2025 Summer Research in Maths (SRIM) Projects

This project would suit three students, who would work on related facets of the problem. The problem at its heart describes convection of water (the ocean) through a porous medium (the oceanic crust), driven by heat at depth, and involves coupled PDEs describing mass conservation, momentum conservation, and energy conservation, which governs heat transfer. One avenue to explore will be predominantly theoretical, at least initially, and will be focussed on questions of what conditions lead to the onset of convective flow in the system: this is the realm of linear stability analysis. Another will involve full numerical simulations, for which some pre-written code can be adapted to suit the problem at hand. Different physics will be incorporated by different students at this point: one project will explore the role of a localised heat source at the base, and what influence this has on heat transfer, patterns of flow and the general mechanics of the problem; another project will investigate the role of varying permeability through the crust, and the influence this has. Depending on progress, we can extend the work to investigate the role of various other physical effects on the problem. All students would be expected to contextualise their work by comparing with geophysical observations and other established models. A successful outcome for this project would be to produce insights into the physics that underlies some of the observable features associated with mid-ocean ridge systems, and to learn something about the way in which qualitatively different observable features correspond to different properties of the underlying system.

The project will involve both applied mathematical techniques, and writing, running and interpreting numerical simulations. Students should have a background in applied mathematics and fluid mechanics up to the level of a third-year mathematics undergraduate, and should have an interest in using mathematical techniques to model real-world problems. No previous numerical experience is required - although it may be helpful - but a willingness to learn is essential.

The project is determinate, with a fixed goal. Although the application of the project is to a certain differential equation, the methods are those of combinatorial geometry. The student will begin by reading the papers of Kodama and Williams and Dimakis and Mueller-Hoissen, in order to understand their work and the idea of the project. The student will then work out how to unify the two approaches by relating certain tropical varieties with cross-sections of tilings of cyclic zonotopes.

The pre-requisite knowledge is only linear algebra. Concepts from tropical geometry will be learnt during the project, as will objects such as zonotopes. The student will first get to grips with a simple example coming from the cube and its cross-sections. The later task will be to extend this example from cubes to cyclic zonotopes in general.

In more detail: the KP equation describes certain kinds of waves, and Kodama and Williams show how the crest of the wave may be approximated by a particular tropical variety. This tropical variety may be described as the dual to a cross-section of a tiling of a cyclic zonotope. Kodama and Williams consider this for three-dimensional tilings, where the cross-sections are dual to certain planar bicoloured graphs. Dimakis and Mueller-Hoissen work for arbitrary-dimensional tilings, but only consider the first cross-section. By considering all cross-sections in arbitrary dimensions, one can unify the two approaches.

The student will prove the results as follows. The relevant tropical variety is part of a particular hyperplane arrangement. The first step will be for the student to understand the relation between this hyperplane arrangement and the tiling of the cyclic zonotope, which is already well-documented in the literature. After this, the student will prove the duality between different tropical varieties coming from the hyperplane arrangement and different cross-sections of the tiling. Applying these results in our particular case will achieve the desired output of the project.

A group G is residually finite (RF) if for every nontrivial element g in G, there is a subgroup H of finite index in G which does not contain g. Many familiar classes of groups are RF (finite groups; free groups; finitely generated abelian groups) but there are also many ways to construct groups which are not RF. The starting point of this project is the behaviour of RF under two product constructions (a "product" here means a method of combining two groups together to give a new group): the free product and the wreath product. The free product of two RF groups is always RF; the wreath product usually isn't. A common generalisation of the free and wreath products is the "graph wreath product", which requires as inputs two groups and a graph on which one of them acts (roughly, the free product is the special case of the graph wreath product corresponding to an edgeless graph, and the wreath product corresponds to a complete graph). The goal of this project is to understand the circumstances under which graph wreath products are RF, in terms both of the group-theoretic properties of the two input groups and the structure of the input graph.

A first course in Algebraic Topology (comparable to the Part II course at Cambridge) is an essential prerequisite.

Geometry/Topology, Algebra/Number Theory, Algebraic Topology

Department / Group

This project will have two computational strands: first, we will simulate decaying MHD turbulence using newly discovered exotic choices of gauge that allow for long-range interactions between local patches of magnetic helicity. These experiments will elucidate the role of correlated fluxes on the conservation (or otherwise) of the magnetic-helicity fluctuation level. Secondly, we shall simulate the breaking of magnetic topology (magnetic reconnection), with the goal of elucidating the effect of viscosity on this process. In doing so, we shall aim to distinguish between two prominent conflicting theoretical descriptions of magnetic reconnection (with possible implications for the evolution of primordial magnetic fields in the early Universe).

The simulations described above will be carried out using codes written in C or C++, but no prior knowledge of these languages is required.

Solving inverse problems with data-driven methods (Machine learning, ML) is an emerging field with many mathematically interesting methods mixed with the most powerful machine learning techniques. In particular, we are interested in these methods for computed tomography (CT) applications. We are building a library of these methods to share with mathematicians, computer scientist and medical physicist alike, called LION (https://github.com/CambridgeCIA/LION).

This project will be a mix of reading and implementing existing scientific articles into LION, and helping with research in the group for new methods. This is an equally maths, machine learning and programming project, so familiarity with python (pytorch/numpy) is desired.

In some counting problems in algebraic geometry, one can spot partial patterns and potentially give closed formulas. This summer project aims to work on concrete examples of this type and unify the partial patterns and partial closed formulas. One approach is to use the method Mishra, Moulik, and Sarkar introduced and realise the partial formulas in their Conjecture Space.

In this project, we are primarily interested in finding patterns in specific integer sequences for which we already have some partial formulas. For example, we consider the sequences of the dimensions of the tangent space at the singular points of specific Hilbert schemes, which are essential in sheaf counting enumerative geometry. Another example would be counting the Weierstrass non-gap sequences associated with curves of a fixed genus, which are important in curve counting enumerative geometry

I am looking for an excellent student with experience in Machine Learning and a keen interest in algebraic geometry. 

C. Mishra, S. Moulik, and R. Sarkar. Mathematical conjecture generation using machine intelligence. arXiv:2306.07277, 2023.

F. Rezaee. Conjectural criteria for the most singular points of the Hilbert schemes of points. Experimental Mathematics, 1–16. Online first. https://doi.org/10.1080/10586458.2024.2400181. arXiv:2312.04520, 2023.

Hilbert schemes are natural and crucial spaces in algebraic geometry. However, they are known to be highly singular, and Murphy's Law has been proven for it: There is no geometric possibility so horrible that it cannot be found generically on some component of some Hilbert scheme [Harris-Morrison].

There has been tremendous growth in research in understanding the singularities of these spaces. Two possible directions for this summer project:

1) Proving some of the conjectures by the supervisor or some weaker versions of them that have been more recently developed (over the last summer projects). Resolving these conjectures will either resolve or be a step towards resolving a long-standing conjecture by Briancon and Iarrobino back in 1987, depending on the progress that will be made over the summer.

2) Providing an even weaker version of these conjectures, which, in turn, will also play a key role in coming up with an idea for proving the long-standing conjecture.

Joel Briancon and Anthony Iarrobino. Dimension of the punctual Hilbert scheme. J. Algebra, 55:536–544, 1978.
D. Eisenbud. Commutative Algebra: With a View Toward Algebraic Geometry. GTM, Vol. 150. New York: Springer-Verlag. 1995.
Joe W. Harris and Ian A Morrison. “Moduli of curves.” (1998). -Hu, X. (2021). On singular Hilbert schemes of points: local structures and tautological sheaves, arXiv:2101.05236.
J. Jelisiejew. (2020). Pathologies on the Hilbert scheme of points. Invent. Math. 220(2): 581–610.
J. Jelisiejew. (2023). Open problems in deformations of artinian algebras, hilbert schemes and around, arXiv:2307.08777.
R. Ramkumar. and A. Sammartano. (2022). On the tangent space to the Hilbert scheme of points in P3. Trans. Amer. Math. Soc. 375: 6179–6203
F. Rezaee. Conjectural criteria for the most singular points of the Hilbert schemes of points, Experimental Mathematics, 1–16. Online https://doi.org/10.1080/10586458.2024.2400181. arXiv:2312.04520.
B. Sturmfels. (2000). Four counterexamples in combinatorial algebraic geometry. J. Algebra 230: 282–294. -R. Vakil. (2006). Murphy’s law in algebraic geometry: badly-behaved deformation spaces. Invent. Math. 164(3): 569–590.

Scattering diagrams, introduced by Kontsevich and Soibelman, play a crucial role in mirror symmetry. Moreover, from the work of Bousseau, there is a correspondence between the scattering diagram of the projective plane and the associated Bridgeland stability space. Inspired by this and transferring the wall crossing in the stability space to the scattering diagram, recently, Gross and the supervisor have completed resolving the problem of divisors on the Hilbert schemes of points in the plane, which has been open for more than a decade (a partial solution was given in the PhD thesis of Huizenga at Harvard in 2012). Along the way, they discovered several picturesque properties related to the geometry of the scattering diagram itself. These observations potentially have applications in enumerative geometry and moduli theory, raising many interesting questions.

This summer project aims to apply these results in moduli theory. For example, one direction is to study some aspects of the moduli space of rank zero objects via the scattering diagram, for which the first step is to find a closed formula for the generating vertices of the rank zero objects.

I am looking for an excellent student with some experience with Algebraic Geometry and a keen interest in Combinatorics.

Neural operators have emerged as a powerful framework for learning mappings between function spaces, particularly for solving partial differential equations (PDEs) and modeling complex systems. However, most existing neural operators rely on supervised learning, which requires large amounts of labeled data, and are typically designed for structured, regular domains. These limitations hinder their applicability to real-world problems involving irregular geometries or spherical domains, such as climate modeling, geophysics, and astrophysics.

This project aims to address these challenges by developing two key advancements:

1. Unsupervised Neural Operators: Leveraging suitable numerical schemes to design physics-informed constraints and self-supervised techniques to reduce dependence on labeled data.

2.Targeted Neural Operators for Irregular and Spherical Domains: Adapting various neural operators to effectively handle inputs on non-standard geometries including irregular domains and spherical surfaces.

The use of machine learning techniques in scientific computing is gaining a lot of popularity, especially in the context of data-driven modelling of physical processes. Despite the fact that we might not have access to the exact equations ruling a given system, we are often aware of important dynamical or geometric properties that its solutions should satisfy. An example is provided by autonomous Hamiltonian systems, which conserve an energy function, a symplectic form, and a volume form. In this context, one can be interested in reproducing such qualitative properties into the neural network used to model them. This is the principle behind SympFlow, which is a symplectic neural network that can approximate the flow map of a Hamiltonian system.

This project aims to expand the class of problems to which SympFlow, see reference below, can be successfully applied. The specifics of which problems could be considered are left open, and we can discuss what to focus on before the project starts. Some options are:

• Exploring more efficient designs and implementations of the architecture to extend its applicability to higher-dimensional ODEs and PDE discretisations.
• Addressing problems where we do not have vast amounts of data but have some knowledge of the equations of motion.
• Learning how the flow map of a Hamiltonian system depends not only on the initial conditions but also on other parameters (e.g., masses, spring constants, or rod lengths).

The cosmic microwave background, which is the oldest light in the Universe, is the brightest signal in the sky at millimetre wavelengths. It has been travelling through the Universe for years, watching structure form and interacting with objects like galaxies. It picks up new signals from these, interactions such as by scattering off galaxies. We can use this signal to learn about where the galaxies are, but it will be much easier if we can separate the background light from the foreground signals.

Many signal separation techniques use frequency information but do not take advantage of the fact that we know something about the statistics of the different spatial distributions of these signals. Certain newly developed techniques have demonstrated such possibilities on other types of signals (such as on the separation of polarised galactic dust emission from the polarised CMB emission), and it would be of great interest to extend these techniques to the new regime we envision. This will be very timely given upcoming measurements of the millimetre sky in unprecedented detail with instruments like Simons Observatory.