2026 Summer Research in Maths (SRIM) Projects

Scientific machine learning has emerged as a powerful set of tools for modelling, simulating, and forecasting complex dynamical systems arising in science and engineering. A particularly promising framework is the Koopman operator, which provides a way to represent nonlinear dynamical systems using linear operators acting on an appropriately chosen space of observables [1]. This yields an effective linear representation of nonlinear dynamics, enabling the application of simple and well-understood linear simulation methodologies to otherwise complex systems.

In recent years, data-driven methods for learning Koopman operators from observations - rather than from explicit governing equations - have gained significant traction. These approaches have been successfully applied to time series forecasting in physical and engineering settings. Software packages such as PyKoopman [3] have further increased the accessibility of these methods to both researchers and practitioners.

A central challenge in many real-world applications is parametric generalisation. In practice, we often have abundant data for certain parameter regimes but only sparse or incomplete data for others. Examples include:

• Weather and climate modelling, where dense sensor data may be available over land but much sparser measurements over oceans.
• Engineering systems where experiments can be run only for a limited set of operating conditions (e.g. Reynolds numbers in fluid flows).

Recent work has begun to extend the Koopman operator framework to handle such parametric dependence, allowing models learned at a finite set of parameter values to generalise to others [2]. However, existing approaches largely assume discrete-time observations sampled at uniform time intervals. This assumption is often unrealistic: real data may be sampled irregularly in time, missing observations, or collected asynchronously. Addressing this limitation requires moving from discrete-time to continuous-time formulations of Koopman-based learning and prediction.

The aim of this project is to investigate parametric generalisation for Koopman operators, with a particular focus on extending existing discrete-time approaches to the continuous-time setting.

The project will have both theoretical and computational components and will proceed broadly along the following lines:

1. Literature review and introduction: review of recent work on parametric Koopman learning and generalisation, with particular emphasis on discrete-time methods; initial experimentation with existing software tools, especially the PyKoopman package.
2. Evaluation of discrete-time approaches: implementation of selected parametric Koopman methods in the discrete-time setting; evaluation of their performance on simple benchmark dynamical systems; understanding of assumptions and limitations related to time discretisation and data regularity.
3. Extension to continuous-time models: exploration of adapting parametric Koopman frameworks to continuous-time dynamics; investigation of approaches for handling parametric generalisation on unevenly spaced observations; preliminary mathematical analysis of how the Koopman operator behaves under parametric variation in continuous time (time permitting).
4. Computational implementation and experiments: development of a prototype implementation extending or interfacing with PyKoopman; evaluation of the proposed approach on the Common Task Framework for SciML [5] and, time-permitting, on real-world observational data from WeatherBench [4] or similar datasets.

There is flexibility for the student to focus their exploration on a desired aspect of this problem or for a group of students to work on complementary components.

Upon successful completion of this project the student is expected to have gained:

• A solid understanding of Koopman operator theory and its role in scientific machine learning.
• Hands-on experience implementing data-driven Koopman methods in Python.
• Insight into the challenges of parametric generalisation and irregularly sampled data.
• Experience producing a short written report summarising the methods, results, and possible future research directions.

This project takes a relatively simple fluid instability (the Kelvin Helmholtz Instability, KHI), such as would normally have been covered in a 2nd or 3rd year undergraduate course on fluid mechanics, and examines it in an exotic context -- that of a binary neutron star merger. The student(s) will study how the KHI can act as a mechanism for magnetic field amplification in the early stages of a neutron star merger. Within the first few milliseconds of impact, the merger of the neutron stars results in a linear shear layer which becomes unstable to the KHI. This is known to create vortices that can concentrate magnetic field. But how effectively can KHI vortices grow a magnetic field, and does the instability have time to saturate over the extremely short (ms) timescales involved? To answer these key questions, the student(s) will carry out 2D (magneto-)hydrodynamic (MHD) simulations of the KHI in special relativity using the astrophysical fluid dynamics code PLUTO. It is expected that the simulations will be run on a local cluster at DAMTP.

The project can be done individually or in a group, with different students working on different aspects of the problem (e.g. one student could carry out isothermal simulations to focus on the dynamics, while the other could focus on observational signatures of the instability by including thermodynamics and radiation, or by developing an analytical theory of relativistic MHD KHI to complement the simulations). Throughout the course of the project the student(s) will learn the basic physics of binary neutron star mergers. In addition the student(s) will also gain experience in computational fluid dynamics (in particular how to set-up and use one of the most popular open-source codes in astrophysical fluid dynamics, the PLUTO code), high performance computing (i.e. how to run simulations in parallel on a supercomputer), coding (particularly in C and Python), and data analysis. No background knowledge on mergers is necessary, though some background in fluid mechanics and astrophysics, and some familiarity with programming would be helpful.

Note that the topic is flexible and can be adjusted depending on the interests of the student(s). Alternative projects, within the confines of computational astrophysical fluid dynamics and accretion disks, are also possible.

This group project is made up of two connected projects:

Theory and Idealized Models: 
The student will tackle the problem using theoretical models and idealized 1D numerical simulations. By seeking a parameterized model—using specific closures and simplifications—this student will work from a high-level perspective to determine how general properties influence the scaling of these mixed layers. A successful project would produce a scaling law where the bottom mixed layer height depends on key physical quantities (such as stratification, the Coriolis parameter, and the turbulent dissipation rate). This result could then be applied to analyze global datasets of bottom mixed layers.

Numerical Simulations: 
The student will focus on high-resolution numerical simulations, likely using the state-of-the-art software Oceananigans. This work will simulate how flows interact with bottom topography (bathymetry) to generate internal waves, as well as the detailed physical processes of wave breaking that mix the bottom layers. A successful model would reveal the lifecycle of these mixing processes. Crucially, it would also quantify the feedback loop between mixing (which weakens stratification) and internal wave generation (which relies on stratification). This will lead to the detailed knowledge of the dynamical process of oceanic bottom mixed layer formation.

This project will look at some problems revolving around divergence-based inequalities motivated by specific applications. One concrete idea is to look at the interesting "reverse-Pinsker's inequality" from this paper: https://arxiv.org/pdf/2201.04735. This inequality underpins the powerful conclusion of this paper that some known "really hard to compute" problems can, in fact, be solved efficiently. Unfortunately, the reverse-Pinsker's inequality relies on an assumption (called "observability" in the paper) which is hard to make sense of, or even verify efficiently computationally. My hope is that one can establish a reverse-Pinsker under a more natural assumption that is easy to check. The project would then involve analysing simple examples or doing simulations to check if the new proposed inequality has any hope of being true (I think this is the case), and then guess and prove the right form of the inequality.

Prerequisite Skills

Other skills used in the project

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 dimensions of the tangent spaces at the singular points of specific Hilbert schemes, which are essential in sheaf-counting enumerative geometry.

There are two potential directions:

1. Using ML to unify the partial patterns: in this case, one approach is to use the method introduced by Mishra, Moulik, and Sarkar and to realise the partial formulas in their Conjecture Space.
2. A more theoretical direction is to combinatorially prove some relevant conjectures and use them to unify the formulas without ML.

Since the theoretical direction is expected to be mostly combinatorial, students with excellent combinatorial intuition or those with experience in Mathematics Olympiads/competitions are particularly encouraged to apply. I am seeking highly talented, motivated student(s), and, most importantly, someone committed to collaborative work, to work with me (and potentially an additional data scientist advisor, if the first direction is taken) on this project.

Psi classes are tautological cohomology classes defined on several fundamental moduli spaces in algebraic geometry. They play a central role in the study of the cohomology rings of these moduli spaces. For instance, the intersection theory of psi classes on the moduli space of stable curves is governed by the celebrated string equation.

In a lecture at the IHES, Rahul Pandharipande proved that psi classes on the moduli space of stable n-pointed curves of genus g are nef. This project investigates the positivity properties of psi classes on moduli spaces of genus zero stable maps to projective spaces. It builds on Pandharipande’s foundational results on intersections of rational divisors on these moduli spaces, as well as the description of their nef and ample cones due to Coskun, Harris, and Starr
 

The project will begin with a detailed study of Pandharipande’s proof of the nefness of psi classes on moduli spaces of stable curves. Building on this, the student will investigate the positivity properties of psi-classes on the moduli spaces of genus-zero stable maps to projective space of fixed degree.

The expected outcome is that suitable linear combinations of psi classes and pullbacks of O(1) via the evaluation morphisms lie in the nef/ample cone. The principal aim of the project is to establish this statement and to determine effective bounds on the minimal amount of twisting required.

Through this project, the student will acquire familiarity with positivity properties of line bundles, the geometry of moduli spaces of curves and stable maps, and tautological cohomology classes on these spaces. A successful outcome will consist of a solid understanding of these concepts and their application to proving the conjectural nefness or ampleness result described above, with the potential to lead to a research paper.

The student will meet with me regularly to discuss progress on the project and the necessary mathematical background. In addition, the student will have the opportunity to engage with Prof. Dhruv Ranganathan and with the Algebraic Geometry group of postdoctoral researchers and PhD students in Cambridge, providing a broader perspective on the problem and exposure to related techniques and ideas.

I am flexible with respect to remote working arrangements, should this be necessary, and this can be discussed and agreed upon at a later stage.

The Einstein equations in the near-singularity regime have been shown by Belinski, Khalatnikov and Lifshitz in the 60-70s to drastically simplify. The dynamics at neighbouring points on a spatial slice decouple and admit a useful description in terms of single particle motion on hyperbolic billiards. The specific billiards that emerge from this approach have peculiar symmetries and display so-called arithmetic quantum chaos. The properties of this type of dynamical systems are in sharp contrast with the usual lore of quantum chaos and could be a hint for fundamental symmetries that underlie gravitational dynamics.

The goal of this project would be to investigate properties of complexity in the hyperbolic billiards that feature arithmetic chaos. During this project, the student will have the opportunity to learn about various topics, such as seminal work on the generic solutions of general relativity close to a spacelike singularity, recently proposed descriptions of complexity of quantum dynamics and its relation to various types of chaos, as well as numerical tools to solve billiard eigenvalue problems. The aspects of this project that go beyond literature study would be mainly numerical at first.