PhD Studentships in Mathematical Sciences

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.

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 S_N of degree N. One speaks of a random permutation if one selects randomly a permutation of S_N. In other words, one endows S_N 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 n^a 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.

This project revolves around the notion of derived localization. Localization is one of the fundamental processes in mathematics; one of its elementary applications is the construction of integers out of natural numbers and rational numbers out of the integers. In the noncommutative context, localization exhibits a variety of subtle phenomena, and is best understood in the context of homotopical algebra and closed model categories. It is relevant to higher categories, stable and unstable homotopy theory, derived and triangulated categories, deformation theory etc. The project will explore foundational aspects of derived localization, particularly in the differential graded context, and its applications.

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.

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 L^p-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.

• Y. Choi and M. Ghandehari, Dual convolution for the affine group of the real line. Preprint, arXiv 2009.05947
• Y. Choi, Constructing alternating 2-cocycles on Fourier algebras. Preprint, arXiv 2008.02226
• Y. Choi, Directly finite algebras of pseudofunctions on locally compact groups. Glasgow Math. Journal 57 (2015) no. 3, 693-707
• Y. Choi and M. Ghandehari, Weak and cyclic amenability for Fourier algebras of connected Lie groups. J. Funct. Anal. 266 (2014), no. 11, 6501-6530

Hyperfiniteness of graph classes plays an important role in analysis, computer science and combinatorics, since these classes are the most amenable for learning and testing. Also, the notion of hyperfiniteness is strongly related to group theory.

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.

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 symplectic 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 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.

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.

This project looks at bounding the maximal decay rate at infinity for solutions to ordinary differential equations with operator coefficients. One example is an elliptic evolution equation of the form (∇_t²-A)u=B_tu, where A is a lower semi-bounded self-adjoint operator, and B_t is some kind of perturbation. Certain types of super-exponentially decaying solutions are possible in general, but this can be limited to at most exponential decay under some assumptions about gaps in the spectrum of A. Such results can be viewed as unique continuation theorems at infinity, and can be obtained from Carleman-type estimates, combined with some spectral analysis of A. There are many potential problems to consider, both generalising the types of operator coefficient ordinary differential equations, and weakening the assumptions on A (possibly by balancing these assumptions against conditions related to the interaction of A and B_t).

One motivation for studying such problems is their application to equations arising in mathematical physics. One example is the Schroedinger equation (-Δ+V)u=0 in R^d. In certain cases where V is a periodic potential a new proof of the absolute continuity of the spectrum of the operator can be obtained. There are possibilities to extend work in this line, and also consider some long standing questions of Landis and Simon.

Let F be a differential field of complex functions with the complex numbers as the subfield of constants. Liouvuille systematically considered the process of solving a linear differential equation with coefficients in F by:

1. solving a polynomial equation with coefficients in F and adjoining the roots;
2. taking the antiderivative of a function in F, and adjoining it to F;
3. taking the exponential of a function in F, and adjoining.

This initiated the differential Galois theory of linear differential equations, which characterizes the ODE that can be solved by repeatedly applying Liouville’s operations.

In the 1980s, Pöppe considered another basic operation, namely taking a Hankel integral operator on L²(0, ∞) with kernel φ(s + x + y) where φ ∈ F and forming the corresponding Fredholm determinant. His motivation was to solve integrable nonlinear partial differential equations such as Korteweg-de Vries, and obtain explicit solutions of in terms of determinants of integral operators. He discovered some differential and multiplication rules for Hankel operators.

In a parallel development, Tracy and Widom considered Hankel integral operators that have symbols given by the Airy and Bessel functions. They introduced Fredholm determinants for products of such Hankel operators, and derived fundamental distributions in random matrix theory. In particular, they introduced a solution q of the Painlevé II differential equation via solution A(x) of Airy’s equation such that q(x, λ) ∼√λA(x) as x → ∞. Then

D₂(s; λ) = exp(−∫_s^∞(x − s)q(x, λ)dx)

and its derivatives with respect to λ can be used to express the cumulative distribution function F₂(s, m) of the mth largest eigenvalue of the Gaussian unitary ensemble. This result has important application in multivariate statistics, where one commonly assumes that the underlying distribution is multivariate normal, and one wishes to consider sampling distributions.

In a paper in press, Blower and Newsham considered Pöppe’s calculations from a more general viewpoint. Any bounded self-adjoint Hankel operator can be realized as the scattering function of a linear system in continuous time. The paper gives multiplication and differentiation rules for algebras of operators associated with linear systems, and related their properties to those of Fredholm determinants. The aim of the proposed PhD project is to analyze the linear systems closely related to the Tracy-Widom models and develop a differential Galois theory for the corresponding nonlinear ordinary differential equations.

• G. Blower and S.L. Newsham, Tau functions associated with linear systems, Operator Theory: Advances and Applications; IWOTA Conference Proceedings, Birkhauser, 2020
• C. P¨oppe, The Fredholm determinant method for the KdV equations, Physica D 13 (1984), 137-160.
• C.A. Tracy and H. Widom, Fredholm determinants, differential equations and matrix models, Comm. Math. Phys. 163 (1994), 33-72

Graph rigidity is concerned with the structural rigidity of discrete objects defined by a set of geometric constraints. These constraints (specifying angles, lengths, 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 given constraints).

Combinatorially, analysing the underlying graph of the system and its rigidity matrix/matroid are crucial in understanding the generic behaviour. This project will use techniques from linear algebra, discrete geometry and combinatorics to develop new results in graph rigidity. In particular, a key direction could be the analysis of rigidity properties for classes of graphs that exclude certain minors.

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.

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.

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.

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.

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. Biostatistics, 14(4), 613-625.

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.

Matching supply and demand are at the core of successful supply chain management. We have all witnessed this importance at the beginning of the COVID-19 crisis when stores ran out-of-stock for important products such as flour and toilet rolls. In this project, we aim to develop novel methods to forecast demand and manage inventories effectively when a supply chain faces disruptions. In particular, we investigate the interface between forecasting, ordering and inventory management. The methods we develop are based on statistical methods for forecasting, stochastic optimisation for inventory management, and an integration of both forecasting and inventory management using robust optimisation.

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.

In recent years, cyber-physical systems have advanced in a way that has made robot-human cooperation and collaboration on the shop-floor inevitable. In this research, the researcher will study complexity theory and, through computational modelling and simulation, will simulate the possible patterns of behaviour of social and automated agents to determine the impact on safety critical systems.

We are flexible with the domain of interest for this PhD project and happy to agree the details during the 1st year of the PhD. Our current ideas, to facilitate a challenging PhD that has high real-world impact, revolve around: Manufacturing, Autonomous Vehicles, Robotics in Healthcare (e.g. Elderly Care), or Operation Theatre Rooms in Hospitals.

This PhD project will involve primary data collection (e.g. surveys or interviews) from the domain of interest; development of a conceptual model of the domain; and development of a computational model in order to perform simulation-based experimentation – we propose System Dynamics (SD) and Agent-Based Modelling and Simulation (ABMS).

Random Regression Trees and Forests are among the most popular tools used to perform non-parametric regression and classification. They generalize linear regression by allowing a piecewise linear regression function, generated using a random tree to partition the covariate space. In the Bayesian version of Decision Trees, the random tree is sampled from the posterior distribution using Markov Chain Monte Carlo (MCMC). However, the MCMC algorithms currently used to infer the underlying tree tend to mix poorly and be inefficient. A second limitation is that there is currently limited understanding of the theoretical properties of the model. The aim of this project is to address computational, methodological, and theoretical improvements of Bayesian trees and forests. Depending on the background and interests of the student, the project can either focus on statistical and computational (MCMC) problems or on theoretical and probabilistic problems related to branching process theory.

Satellite altimeters are used to monitor the response of the polar ice sheets to climate change. Radar altimeters operate by firing a microwave pulse towards Earth’s surface and listening to the returned echo. To convert this raw signal into climatological information requires a sequence of processing steps; and these have remained largely unchanged and unchallenged for several decades. This current approach is, however, subject to several limiting factors; (1) it relies upon overly simplistic assumptions relating to the physical signal being observed (namely the characteristics of the returned echo), and (2) it reduces the dimensionality of the data (from a vector to a scalar). As a consequence, hard-to-quantify errors are introduced into the end products, and useful information is lost along the way. This project aims to develop a new approach to processing altimeter echoes to infer the observed surface elevation. The project will develop and apply techniques from functional regression (regression where the predictors and/or responses are functions instead of numbers) and inverse problems (when one observes “y” and wants to predict “x”) to address the challenges posed by the data.

Infinite-server queueing models are well-suited to capacity planning contexts where operational decisions need to be made if there is significant chance of queues occurring, e.g. hospital operations need to be postponed if ward capacities are likely to be exceeded. Such systems can be modelled mathematically or by computer simulation, with the former approach being computationally faster and more insightful, and the latter being more flexible. This research is to build on previous research at Lancaster, to extend the range of systems that can be tackled analytically and to investigate possible ways in which the mathematical and simulation approaches can complement one another.

Various probabilistic representations exist for multivariate extreme values, each lending themselves to different statistical methodology. The success of each approach depends on the (unknown) underlying dependence structure and part of the statistical task is to determine which method is likely to yield the best results. This project will draw on new insights into how these representations are linked (https://arxiv.org/abs/2012.00990) to work towards a connected strategy for statistical exploitation.

For networks in which pairwise interactions between individuals are concerned, various classes of statistical models, called random graph models collectively, have been developed for different purposes, ranging from understanding how networks evolve and grow, to clustering the nodes into communities, and predicting unobserved edges between nodes. Computational advances have also been made for statistical inference of random graph models, by taking into account the uniqueness of network data.

There also exist networks in which more than two individuals can be involved in an interaction. An example is co-authorship networks, in which an academic article can be written by more than two authors. The quest for modelling this kind of networks presents some interesting challenges. On one hand, applying random graph models directly will lead to loss of information. On the other hand, hypergraph models, which are the generalisation of random graph models, are more natural for co-authorship networks but have not received the same level of attention. In this project, we will explore how we can apply the existing methods and techniques for random graph models, by treating co-authorship networks as bipartite networks.

Reference: Lung, R.I., Gaskó, N. & Suciu, M.A. A hypergraph model for representing scientific output. Scientometrics 117, 1361–1379 (2018). https://doi.org/10.1007/s11192-018-2908-2

Autonomous decision-making agents for complex systems need to classify different system states and recommend actions. For instance, in telecommunications networks different traffic patterns are indicative of different issues and should be brought to the attention of different parties. Similar problems arise in triaging hospital patients, monitoring potentially fraudulent credit card transactions, and many other settings. This project would explore online learning schemes for such classification problems, and focus on the development of algorithms which can handle limited feedback – i.e. when the true class of a state is only infrequently observed, and this observability may depend on the action taken. The work will build on ideas from ‘multi-armed bandits’ as well as classification algorithms and could have a substantial theoretical as well as computational component depending on the interests of the student.