PostgraduateResearchDay

May 12, 2026
MCS 0001

The Durham Maths postgraduate research day is a celebration of the research being done by postgraduates in the department. It is an opportunity for students and postdocs to present their work to their peers via short accessible talks and to learn about the research being done by others.

Schedule

Abstracts

Yue Jiang - A sub-Riemannian flavored dynamic optimal transport problem

In this talk, I shall present a dynamic optimal transport problem arising from the linear control system (something more applicable to real life) which is endowed naturally a sub-Riemannian structure. I will also present its equivalence to a static optimal transport problem. Moreover, it turns out that the equivalence extends to the uniqueness of minimiser of the two variational problems. Basics on optimal transport theory and sub-Riemannian geometry will be briefly revisited.

Hei Jie Lam - Game Theory with a lot of players

In this talk I will explain the basic concepts of Nash equilibria in games with finite number of players and the corresponding limiting problem when we would like to approximate a game with many players using a mean field approach. We will then discuss some real world applications of Nash equilibrium theory.

Patrick Creagh - The weight part of Serre's conjecture

In 1987, Serre conjectured a deep connection between 2 dimensional, mod-p representations of the absolute Galois group over the rational numbers and modular forms and gave a recipe for this correspondence. This has become known as Serre's weight conjecture and Serre's original statement was proven in 2008 by Khare and Wintenberger. In the modern day, work is being carried out to generalise this to higher dimensional representations and even more generally to reductive groups. Finding the correct `weight' for a higher dimensional representation has proven to be the most difficult part of the problem. In this talk I hope to give an indication of the weight part of the problem for GL2 from a representation theoretic point of view with a view to how this generalises for higher dimensional representations.

Samuel Shepherd - Geometric and Arithmetic Degrees on Unitary Shimura Varieties

We discuss the problem of constructing generating series for the geometric and arithmetic degrees of special cycles on locally symmetric spaces associated to unitary groups. Via the machinery of the Siegel--Weil formula, these generating series are given as Eisenstein series for SL(2). We mention briefly the significance of such results in the broader landscape of the Kudla programme and arithmetic geometry.

Victoria Pelayo Alvaredo - Concentration and Median Leaves

We briefly recall the Lévy–Gromov concentration phenomenon under a positive lower Ricci curvature bound. We then discuss how, in the setting of codimension one singular Riemannian foliations, concentration can be localized near distinguished regular leaves.

Ryoki Endo: The second Dirichlet eigenvalue is simple on every non-equilateral triangle

The interaction between Laplacian eigenvalues and the shape of the underlying domain lies at the heart of spectral geometry. In this talk, I will present a computer-assisted proof, joint with Xuefeng Liu, of a conjecture of R. Laugesen and B. Siudeja recorded as Conjecture 6.47 in Henrot's Shape Optimization and Spectral Theory: the second Dirichlet eigenvalue is simple on every non-equilateral triangle.

Jost Pieper: Some recent advances in Rough Stochastic Analysis (with jumps)

The study of Rough stochastic differential equations (initiated by [FrizHocquetLê2021]) is a recent development at the interface of stochastic analysis and rough path theory and has led to several active research directions. It especially is a natural way to study conditioned doubly stochastic systems with applications in stochastic control theory as well as the study of robustness results for stochastic filtering and the rough counterparts of the associated SPDEs. In this talk, I will introduce some basic ideas from rough path theory and explain how rough SDEs allow one to benefit from the stability guarantees of rough path theory in conditioned (doubly) stochastic systems. If time permits, I will also discuss some recent work of mine concerning rough stochastic differential equations with jumps, including challenges related to pathwise formulations, maximal solutions, as well as Malliavin calculus for rough SDEs.

Yanpeng Zhi - Elliptic Phi_4^4 Model with Boundary Blow-up: Triviality and Renormalization

Phi^4 model refers to a class of stochastic PDEs related to construction of Phi^4 Euclidean quantum field theory. They involve cubic non-linearity, driven by white noise and usually called stochastic quantization equations. There has been extensive research on parabolic and hyperbolic Phi_2^4 models, where the subscript "2" means 2-dimensional domains. In this talk, I will present my recent work for elliptic models on the 4-dimensional hypercube, imposed with an infinity boundary condition. Without renormalization, the regularised equations will converge to something deterministic as white noise recovers, which is the triviality property shared with parabolic and hyperbolic settings. The limit is a PDE if noise intensity is weak, and is zero if noise strength is strong. To find a meaningful stochastic object solving elliptic models, we will prove existence of renormalized solutions. This requires renormalizing suitably the non-linear term in equations, and yields non-trivial limits as noise regularisation is removed.

Zhaocheng Li - Bayes Linear Analysis to Models with Multi-partial Discontinuities with UQ

The simulation for real-world systems can be computationally intensive or even infeasible to compute the acceptable parameters. We analytically build an emulator using the Bayes Linear Analysis to adjust the expectation of our output, provide the uncertainty quantification, and achieve the feasibility to surrogate the simulator. This methodology hardly works with the scenario of models that are partially discontinuous, as its exponentialized kernel functions are at least continuous and more-or-less differentiable. We will provide some possible extensions with statistical estimates to adjust our running input design to solve some typical cases.

Tianlin Yang - A Distribution-to-Distribution Neural Framework for Probabilistic Forecasting of Dynamical Systems

This talk presents a distribution-to-distribution neural framework for probabilistic forecasting in dynamical systems. Instead of predicting individual trajectories, the method models the evolution of predictive distributions directly, allowing uncertainty to be represented and propagated in a unified way. I will also briefly show results from the Lorenz63 system to illustrate its ability to capture nonlinear distributional dynamics.

Shrog Albalawi - Assessing variable influence on change points in sparse high-dimensional data

Detection of change points in sequences of high-dimensional observations remains a challenging problem, especially in sparse settings where only a small subset of variables may be affected. Beyond detecting whether a significant change point is present and estimating its location, the main objective is to identify which variable, or group of variables, drives the detected change point. This work studies the problem in a post-detection setting, where variables are grouped according to a chosen criterion and their relation to the detected change point is then examined. Such grouping provides a natural way to reduce dimensionality from p variables to K groups, with K < n ≪ p, while improving the interpretability of sparse high-dimensional data. Numerical simulations illustrate the performance of this framework and indicate improved results as the sample size and signal strength increase.

Ngoc Nguyen - Data distribution valuation using Generalised Bayesian Inference

We investigate the data distribution valuation problem, which aims to quantify the values of data distributions from their samples. This is a recently proposed problem that is related to but different from classical data valuation and can be applied to various applications. For this problem, we develop a novel framework called Generalized Bayes Valuation that utilizes generalized Bayesian inference with a loss constructed from transferability measures. This framework allows us to solve, in a unified way, seemingly unrelated practical problems, such as annotator evaluation and data augmentation. Using the Bayesian principles, we further improve and enhance the applicability of our framework by extending it to the continuous data stream setting. Our experiment results confirm the effectiveness and efficiency of our framework in different real-world scenarios.

Joseph Marshall - A Flat Pack Universe

In particle physics we have a problem: our theories are too good. For around 100 years now, quantum mechanics (and later, quantum field theory (QFT)) have provided us with the framework to make predictions for phenomena to very high energies with very high precision, and, barring a few notable exceptions, the experiments keep agreeing with theory as well as could be hoped. This sounds like a nice problem to have, but of course we also know that QFT can’t be the whole story, as it doesn’t provide a consistent quantum description of gravity. One place to look that may give us some insight into how the two interact is the very early universe, when gravity was incredibly important and small quantum fluctuations were blown up to large scales, seeding perturbations that can be observed today. In this talk, I aim to give a (very) brief and introductory look at quantum mechanics and cosmology, followed by a surface overview of the framework developed in my research in which quantum objects in the early universe can be directly related to those in “flat space” (in the absence of gravity). I aim to review these topics at a very basic and pedagogical level, assuming only a familiarity with linear algebra.

Samuel Gunatilleke - Categorical Symmetries

Categorical (or 'generalised', to a less mathematical audience) symmetries represent a paradigm shift in the way physicists talk about symmetry; gone are the days where a symmetry is a collection of associative operations with a unit and inverses (i.e. a group). In this talk I will introduce the language and definitions of categorical symmetries from the perspective of a (physical) mathematician, and present some of the key applications for the field, as well as a selection of areas anticipating further development. Neither categorical nor physical knowledge will be required.

Sam Kay - Switchbacks 101

I introduce and present the latest knowledge on mysterious structures in the solar wind known as magnetic switchbacks, and how I aim to model them in order to better understand them.

Jeremy Cai - Mathematics of SAXS with an application to Protein RNA docking

Proteins and their complexes participate in almost all biological processes required for life, such as, molecular binding, transport and chemical catalysis. Recently deep learning methods, mainly trained on known, X-ray crystallographic structures, such as AlphaFold 3 have predicted static protein and protein complex structures when given amino and nucleic acid sequences as input. However, these methods perform poorly with protein RNA complexes since there is a lack of experimentally resolved structures at atomic resolution and molecular flexibility. Small Angle X-ray Scattering (SAXS) is a low resolution experimental technique which observes biomolecules under near native solution conditions. This gives an underdetermined scattering curve which encodes information on the size and shape of the biomolecule as well as a pairwise distance distribution of electron density. In this talk we present some work on Protein RNA docking using SAXS data. Firstly, we introduce the mathematics behind SAXS scattering calculations and curve fitting. Then, given initial structures of high atomic resolution, we propose a method for the protein RNA docking search. Finally, we test our method on a known complex: NS1 protein of influenza B virus bound to 5’ triphosphorylated dsRNA.

Xende Rivero-Bowers - Using the Elliptic Gamma Function to find abelian extensions of cubic fields

In this talk, I will discuss the problem in number theory of finding abelian extensions of number fields. This has been solved in the quadratic case but is still open for cubic fields and higher. The standard method is to use Artin L-functions that arise from the number fields to identify units called Stark Units which can be used to generate the field extensions, but there is no easy way to identify the Stark Units. My research involves building off the work of a 2023 paper by Bergeron, Garcia and Charollois. Their paper shows that a three-dimensional version of the theta function known as the Elliptic Gamma Function can be used on certain candidate elements of cubic fields to produce numbers that have similar properties to the Stark Units. If time permits, I will also prove the Riemann Hypothesis as an encore.

Xintong Wang - From a Conjecture of Andrews to Almost Alternating Sign Patterns

In 1986, Andrews conjectured striking sign patterns and growth behaviour for the coefficients of several partition-theoretic q-series appearing in Ramanujan’s Lost Notebook. While the most famous example was later connected to quadratic fields, Maass waveforms, mock modular forms, and quantum modular forms, the remaining series were only recently understood. In this talk, I will discuss recent progress on these conjectures, including results on almost alternating sign patterns and asymptotic formulas for the coefficients. Using analytic techniques such as the circle method and steepest descent, we explain the oscillatory behaviour responsible for these phenomena and introduce new families of q-series exhibiting similar behaviour. The emphasis will be on the main ideas and connections rather than technical details.

Organizers

Feel free to direct any questions or comments to:

• Tahani Coolen-Maturi
• Jack Shotton
• Peter Wyper
• Sam Jackson