2024 Summer Research in Maths (SRIM) Projects

A *commutator* is an element [a,b]:=aba^{-1}b^{-1} in a group G. The notation num(g) denotes the number of ways that an element g can be written as a commutator. (The full definition involves an equivalence relation, and is not completely obvious.) The purpose of this project is to investigate the behaviour of num(g) in the case of a non-abelian free group F. More precisely, we will investigate the maximum M(F)=max(num(g)) as g ranges over all non-trivial elements of F.

The first result about M(F) is a 1981 theorem of Lyndon—Wicks, who proved that M(F) is at least 2. This remains the only known lower bound.

The first upper bound comes from a general theorem of Sela from around 2000, which implies that M(F) is finite. However, the proof is non-constructive, and gives no finite upper bound. Beyond this, the state of the art is an unpublished result of Louder, who showed that M(F) <= ~10,000.

The goal of this project is to investigate what’s known about M(F), and to try to improve the best known bounds. Although the definition of M(F) is algebraic, it can be rephrased in terms of maps from surfaces (specifically, a torus with one boundary component) to graphs. The techniques used will probably be combinatorial and geometric, involving drawing pictures of labelled graphs. Some experience with coding may also be useful in trying to improve Louder’s upper bound.

Background reading on free groups and graphs:
Stallings, J.R. Topology of finite graphs. Invent Math 71, 551–565 (1983).

The state of the art on this problem:
Bestvina and Feighn, Counting maps from a surface to a graph, https://arxiv.org/abs/math/0505363v1

The phenomenon of motile microorganisms in fluids, such as bacteria, congregating close to (or away from) surfaces has been widely observed in many different contexts. The surface accumulation or depletion of these motile microorganisms, or microswimmers, is often attributed to drive biofouling and biofilm formation. While it seems to be a generic behaviour among many species, the specific processes underlying this behaviour can vary depending on the flow condition, type of surfaces and the microorganisms' swimming mechanism, speed and size. Mechanisms that can give rise to surface accumulation include, but are not limited to, biological taxes, far-field hydrodynamic interactions with the surface [1], coupling of stochastic swimming direction and flow [2], coupling between diffusion and swimming [3], and shape of the microswimmer and the wall [4]. Despite the distinct length scales and timescales associated with each process, a comprehensive study encompassing all these mechanisms and a definitive guideline for discerning between them based on observations are still lacking.

In this project, student will be given an existing code (in JULIA) that simulate microswimmers in a long, narrow channel with a Poiseuille background flow. The first part of the project is to understand the code and replicate the result of [2] and [3]. We shall then study how varying parameters such as channel width, microswimmers' size, shape and stochasticity can give rise to or suppress different accumulation mechanism. Lastly, the student is expected to write up a short report on how to distinguish different mechanisms driving accumulation based on given parameters.

This project is designed for a student with a strong background or interest in coding, numerics and stochastic processes. While prerequisite to JULIA is optional, the student embarking on this project is expected to learn the language and the Git workflow on the fly. Students who have pre-existing knowledge on Stochastic Ordinary Differential Equations (SDEs), Brownian motion and diffusion, and fluid dynamics are preferred.

1. Lauga, E. The Fluid Dynamics of Cell Motility
2. Vennamneni, L., Nambiar, S. & Subramanian, G. (2020). J. Fluid Mech., 890, A15.
3. Ezhilan, B. & Saintillan, D. (2015). J. Fluid Mech., 777, pp. 482-522.
4. Chen, H. & Thiffeault, J.-L. (2021). J. Fluid Mech., 916, p. A15.

Papers below are not required as we use elementary methods (books will be useful though), but I just provide some references just in case the student wants to have a look at some texts in the literature to motivate what a subtle object we propose to compute via elementary tools (4-7).

The first three are books for commutative algebra and toric geometry background. Others are more advanced papers (the last two are more relevant).

1. M.F. Atiyah, .G.MacDonald,Introduction to commutative algebra,Addison-Wesley Publishing Company, 1969.
2. David Eisenbud, Commutative algebra, Graduate Text in Mathematics,vol.150,Springer-Verlag,1995.
3. David A. Cox, John B. Little, Henry K. Schenck: “Toric Varieties”
4. Kai Behrend, Donaldson–Thomas type invariants via microlocal geometry, Ann. of Math. 2 (2009), no. 170, 1307–1338.
5. Kai Behrend and Barbara Fantechi, The intrinsic normal cone, Invent.Math.128(1997),no.1,45–88.
6. F. Rezaee, A quotient formula for the Behrend function and its combinatorial interpretation, preprint (available upon request)
7. M. Graffeo, A. Ricolfi, On the Behrend function and the blowup of some fat points, Advances in Mathematics, Volume 415 (2023), 108896.

Bacteria typically form surface-associated communities (biofilms) that are critical for microbial biology and their impact on us, including the ability to promote health or disease in animals and plants. While biofilm research has been ever increasing, most studies on biofilm growth have focused on unconfined conditions, particularly on the expansion of bacterial colonies above solid or semi-solid surfaces (e.g., 1, 2). In natural and clinical conditions, however, bacteria are often exposed to spatial confinement such as the vascular systems of their hosts, and we still know little about the governing dynamics of biofilm growth under confinement, namely the interplay between radial expansion and vertical pushing.

We have recently developed an experimental and theoretical framework to study biofilm growth under elastic confinement, which considers one of the simplest confined geometries, a bacterial biofilm growing in the space between a solid surface and an overlying elastic sheet (3). In particular, we considered a linear poroelastic model that led to a governing equation for the radius of the biofilm, whose dynamics matches our experimental data of biofilms growing monotonically outwards. More recently, we developed an experimental and image analysis pipeline that allowed us to correlate the radial (lateral) expansion with the pushing (vertical) dynamics of confined biofilms, and found that these two components are intrinsically linked, one determining the other (4), which further validates our analytical model that had predicted this dependence. This research project aims to expand on this work to further our understanding of the interplay between radial expansion and vertical pushing of biofilm growth under confined conditions where growth, namely under conditions where growth is not monotonic as we have considered so far but oscillatory (5).

While spectral methods for time-dependent PDEs have been around for a long time, only recently we are gaining broader understanding on their design and analysis. This will be the subject matter of this project.

A spectral method starts by specifying an orthonormal basis of the underlying Hilbert space – once this is done, we can convert a time-dependent PDE into an infinite-dimensional ODE system by Galerkin conditions and solve its finite-dimensional truncation. Thus, everything starts with an orthonormal basis, which need be selected while balancing a number of criteria:
• Stability and well-posedness
• Fast convergence
• Preservation of qualitative features, e.g. mass or dissipativity
• Ease of expansion
• Ease of linear algebra computations

The project will survey a number of contemporary approaches in the choice of orthonormal bases, in particular T-systems and W-systems. Otherwise it is fairly open ended and might pursue a number of possible directions, depending on the interests of the student. It might be theoretical or computational in flavour and might explore some open problems, in particular applications to the computation of dispersive equations of quantum mechanics.

The realm of numerical partial differential equation (PDE) solutions remains a perpetual focal point in the domain of numerical analysis, brimming with mathematical intricacies. This is most pronounced when tackling nonlinear PDEs, PDEs residing in high-dimensional spaces, and PDEs defined over intricate domains - scenarios frequently encountered in real-world problem spanning from turbulent flows and mechanical design to modelling the stock market. The crux of the matter invariably revolves around achieving a fine-grained discretisation capable of capturing the underlying phenomena or objects accurately.

Conventional solvers, typified by finite element methods, perpetually pose a conundrum, demanding a delicate balance between computational load and precision. In recent years, an avant-garde solution to this age-old problem has emerged in the form of machine learning (ML), with deep neural networks. Herein, the solution to a PDE is approximated by a deep neural network, its parameters fine-tuned through training to minimise a devised loss function rooted in the essence of the PDE. This approach has, in the last few years, ignited a surge of interest within the mathematical community. It has not only outperformed classical techniques like finite element methods for certain categories of PDEs, notably high-dimensional ones, but has also excelled in specific numerical regimes, especially when swiftness in approximating a solution is imperative.

One pioneering concept that has emerged along these lines is the notion of physics-informed neural networks (PINNs) [1,2]. However, PINNs grapple with the computational overhead associated with their training phase, entailing the resolution of a per-instance optimisation challenge. Conversely, another cluster of techniques has steered away from the intricacies of mesh-bound approaches, as exemplified in [3,4]. Recent strides [5,6] have approached the problem by casting it as a learning targeting infinite-dimensional operators - such as solution operators intrinsic to PDEs.

The rise of Machine Learning (ML) means that many fields, including mathematics, have been exploring uses for solving complex non-convex optimization problems. In particular, this project touches on the use of ML for Computed Tomography (CT) reconstruction, an inverse problem with a very common application in medicine. The use of ML for the task of CT reconstruction is quite new, and therefore many methods exist in a seemingly chaotic state of the literature.

Our research group Cambridge Image Analysis (CIA) (lead by Prof. Carola-Bibiane Schonlieb) has been an active part of the developments of these methods. Currently, a big push of the project is to provide a unified repository of methods that are comparable, with the aim of breaching the gap between Mathematics research and applications. Project named Learned Iterative Optimization Networks (LION, https://github.com/CambridgeCIA/LION).

The task within the LION framework would be to study and implement a Mixed Scale-Dense (MS-D) network in the latest Python/Pytorch versions, and integrate it to LION. Then, to test its performance with supervised and unsupervised ML CT tasks, such as Noise2Inverse.

The project would require to learn about ML models, in particular MS-D model, and successfully learn how these are implemented in practice. Then learning how to use medical data to train such models to produce successful reconstructions, and compare it to other existing methods.

This project would teach a student everything they need to successfully train Machine Learning methods. It is a very applied project, in where most of the work will be programming and testing ML methods, rather than developing new mathematical formulations, but there is space to do that if the implementation is finalized in an early stage.

https://www.pnas.org/doi/10.1073/pnas.1715832114
https://arxiv.org/abs/2001.11801

The emergence of deep learning has transformed the approach to optimisation problems in applied sciences and mathematics. The concept of algorithm unrolling is a focal point within the paradigm of learning to optimise (L2O) method, recognising the burgeoning interest due to their potential in crafting efficient, high-performance, and interpretable network architectures from reasonably sized training sets. It takes the form of x^{k+1} = f(x^k; \theta^k), for k = 1, 2, …, K-1 [1,2], which unfolds a truncated optimisation algorithm within the architecture of a neural network. Traditional learning to optimise technique enjoys deep learning models with redundant training, while recent work[3] has introduced the single-shot concept in the field, which minimises the need for extensive training data. This concept aligns well with algorithm unrolling, as it necessitates end-to-end training.

The aim of this project is to open a new research line for algorithm unrolling, drawing attention for minimal training data. The primary emphasis will involve exploring a new single-shot strategy for neural networks, followed by seamlessly integrating this innovative strategy into the algorithm unrolling framework.

Summer projects: Inverse problems are concerned with reconstructing unknown model parameters from indirect and noisy measurements and arise in practically all imaging applications. In the last decade, thanks to deep learning, processing of imaging data has undergone a paradigm shift from knowledge driven approaches, deriving imaging models from first principles, to purely data driven approaches, instead deriving models from data. Most inverse problems of interest are ill-posed and require appropriate mathematical treatment for recovering meaningful solutions and while purely data-driven learning have been able to achieve remarkable empirical success for image reconstruction, they often lack rigorous reconstruction guarantees. Of particular interest are therefore methods that operate at the interface of these paradigms and feature both a knowledge driven (mathematical modelling) and a data driven (machine learning) component.

The precise details of the project will be tailored to the student's interest and skill set, but possible topics/projects for investigation include:

* Acquisition (operator)-aware learned reconstruction [1,2]
* Combining physics informed neural networks with learned regularisation [3,4,5,]
* Structured non-convex learned regularisation [6,7]

Gaussian processes are powerful probabilistic machine learning models, with particular prominence among many scientific disciplines, like in cosmology, molecular search, and climate forecasting. One reason for their prominence is the possibility they provide of directly encoding prior scientific knowledge into otherwise purely data-driven machine learning tasks. In particular, The choice of the Gaussian processes’ kernel function controls the covariance of the learned processes, with specific choices of kernel corresponding to different physical prior knowledge, e.g. conservation laws, symmetries, smoothness, or the satisfaction of particular differential equations.

However, when constructing predictive models for systems or quantities with high-dimensional inputs, particularly in scenarios with limited data, the necessity arises to incorporate additional structural assumptions. This project aims to explore methods for efficiently encoding such structures by leveraging well-established mathematical expressions, reminiscent of the renowned Girard-Newton identity.

The selected student will become an expert in Bayesian ML ( e.g. Gaussian processes), while concurrently scouring the mathematical literature to identify relevant extensions to the Girard-Newton identity. A key part of the project will be in building open-source tooling to enable the use of the project’s outcomes by the wider climate community.