PostgraduateResearchDay 2025

May 12, 2025
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 (tentative)

Registration

To register, please send an email to the organizers.

Abstracts

Alexander Jackson - Representations of GL_n(o/p^r) and related groups

We discuss the problem of finding the representations of GL_n(o/p^r), where n,r are integers and o is the ring of integers of a non-Archimedean local field with maximal ideal p. In particular, I will explain how studying automorphism groups of finite o-modules relates to the representation theory of GL_n(o/p^2), and explain recent work giving progress towards a solution for n=5. This extends previous work of Singla ('10) and Onn ('08).

Rafail Psyroukis - From classical to Siegel modular forms

In this talk, we will discuss how the classical theory of modular forms generalises to the theory of Siegel modular forms. We will present analogues for both cases but also discuss fundamental differences, which makes the theory of Siegel modular forms so interesting to explore.

Julio Ignacio Quijas Aceves - Constructing fundamental Polyhedra for Complex Hyperbolic Space

Given a geodesic triangle in hyperbolic space with rational angles, we can construct a discrete group by taking the group generated by reflections on the sides of the triangle. This leads to the well-known beautiful triangular tessellations of the hyperbolic plane. We generalize this construction to the Complex Hyperbolic plane, where the construction increases in complexity due to the non-constant curvature of the Complex Hyperbolic plane.

David Lanners - Emergent Continuous Symmetries from Discrete Systems

We define Z_N lattice gauge theory as a model of N discrete spin values sitting on the edges of a lattice. Surprisingly, for N>4, this discrete model exhibits a phase with an emergent continuous U(1) symmetry, reminiscent of the photon phase in standard electromagnetism. What makes this phase particularly intriguing is its robustness — it persists over a broad and continuous range of system parameters. This allows for stable, long-range interactions even in this discrete setting. This talk explores the model and, time permitting, presents how we were able to open up this photon phase for N=3 by suppressing string-like objects called monopoles.

Yanpeng Zhi - Why People Get Lazy in the Long Run?

Being lazy seems inevitable as humans, and we are not tireless robots after all. In this talk, I will defend ourselves by considering an ergodic control problem. It can be proved that an optimal control turns out of a lazy type. This provides mathematical evidence for our daily behaviour.

Jost Pieper - An application of rough SDEs to robust filtering with jumps

Finding a robust representation of the conditional distribution of a signal given a noisy observation is a classical problem in stochastic filtering. Such representations are of interest as they justify the use of discrete observation data and ensure robustness of the signal approximation to slight model misspecification. When the signal and observation are correlated through their noise, Crisan, Diehl, Friz, and Oberhauser (2013) showed that such a robust representation typically cannot exist as a functional on the space of continuous paths, but must instead be formulated on the space of geometric rough paths. In this talk, I will discuss how to extend these results to stochastic filtering problems involving correlated multidimensional jump diffusions, using the theory of rough stochastic differential equations (RSDEs) with jumps.

Adam Stone - AddiVortes: (Bayesian) Additive Voronoi Tessellations

Many statistical models, such as Random Forests, Gradient Boosted Machines, and Bayesian Additive Regression Trees (BART), rely on decision trees to partition the covariate space. While effective, tree-based methods often struggle due to their rigid structure and limited flexibility in capturing relationships between features. We introduce AddiVortes, a novel model that uses Voronoi tessellations instead of decision trees for more adaptive partitioning. AddiVortes builds on BART's framework, with performance demonstrated through comparisons on multiple datasets and simulations verifying its properties. By leveraging Voronoi tessellations, AddiVortes is able to capture complex relationships, offering a powerful and interpretable alternative to traditional tree-based models.

Lupeng Zhang - Some discussions and proposals for high dimensional change point detection

Change point analysis aims to detect significant changes in the distribution of a data sequence. It holds critical importance across modern statistical applications such as economics, finance, quality control, genetics, and medical research. While change point detection for low dimensional data is thoroughly studied in extensive literature, change point detection is challenging in high dimensional data where the number of variables is much larger than the number of observations. Hence high dimensional change point analysis has become a vital focus of recent research. In the present work, we discuss some main challenges with high dimensional change points and introduce a nonparametric approach to deal with those challenges. The proposed method applies the dissimilarity distance and cumulative sum (CUSUM) statistics to detect high dimensional change points. It can detect changes in both the mean and variance of high dimensional observations, as well as other distributional changes. We also present its simulation results and data application on the S&P index to demonstrate its effectiveness in addressing challenges for high dimensional data.

Germaine Uwimpuhwe - On testing random effects in linear and non-linear mixed effects models

Linear and non-linear mixed models (LMMs and NLMMs) are frequently used for analysing longitudinal data with nested structures. Unlike LMMs, testing the necessity of random effects in NLMMs remains challenging due to the limitations of traditional likelihood-based methods, which are constrained by assumptions of boundary issues and normality, as well as convergence issues and difficulties with correlated random effects. To address those issues, we develop and extend a permutation-based test, originally designed for linear models, to the non-linear context. Our approach uses non-parametric estimation techniques, including variance least squares (VLS), method of moments (MM), and first-order approximation. Through extensive empirical analyses, we compare these tests with the traditional mixture of chi-square tests (MLRT) based on empirical power, Type I error rate, and bias. Our results show that test based on VLS achieves power between 92% and 100%, while MM-based methods range from 88% to 100%, with a controlled Type I error rate. Our tests stable convergence rates across scenarios, in contrast to the MLRT method which suffers from convergence issues. We are developing the R package “TestREnlme” for implementation of these methods, facilitating non-parametric estimation and permutation tests for random effects.

Finn Gagliano - Categorical Symmetries and Phase Transitions

In the past decade, there has been a major shift in the way physicists view the symmetries of a quantum field theory. Extending the usual group-theoretical notion of a symmetry, categorical symmetries have led to new insights in many different areas of quantum field theory. In this talk, I will discuss one particular aspect, namely how we can use the tools of category theory to distinguish (de)confined phases of gauge theories.

Luci Mullen - Nonlinear analysis of the Keller-Segel system with logistic growth

The Keller-Segel model of chemotaxis has been proposed to describe spatial and spatiotemporal pattern formation observed in nature. Example applications include pigmentation on snake skin, and intricate patterns of bacteria, slime moulds, and other microorganisms. While historically useful to understand pattern formation in such systems, linear stability analysis of spatially homogeneous states can fail to capture the emergent dynamics. In this talk, I use multiple scales asymptotics to develop a weakly nonlinear theory that predicts the structure of solutions. Focusing on the Keller–Segel model with logistic growth, I numerically show that solutions can exhibit oscillatory, and even chaotic, behaviour and consider the theory of competing modes with slowly varying amplitudes to explain the emergence of oscillations.

Organizers

Feel free to direct any questions or comments to:

• Mathew Bullimore
• Tahani Coolen-Maturi
• Yue Ren
• Shrog Albalawi