorithms and Quantum Information Theory
• Quantum Information Theory: Study of Quantum Communication and Computation
• Quantum Error Correction: Methods for Protecting Quantum Information from Errors
• Topological Quantum Computing: Using Topological Properties for Quantum Computation
• Quantum Algorithms: Efficient Algorithms for Quantum Computers
• Quantum Cryptography: Secure Communication Using Quantum Key Distribution
• Topological Data Analysis: Analyzing Shape and Structure of Data Sets
• Persistent Homology: Topological Invariants for Data Analysis
• Mapper Algorithm: Method for Visualization and Analysis of High-Dimensional Data
• Algebraic Statistics: Statistical Methods Based on Algebraic Geometry
• Tropical Geometry: Geometric Methods for Studying Polynomial Equations
• Model Theory: Study of Mathematical Structures and Their Interpretations
• Descriptive Set Theory: Study of Borel and Analytic Sets
• Ergodic Theory: Study of Measure-Preserving Transformations
• Combinatorial Number Theory: Intersection of Combinatorics and Number Theory
• Additive Combinatorics: Study of Additive Properties of Sets
• Arithmetic Geometry: Interplay Between Number Theory and Algebraic Geometry
• Proof Theory: Study of Formal Proofs and Logical Inference
• Reverse Mathematics: Study of Logical Strength of Mathematical Theorems
• Nonstandard Analysis: Alternative Approach to Analysis Using Infinitesimals
• Computable Analysis: Study of Computable Functions and Real Numbers
• Graph Theory: Study of Graphs and Networks
• Random Graphs: Probabilistic Models of Graphs and Connectivity
• Spectral Graph Theory: Analysis of Graphs Using Eigenvalues and Eigenvectors
• Algebraic Graph Theory: Study of Algebraic Structures in Graphs
• Metric Geometry: Study of Geometric Structures Using Metrics
• Geometric Measure Theory: Study of Measures on Geometric Spaces
• Discrete Differential Geometry: Study of Differential Geometry on Discrete Spaces
• Algebraic Coding Theory: Study of Error-Correcting Codes
• Information Theory: Study of Information and Communication
• Coding Theory: Study of Error-Correcting Codes
• Cryptography: Study of Secure Communication and Encryption
• Finite Fields: Study of Fields with Finite Number of Elements
• Elliptic Curves: Study of Curves Defined by Cubic Equations
• Hyperelliptic Curves: Study of Curves Defined by Higher-Degree Equations
• Modular Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Number Theory
• Zeta Functions: Analytic Functions with Special Properties
• Analytic Number Theory: Study of Number Theoretic Functions Using Analysis
• Dirichlet Series: Analytic Functions Represented by Infinite Series
• Euler Products: Product Representations of Analytic Functions
• Arithmetic Dynamics: Study of Iterative Processes on Algebraic Structures
• Dynamics of Rational Maps: Study of Dynamical Systems Defined by Rational Functions
• Julia Sets: Fractal Sets Associated with Dynamical Systems
• Mandelbrot Set: Fractal Set Associated with Iterations of Complex Quadratic Polynomials
• Arithmetic Geometry: Study of Algebraic Geometry Over Number Fields
• Diophantine Geometry: Study of Solutions of Diophantine Equations Using Geometry
• Arithmetic of Elliptic Curves: Study of Elliptic Curves Over Number Fields
• Rational Points on Curves: Study of Rational Solutions of Algebraic Equations
• Galois Representations: Study of Representations of Galois Groups
• Automorphic Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Automorphic Forms
• Selberg Trace Formula: Tool for Studying Spectral Theory and Automorphic Forms
• Langlands Program: Program to Unify Number Theory and Representation Theory
• Hodge Theory: Study of Harmonic Forms on Complex Manifolds
• Riemann Surfaces: One-dimensional Complex Manifolds
• Shimura Varieties: Algebraic Varieties Associated with Automorphic Forms
• Modular Curves: Algebraic Curves Associated with Modular Forms
• Hyperbolic Manifolds: Manifolds with Constant Negative Curvature
• Teichmüller Theory: Study of Moduli Spaces of Riemann Surfaces
• Mirror Symmetry: Duality Between Calabi-Yau Manifolds
• Kähler Geometry: Study of Hermitian Manifolds with Special Symmetries
• Algebraic Groups: Linear Algebraic Groups and Their Representations
• Lie Algebras: Study of Algebraic Structures Arising from Lie Groups
• Representation Theory of Lie Algebras: Study of Representations of Lie Algebras
• Quantum Groups: Deformation of Lie Groups and Lie Algebras
• Algebraic Topology: Study of Topological Spaces Using Algebraic Methods
• Homotopy Theory: Study of Continuous Deformations of Spaces
• Homology Theory: Study of Algebraic Invariants of Topological Spaces
• Cohomology Theory: Study of Dual Concepts to Homology Theory
• Singular Homology: Homology Theory Defined Using Simplicial Complexes
• Sheaf Theory: Study of Sheaves and Their Cohomology
• Differential Forms: Study of Multilinear Differential Forms
• De Rham Cohomology: Cohomology Theory Defined Using Differential Forms
• Morse Theory: Study of Critical Points of Smooth Functions
• Symplectic Geometry: Study of Symplectic Manifolds and Their Geometry
• Floer Homology: Study of Symplectic Manifolds Using Pseudoholomorphic Curves
• Gromov-Witten Invariants: Invariants of Symplectic Manifolds Associated with Pseudoholomorphic Curves
• Mirror Symmetry: Duality Between Symplectic and Complex Geometry
• Calabi-Yau Manifolds: Ricci-Flat Complex Manifolds
• Moduli Spaces: Spaces Parameterizing Geometric Objects
• Donaldson-Thomas Invariants: Invariants Counting Sheaves on Calabi-Yau Manifolds
• Algebraic K-Theory: Study of Algebraic Invariants of Rings and Modules
• Homological Algebra: Study of Homology and Cohomology of Algebraic Structures
• Derived Categories: Categories Arising from Homological Algebra
• Stable Homotopy Theory: Homotopy Theory with Stable Homotopy Groups
• Model Categories: Categories with Certain Homotopical Properties
• Higher Category Theory: Study of Higher Categories and Homotopy Theory
• Higher Topos Theory: Study of Higher Categorical Structures
• Higher Algebra: Study of Higher Categorical Structures in Algebra
• Higher Algebraic Geometry: Study of Higher Categorical Structures in Algebraic Geometry
• Higher Representation Theory: Study of Higher Categorical Structures in Representation Theory
• Higher Category Theory: Study of Higher Categorical Structures
• Homotopical Algebra: Study of Algebraic Structures in Homotopy Theory
• Homotopical Groups: Study of Groups with Homotopical Structure
• Homotopical Categories: Study of Categories with Homotopical Structure
• Homotopy Groups: Algebraic Invariants of Topological Spaces
• Homotopy Type Theory: Study of Foundations of Mathematics Using Homotopy Theory

In conclusion, the world of mathematics is vast and multifaceted, offering endless opportunities for exploration and discovery. Whether delving into the abstract realms of pure mathematics or applying mathematical principles to solve real-world problems, mathematicians play a vital role in advancing human knowledge and shaping the future of our world.

By embracing diverse math research topics and interdisciplinary collaborations, we can unlock new possibilities and harness the power of mathematics to address the challenges of today and tomorrow. So, let’s embark on this journey together as we unravel the mysteries of numbers and explore the boundless horizons of mathematical inquiry.

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Research Areas

It is possible to apply mathematics to almost any field of human endeavor. Here are some of the fields we’re working on now.

Scientific Computing and Numerical Analysis

Researchers : Loyce Adams , Bernard Deconinck , Randy LeVeque , Ioana Dumitriu , Anne Greenbaum , James Riley

Many practical problems in science and engineering cannot be solved completely by analytical means. Research in the area of numerical analysis and scientific computation is concerned with the development and analysis of numerical algorithms, the implementation of these algorithms on modern computer architectures, and the use of numerical methods in conjunction with mathematical modeling to solve large-scale practical problems. Major research areas in this department include computational fluid dynamics (CFD), interface and front tracking methods, iterative methods in numerical linear algebra, and algorithms for parallel computers.Current research topics in CFD include: 

• high resolution methods for solving nonlinear conservation laws with shock wave solutions
• numerical methods for atmospheric flows, particularly cloud formation
• Cartesian grid methods for solving multidimensional problems in complicated geometries on uniform grids
• spectral methods for fluid stability problems
• front tracking methods for fluid flow problems with free surfaces or immersed interfaces in the context of porous media flow (ground water or oil reservoir simulation) and in physiological flows with elastic membranes.
• nonequilibrium flows in combustion and astrophysical simulation
• immersed interface methods for solidification or melting problems and seismic wave equations with discontinuous coefficients that arise in modeling the geological structure of the earth.

Another research focus is the development of methods for large-scale scientific computations that are suited to implementation on parallel computer architectures. Current interests include:

• preconditioners for the iterative solution of large linear or nonlinear systems
• methods for the symmetric and nonsymmetric eigenvalue problems
• methods for general interface problems in complicated domains.

The actual implementation and testing of methods on parallel architectures is possible through collaboration with the Department of Computer Science, the Boeing Company, and the Pacific Northwest Labs.

Nonlinear Waves and Coherent Structures

Researchers : Bernard Deconinck , Nathan Kutz ,  Randy LeVeque

Most problems in applied mathematics are inherently nonlinear. The effects due to nonlinearities may become important under the right circumstances. The area of nonlinear waves and coherent structures considers how nonlinear effects influence problems involving wave propagation. Sometimes these effects are desirable and lead to new applications (mode-locked lasers, optical solitons and nonlinear optics). Other times one has no choice but to consider their impact (water waves). The area of nonlinear waves encompasses a large collection of phenomena, such as the formation and propagation of shocks and solitary waves. The area received renewed interest starting in the 1960s with the development of soliton theory, which examines completely integrable systems and classes of their special solutions.

Mathematical Biology

Researchers : Mark Kot , Hong Qian , Eric Shea-Brown , Elizabeth Halloran , Suresh Moolgavkar , Eli Shlizerman , Ivana Bozic

Mathematical biology is an increasingly large and well-established branch of applied mathematics. This growth reflects both the increasing importance of the biological and biomedical sciences and an appreciation for the mathematical subtleties and challenges that arise in the modelling of complex biological systems. Our interest, as a group, lies in understanding the spatial and temporal patterns that arise in dynamic biological systems. Our mathematical activities range from reaction-diffusion equations, to nonlinear and chaotic dynamics, to optimization. We employ a variety of tools and models to study problems that arise in development, epidemiology, ecology, neuroscience, resource management, and biomechanics; and we maintain active collaborations with a large number and variety of biologists and biomedical departments both in the University and elsewhere. For more information, please see the  Mathematical Biology page .

Atmospheric Sciences and Climate Modeling

Researchers : Chris Bretherton , Ka-Kit Tung , Dale Durran

Mathematical models play a crucial role in our understanding of the fluid dynamics of the atmosphere and oceans. Our interests include mathematical methods for studying the hydrodynamical instability of shear flows, transition from laminar flow to turbulence, applications of fractals to turbulence, two-dimensional and quasi-geostrophic turbulence theory and computation, and large-scale nonlinear wave mechanics.We also develop and apply realistic coupled radiative- chemical-dynamical models for studying stratospheric chemistry, and coupled radiative-microphysical-dynamical models for studying the interaction of atmospheric turbulence and cloud systems These two topics are salient for understanding how man is changing the earth’s climate.Our work involves a strong interaction of computer modelling and classical applied analysis. This research group actively collaborates with scientists in the Atmospheric Science, Oceanography, and Geophysics department, and trains students in the emerging interdisciplinary area of earth system modeling, in addition to providing a traditional education in classical fluid dynamics.

Mathematical Methods

Researchers : Bernard Deconinck , Robert O'Malley ,  Jim Burke , Archis Ghate , John Sylvester , Gunther Uhlmann

The department maintains active research in fundamental methods of applied mathematics. These methods can be broadly applied to a vast number of problems in the engineering, physical and biological sciences. The particular strengths of the department of applied mathematics are in asymptotic and perturbation methods, applied analysis, optimization and control, and inverse problems.

Mathematical Finance

Researchers : Tim Leung , Matt Lorig , Doug Martin

The department’s growing financial math group is active in the areas of derivative pricing & hedging, algorithmic trading, portfolio optimization, insurance, risk measures, credit risk, and systemic risk. Research includes collaboration with students as well as partners from both academia and industry.

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Setting Up High-Impact Tasks in Elementary School Math Centers

Allowing students to select math centers based on interest instead of skill level provides opportunities for them all to grow.

In our work as math coaches and consultants, we are often asked to help teachers structure small group practice time. Teachers who are required to implement “what I need” (WIN) groups or small group math time have questions about how to put it into practice so that the wide range of students’ needs are being met. We invite you to consider how prioritizing equity-based principles and providing high-impact tasks can offer a path to differentiating instruction, deepening skills and concepts, and strengthening problem-solving. 

differentiation in math centers

Elementary teachers are often encouraged to use math workshop (also called math center time or math rotations) to differentiate. In this model, teachers create structures for small groups of students to move from one task to another in timed rotations to complete activities that the teacher has assigned and prepared for them. In this model, there is almost always a “teacher table,” where the teacher works closely with small groups of students on “what they need.” 

This model can send unintended messages to students about what it means to be a successful, competent learner of mathematics. It can result in a math classroom that is hierarchical and leveled rather than one that supports students with multi-abilities. Students may begin at early ages to feel the impact of being identified and tracked . So what is the alternative?

designing better Tasks for math centers

Math centers offer ideal opportunities to go deep with the mathematics . The choice of tasks and the ways that teachers interact with students during the workshop impact essential equity standards. We prioritize activities or games with a low floor and high ceiling and that have a high cognitive demand with multiple solution strategies. These games and activities encourage students to make conjectures, to reason through multiple solutions, and to practice important mathematical and problem-solving skills . 

One example we often begin with is counting collections . The task focuses on significant mathematics, and yet the directions are uncomplicated. Students choose a collection of items and then figure out how many items are in the collection . The complexity and rigor come from students having to figure out how to count and then how to represent their count. Students develop essential skills such as counting, sorting, grouping, and problem-solving.  

Role of the teacher: Instead of having a teacher be stationary at a table where they supervise and lead students through a task, they move around the classroom listening to the students as they engage in the activities. They press on ideas, nudge and notice how students respond and interact. They ask probing questions to help surface mathematical ideas, and they take notes on what they observe and how students respond. Then they use their observations to assess student understanding and inform planning.

Teachers can still gather a small group of students together to bring forward some aspect of their work. In the example of counting collections, we sometimes bring together students to practice ideas related to one-to-one correspondence or extend skills related to number sense.

Grouping Students for centers

We plan tasks for math centers that allow us to leverage multiple competencies among learners and challenge spaces of marginality. In small groups or partnerships, students with different strengths learn with and from each other as they collaborate on activities. We see the variation in student abilities within a group as benefiting all group members, as it allows for greater richness of ideas and knowledge mobility . Students with varying skills and solution strategies work side-by-side using each other as equal thought partners who are able to engage in the mathematics as sense-makers. 

Many math games, like the classic “compare” games , offer the kind of richness that makes them well-suited as tasks that leverage the multiple competencies of our students during math workshop. For instance, when they play “multiplication compare,” the game directions are routine: Draw the number of cards needed, figure out the product, and compare it to the product of your partner’s hand. We use sentence starters to support partner talk for all learners and develop protocols for partner decision-making .

The success of multi-ability groups depends on how the teacher establishes an equitable math learning community and how that community is nurtured and maintained throughout the year. We pair students randomly in order to disrupt any narrative that only certain kinds of learners are capable of engaging in deep mathematics. Random groupings position all learners as competent and capable.

Role of the teacher: Following a class period of math rotations, we debrief with our students , not only about the mathematical content they have been engaged in, but also about aspects of their group work and interactions. We help students acknowledge and describe how a partner’s solution offered something new and productive to consider.  

Affirming Learners’ Math Identities

Because we see math centers as opportunities to position all students as competent, we prioritize student agency and expand access for all . We select games and activities that support students’ independence, interdependence, and decision-making. Choice is an essential part of the math workshop we are advocating for. When we reposition our students as competent independent learners and give them opportunities to make mathematical decisions, students will rise to the challenge in ways that surprise and excite their teachers as they develop more positive identities as math learners.

Role of the teacher: As we interact with students during math workshop, we press on important mathematical ideas and help shape how students view mathematics and how they view their relationship to mathematics. For example, during “counting collections,” we might say, “You have shown one way to count this quantity. Is there another way you and your partner could count this collection and represent it so that other people would know easily how many items there are? Mathematicians often make several attempts at representing their ideas in order to communicate them clearly to others. See what you might come up with for a second attempt.”

Or, after observing students at a table playing “multiplication compare,” we might say, “I noticed that your group had a few ways of figuring out who had the greater product. When we meet at the end of workshop time, it would be helpful for your classmates to hear your ideas. Why don’t you talk together now about how you might present your ideas to the whole group?”

What’s Next?

We believe it’s time to reconsider some of the assumptions and expectations that educators typically bring to the design and implementation of math workshop time. We suggest prioritizing the development of positive math identities for all students by providing opportunities for access, agency, collaboration, and independence by giving them choice and voice.

We have found that when we rethink the role of the teacher, the kinds of mathematically rich tasks we offer, and the way in which we group students, math workshop can become integral to the creation of equitable math classrooms and be a place for students to develop strong habits of mind alongside math competencies.