PhD Studentships in Mathematical Sciences

Following substantial recent investment in the Mathematical Sciences at Lancaster University, we invite applications for up to 6 fully funded PhD studentships in Pure Mathematics, Statistics and Operational Research for entry in October 2021.

Research projects

Any research areas in Pure Mathematics, Statistics and Operational Research that are consistent with those of staff members in the departments of Mathematics and Statistics and Management Science will be considered. It is not necessary to submit a full research proposal but applications should indicate a preferred supervisor or supervisors and describe how the application aligns with one of the following strategic areas.

  • Pure Mathematics – Noncommutative algebra and analysis, including cluster algebras, representation theory, symplectic geometry, homological and homotopical algebra, operator algebras, noncommutative probability, algebraic quantum field theory and allied topics. We are looking to build a cohort of students around this topic who can collaborate and learn from one another.
  • Statistics and Operational Research – Applied probability, computational statistics, forecasting and predictive analytics, medical statistics, statistical learning, optimisation, simulation, time-series analysis.

We particularly welcome proposals that also align with applications in healthcare, energy, transportation, logistics, marketing or telecommunications.

Details of indicative projects being offered are listed below.

Pure Mathematics

Accordion

  • Symplectic geometry and applications (Jonny Evans)

    My own interests include: Floer theory, the topology of Lagrangian submanifolds, the symplectic topology of algebraic varieties and their singularities, Lagrangian torus fibrations, and symplectomorphism groups. I'm particularly interested in applications of symplectic geometry to other areas (e.g. algebraic geometry, calibrated or Riemannian geometry, representation theory). I'm happy to supervise projects in symplectic geometry. If you want to see what my former students have worked on, you can read their theses here: http://discovery.ucl.ac.uk/1570398/http://discovery.ucl.ac.uk/id/eprint/10077487.

  • Approximation of random permutations with polynomial cycle weights by independent random variables (Dirk Zeindler)

    A permutation is a rearrangement of an ordered set. Permutations occur, in more or less prominent ways, in almost every branch of mathematics. An important case is the group of all permutations of the set {1,2,3,..,N}. This group is called the symmetric group SN of degree N. One speaks of a random permutation if one selects randomly a permutation of SN. In other words, one endows SN with a probability measure.

    One of the most natural measures is uniform measure. This has been studied extensively and is well understood. In recent years, there has also been much interest in weighted random permutations. In these models, one assigns each cycle in a permutation a weight depending on its length and takes the product over all cycles to obtain a probability measure. In particular of interest are polynomial cycle weights. It is know that a typical cycle under this measure has order of magnitude na with a known. The aim of this project is to study this measure more carefully and in particular to invest how good this measure can be approximated by independent random variables.

     

  • Noncommutative global Koszul duality and homotopy theory (Andrey Lazarev)

    Koszul duality is a fundamental phenomenon playing a central role in many branches of algebra and geometry: rational homotopy theory, deformation theory, representation theory, algebraic geometry and others. It is usually formulated as an equivalence of categories of algebras and conilpotent coalgebras. The noncommutative version of Koszul duality allows a globalization by removing the condition of conilpotency on the coalgebra side. The resulting global theory exhibits various subtle and unexpected features of homotopy theoretical nature. This project will explore these phenomena and apply it to the study of moduli spaces of algebraic and topological origin.

  • Projective modules over function algebras (Gordon Blower)

    One of the most basic algebras in function theory is the disc algebra A(D) of complex functions that are continuous on the closed unit disc and holomorphic on the open unit disc. Carlson et al [CCFW] gave examples of Hilbert modules over the disc algebra that are nonzero and projective. Answering a well-known question of Halmos, Pisier [P] constructed a Hilbert module over A(D) that is generated by an operator T that is not similar to a contractive operator. One of the main aims of the project is to produce criteria for Hilbert modules over the disc algebra to be projective. In an apparently separate line of development, Cuntz and Quillen [CQ] introduced the class of quasi- free algebras. An algebra A is quasi-free if the space of noncommutative differential forms is a projective A-bimodule. A fundamental example is the algebra of coordinate functions on an algebraic curve.

    The proposed PhD project will investigate projective modules over function algebras and the connection with and the results of Cuntz and Quillen. The work of Cuntz and Quillen has not yet been absorbed into the literature of operator space theory or fully realized in multivariable operator theory.

    [CCFW] J.F. Carlson, D.N. Clark, C. Foias, and J.P. Williams, Projective Hilbert A(D) modules, New York J. Math. 1 (1994/5) 26-38.

    [CQ] J. Cuntz and D.G. Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8 (1995), 251-286.

    [P] G. Pisier, A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc. 10 (1997), 351-369.

  • Function algebras and operator algebras arising in noncommutative harmonic analysis (Yemon Choi)

    Classical Fourier analysis on the circle, or for periodic functions on the line, rests on the duality between spatial symmetries (rotations) and symmetries on the frequency side (shifts). In more general situations the spatial symmetries need not commute with each other: think of the rotation group in three or more dimensions, or the group generated by translations and dilations on the line. Generalizing Fourier analysis to this setting leads to two kinds of algebraic structures: operator algebras generated by group representations (living on the frequency side); and function algebras generated by the "matrix coefficients" of these representations (living on the spatial side). The bridge between the two sides is a suitable generalization of the classical Fourier transform and Plancherel theorem. For the two examples mentioned above, this abstract theory underlies tools such as spherical harmonics and the wavelet transform.

    A PhD project in this area would investigate structural properties of these algebras using methods from functional analysis and representation theory, with a focus on understanding examples attached to particular families of groups. Within this broad remit there are distinct but related possible topics of research. One possible direction is to study the local structure of Lp-operator algebras generated by representations of Lie groups, where for p ≠ 2 much less is known than for C*-algebras. A different direction would attempt to compute or estimate cohomological invariants associated to function algebras on Lie groups; here is there considerable scope for computer-aided experimentation, although rigorous proofs will ultimately be required.

    For some of my own work in these directions, see the references below.

  • Hyperfiniteness and amenability (Gabor Elek)

    One of the most mysterious and untouchable conjectures of mathematics is the "amenability implies hyperfiniteness" problem. The question is answered in the measurable category, and earned a Fields medal for Alain Connes in the case of von Neumann algebras.  However, the Borel version is wide open. It seems that there are very strongly amenable Borel equivalence relations for which hyperfiniteness can be deduced, also for some other amenable relations a weaker version of hyperfiniteness can be proved. The topic is in the crossroads of analysis, group theory and operator algebras.

  • Dependence structures in random growth (Amanda Turner and Azadeh Khaleghi)

    The focus of this project is on understanding the long-time behaviour of planar random growth processes. These arise in physical and industrial settings, from cancer to polymer creation. Such models are important, not only due to potential applications arising from their physical prevalence, but also because of connections with new and rapidly evolving areas of mathematics such as Schramm-Loewner Evolution (SLE) and regularity structures. However, as many of these models exhibit complicated long-range dependencies, mathematical analysis has yielded very little progress. One approach to modelling random growth is to represent clusters as compositions of conformal mappings. This enables the generation of clusters using computer simulation and indicates the existence of phase transitions from clusters that converge to disks, to clusters that converge to straight lines. Preliminary simulations suggest that off-critical limits, near the point of transition to straight lines, are simple paths which can be described as stochastic processes. The aim of this project is to undertake a systematic numerical study of clusters near this point of transition. The objective is to understand the dependency structure of these stochastic processes and in particular to identify a relationship between the parameter describing the cluster, and the dependency structure. This project has the potential to be taken in a mathematical direction, or in a statistical direction, depending on the strengths and interests of the student.

  • Nilpotent singularities in positive characteristic (Paul Levy)

    The geometry of the set of nilpotent elements, or nilpotent cone of a simple complex Lie algebra has been intensively studied in Lie theory since at least the 1960s. One important area of study concerns the isolated singular varieties which are obtained for each pair of adjacent nilpotent orbits in the closure ordering. These are important examples of sympletic singularities.

    In positive characteristic, some work has been done for classical Lie algebras which generalizes the above picture for complex Lie algebras. However, some of the most interesting symplectic singularities occur in relation to the exceptional Lie algebras. This project will focus on the isolated singularities one obtains for exceptional Lie algebras in positive characteristic (possibly including the small characteristics which are known as the bad primes).

  • Graph rigidity and discrete geometry (Bernd Schulze)

    Graph rigidity is concerned with the structural rigidity of discrete objects defined by a set of geometric constraints. These constraints (specifying lengths, angles, directions, etc.) on a set of rigid objects (points, line segments, polygons, etc.) are modelled by systems of non-linear real polynomial equations. The nature of the set of solutions to these equations determines if the object is globally rigid (unique solution up to isometries), rigid (finitely many solutions up to isometries) or flexible (infinitely many solutions; hence can be continuously deformed while satisfying the constraints).

    Since the rigidity and flexibility properties of a structure - either man-made, such as a building, bridge or mechanical linkage, or found in nature, such as a biomolecule, protein or crystal - are critical to its form, behaviour and functioning, graph rigidity has many practical applications in fields such as engineering, robotics, materials science, biophysics and Computer-Aided Design. Graph rigidity also has connections to a number of other areas in pure mathematics, such as polytope theory, polyhedral scene analysis, low rank matrix completion, and topological graph theory.

    The rigidity and flexibility of geometric constraint systems that are in generic position can be analysed using techniques from graph theory, combinatorics, matroid theory and linear algebra. Key tools include the rigidity matrix/matroid and the stress matrix of the system. For the rigidity analysis of constraint systems in special geometric positions, such as symmetric or periodic systems, techniques from group representation theory and algebraic or projective geometry become relevant.

  • Operators on Banach spaces (Niels Laustsen)

    My main line of research is to explore the interplay between a Banach space X on the one hand and the associated Banach algebra B(X) of bounded operators acting on Xon the other.

    The topic that I have studied most extensively is the closed ideal structure of B(X) for various Banach spaces X, both “classical” and “purpose-built”. This has recently become a very active area of research internationally. I am happy to supervise projects in it, or on a number of related topics involving Banach spaces and their operators.

    Previous knowledge of Banach spaces, operators and/or basic functional analysis is an advantage, but not a pre-requisite, as the MAGIC network provides excellent training opportunities for students during their first year.

Statistics and Operational Research

Accordion

  • Robust decision-making in an uncertain world: Monte Carlo methods for stochastic programming (Jamie Fairbrother and Christopher Nemeth)

    Stochastic programming is a tool for making decisions under uncertainty by minimizing expected losses. It is commonly used in areas including energy, telecommunications and finance. There is typically a trade-off in these problems in how one represents uncertainty: the more detailed the representation, the more reliable the output, but the more computationally intensive the problem becomes to solve. In this project, we will develop new Monte Carlo methods for stochastic programming which exploit the problem structure to more efficiently represent uncertainty. As well as allowing one to solve more complex and larger problems, the project will also lead to theoretical contributions in statistics and optimization.

  • Algorithmic Developments for Scalable Model Selection in High-Dimensions (Alex Gibberd)

    In many statistical applications there can be a relatively high number of model parameters compared to the number of data-points that we observe. In such settings, one typically aims to restrict parameters to some sub-space using regularisation methods, or explicitly allow only a small subset of parameters to active. Popular examples of these respective methods are the least-absolute shrinkage and selection operator (lasso), and exact selection methods such as AIC, BIC, matching-pursuit. This project will investigate algorithmic developments at the interface of these two model-selection paradigms to enable both low-bias estimation and accurate selection properties in low signal-to-noise statistical environments.

  • Large scale statistics (Steffen Grunewalder)

    One of the most ubiquitous themes in the 21st century is to make sense of the gigantic amount of data that is produced daily, in order to obtain potentially life changing insights into different important phenomena in such various fields as medicine, technology, environment only to name a few. So far, the traditional approach to organise, analyse, interpret and present data, has been to turn to classical statistical tools. Centuries of research in the field has given rise to an impressive toolbox of methods which go far beyond simple curve fitting. Yet existing statistical methods only play a minor role in the analysis of modern large scale data due to computational limitations. The lack of traditional methods for such data left an opening for ad-hoc methods which cannot offer the same amount of rigour and insight that traditional statistical methods provide but which are able to process extremely large amounts of data. This project is about developing non-parametric methods that can be applied to extremely large datasets. Their flexibility makes these non-parametric methods vastly superior to parametric methods for such data while their solid footing in statistical theory makes them superior to ad-hoc methods.

  • Developing time-varying parameter models in an exponential smoothing framework (Ivan Svetunkov and Alisa Yusupova)

    Exponential smoothing is one of the most popular model families in forecasting. The existing exponential smoothing framework is based on the fixed smoothing parameters, assuming that the update of level/trend/seasonal components is done in the same way for all the available data. In the past, there were some studies on the topic of adaptive smoothing parameter, but these studies were limited to the simplest exponential smoothing model and have not performed well in real life evaluations. In this project we expect the student to develop the mechanism for time varying parameters for the existing 30 exponential smoothing models.

  • Adaptive enrichment designs using machine learning techniques for subgroup identification (Fang Wan)

    Modern drug development is increasingly focused on identifying the most promising subpopulation to respond to a treatment, rather than a `one-size fits all’ approach. Adaptive enrichment designs have been developed for this purpose, but currently mainly focus on predefined subgroups or regression-based methods. The project will investigate the application of machine learning classification techniques, e.g. support vector machines; random forests; neural networks, to aid the identification of the promising subpopulation, while still ensuring family-wise error control. A starting point could be to build upon the general design proposed by Simon and Simon (2013).

    Reference: Simon, N. and Simon, R., 2013. Adaptive enrichment designs for clinical trials. Biostatistics14(4), 613-625.

  • Data-Driven Robust Optimization and Applications (Vikram Dokka and Matthias Ehrgott)

    Distributionally Robust Optimization (DRO) aims at finding solutions to decision-making problems that perform well under uncertainty/ambiguity, by encoding the possible realizations of uncertain parameters in a set referred to as the uncertainty/ambiguity set. Once this set is constructed, robust counterparts of the deterministic problems (when there is no uncertainty) are formulated to find robust solutions. This project is concerned with developing computationally efficient methods for modelling and solving data-driven DRO problems. As a start, the project will explore algorithms for DRO with mixed uncertainty sets. Secondly, we will explore modelling approaches for embedding tighter risk bounds within robust problems with a goal of constructing less conservative robust solutions. As a larger goal, the project will look into robust multi-objective optimization and will explore interfacing DRO with machine learning.

  • Spare parts inventory management using robust optimization (Anna-Lena Sachs and John Boylan)

    Managing inventories of spare parts is an important task. If a machine breaks, spare parts should be in-stock so that it can be repaired. This is of particular importance in industries such as healthcare and aviation. The demand for spare parts is often sporadic and difficult to predict. Some spare parts are very expensive and holding them in-stock in different locations is costly. The objective of this project is to develop new algorithms to solve this challenge using robust optimization. Robust optimization allows to consider uncertainty in the parameters and does not require distributional assumptions which can be an advantage for spare parts when it is difficult to fit a distribution.

  • Reinforcement learning algorithms for stochastic routing problems (Rob Shone)

    Machine learning (ML) algorithms are now widely used for teaching computers how to classify emails, recognise people’s faces and even play board games (Chess, Go). This project is about using a particular type of ML algorithm, known as “reinforcement learning” (RL), to search for optimal decision-making policies in rapidly-changing, probabilistic environments – which could be based on the routing of data packets in communications networks, mitigation of road traffic congestion, etc. From a mathematical perspective, this will require the use of Monte Carlo simulation and the development of appropriate metamodels to provide functional approximations of the costs and benefits associated with different action choices under different scenarios.

Remote supervision

Given the ongoing challenges created by COVID-19, supervision may be provided remotely for an initial period where necessary, with IT support from Lancaster University.

Entry Requirements

Applicants are expected to have, or shortly to be awarded, a minimum of an upper-second class honours degree, or its equivalent, in Mathematics, Statistics or a related field. Preferably applicants will hold, or be expected to achieve, a first-class honours degree in MSci or MMath Mathematics, a distinction in MSc Statistics or MSc Operational Research, or an equivalent qualification. Those who have excelled, or are excelling, at BSc level will also be considered positively.

Funding eligibility

These studentships cover up to four years of full payment of tuition fees at the Home/EU level plus an annual stipend for living expenses. Please note that we are unable to consider overseas candidates at this time.

Application process

All applications received before 1200 on Friday 20th November 2020 will be considered equally; applications received by this date are more likely to receive earlier decisions. Those received after that date are also welcome and will be assessed on a rolling basis. Guidelines on the application process may be found at on the university "How to Apply" pages. When applying please select "Applying for specific award" when asked about funding and write "Mathematical Sciences studentships 2021" in the free text box.

Commitment to Equality, Diversity and Inclusion

We are committed to family-friendly and flexible working policies, and seek to promote a healthy work-life balance. The University is a charter member of Athena SWAN and has held a Bronze award since 2008, in recognition of good employment practice to address gender equality in higher education and research. The Department achieved its own Athena SWAN Bronze award in 2017 and is a registered supporter of the London Mathematical Society’s Good Practice Scheme. We welcome applications from people in all diversity groups.

Contact

Those interested are encouraged to discuss their application with potential supervisors or the relevant contact: