3. From Hamiltonian Mechanics to Symplectic Topology

Haoran Shi, Mathematics and Statistics, 1-2pm

Symplectic geometry and topology originated in Hamiltonian mechanics, where a physical system is described by its evolution in phase space, with coordinates given by position and momentum. The mathematical structure of phase space encodes key features of dynamics and leads naturally to geometric questions about motion and invariants. My research studies the geometry and topology of symplectic manifolds, which generalise phase spaces in Hamiltonian mechanics. A central problem is to understand the dynamics of Hamiltonian diffeomorphisms through their fixed points. In symplectic topology, this is closely related to the Arnold conjecture, a counterpart of classical fixed point theory in algebraic topology. A key tool is Floer homology, which can be viewed as an infinite-dimensional analogue of Morse theory on the loop space of a symplectic manifold.

11. Dynamical Dark Energy emerging from the QCD vacuum

Dong Ha Lee, Mathematics and Statistics, 1-2pm

Dark Energy is the name given to the phenomenon driving the late-time acceleration of our universe. Despite being responsible for nearly 70% of the current energy density of the universe, there is little consensus on the physics behind this phenomenon. 

Here, we try to describe it from within the Standard Model of physics via a dynamically emerging vacuum energy of the QCD gauge field. Hence, we attempt to simultaneously tackle the problems of the fundamental nature of dark energy, the requirement for beyond Standard Model physics and the potential dynamic nature of dark energy, a problem that has come into the forefront of cosmology in the past decade.

18. Black Hole Separability: Umbilical Foliations and the Generalised Killing-Yano Polyform

Amin Omarouayache, Mathematics and Statistics, 1-2pm

An investigation into the mathematical machinery underpinning the description of a broad class of black holes. The project aims to give a geometric origin for the existence of the remarkable symmetry properties we observe in black hole theories.

26. Category Theory and Convex Analysis

Lily Bennett, Mathematics and Statistics, 1-2pm

If we focus on one area of mathematics, we may see things behaving and relating to each other in certain ways. Moving onto another area of mathematics and dealing with different things, we can sometimes still spot them behaving and relating to each other in similar ways. Category theory can be used to express patterns of behaviour across different areas of mathematics, and this poster aims to explain how we can spot some known behaviours in the area of convex analysis.

30. The effects of vertical transmission on a spatially-structured host-parasite model

Jack Woodruff, Mathematics and Statistics, 1-2pm

Vertical transmission occurs when an infected host reproduces and passes the infection on to their offspring, whereas horizontal transmission (usually) occurs via contact between an infected and susceptible host. The role of vertical transmission in host-parasite dynamics is explored through analysis of a spatially-structured mathematical model. The key questions of the model are, what are the different long-term outcomes of the model? Can vertical transmission determine invasion? How does spatial structure affect the dynamics? A second model is then considered, where a pathogen that has horizontal transmission only is prevalent. A mutant strain then emerges that can also be transmitted vertically. The questions here are, does initial growth of the mutant strain predict the long-term dynamics? How likely is it that the mutant strain will invade? How do the dynamics change when there is a trade-off between vertical and horizontal transmission?

38. Representations of Semi-Direct Products of Lie Algebras in Prime Characteristic

Mona Aljuaid, Mathematics and Statistics, 2-3pm

We consider a Lie algebra 𝐶 with basis {𝐻, 𝐸, 𝑋, 𝑌 }, where these elements satisfy certain relations. This Lie algebra can be built from smaller pieces using a semidirect product, where one part acts on the other through the Lie bracket. To study it more deeply, we consider its universal enveloping algebra 𝑈 (𝐶). This is an associative algebra constructed from the Lie algebra 𝐶. In characteristic zero, Bavula and Lu studied this algebra. They classifiedits simple modules, prime, primitive, and maximal ideals.In this work, we study the same Lie algebra and its universal enveloping algebra over fields of prime characteristic. Our main result is a description of the center of 𝑈 (𝐶). We obtain this result by relating the algebra to the Weyl algebra using a localization method.These results help us better understand the structure of 𝑈 (𝐶) in prime characteristic and reveal new behaviors that do not appear in the characteristic zero case

48. Surface alignment to improve observational constraint and reduce predicted aerosol radiative forcing uncertainty

Iain Webb, Mathematics and Statistics, 2-3pm

Observations of an observable output of a complex climate model produce a sizeable reduction in this variable's own uncertainty, but don’t necessarily reduce the uncertainty in an unobservable output of the same model. Could the reason why lie in the alignment of the two outputs' response surfaces relative to one another? Can a simple measure offer the means to check the potential for uncertainty reduction prior to collection of data? 

A measure involving the comparison of partial derivatives for a pair of response surfaces will be presented. An overview will be given of both Gaussian processes and the method used for obtaining estimates of the derivatives required for the measure, as well as the computational implications of calculating the measure at thousands of gridboxes across the globe. 

The application of the measure to a pair of response variables for a particular climate model will be presented, and avenues for future work outlined.

52. Quantifying Notch-regulated cell state transitions in the human enteric nervous system

Nikolaos Stefanidis, Mathematics and Statistics, 2-3pm

The enteric nervous system (ENS) develops through a tightly regulated process in which bipotent progenitor cells differentiate into neurons and glia. Although the Notch signalling pathway is known to influence these transitions, the timing and rates of these changes in human cells remain poorly understood.

Using an in vitro human pluripotent stem cell differentiation platform, we generated longitudinal data on ENS development. We then built a computational model based on coupled ordinary differential equations to describe transitions rates between cellular states, and used Bayesian inference to quantify how Notch signalling affects these rates.

Our results show that Notch acts as a critical brake on differentiation. When Notch signalling is reduced, the balance of the system shifts, leading to premature ENS differentiation. This quantitative framework helps explains how altered developmental timing may contribute to neurological gut disorders such as Hirschsprung’s disease.

59. Topological Hochschild Homology Of HZ Relative To MU

Henry Rice, Mathematics and Statistics, 2-3pm

The complex cobordism spectrum is a ring spectrum central to chromatic homotopy theory. In recent years, topological Hochschild homology relative to to this spectrum has played a role in our understanding of the famous redshift and telescope conjectures. Our aim is to obtain a better understanding of the homotopy of THH^MU(HZ). Indeed, in several similar cases, this ring is polynomial and one hopes that this is true in our case as well. One method to attempt to study this ring is by computing the Künneth spectral sequence associated to the enveloping algebra construction of THH. Although this spectral sequence collapses, we are left with non-trivial extension problems to solve in this case. Our method instead utilises the Hurewicz homomorphism to induce a morphism to the homology of BSU. This ring is the universal example of a ring with a symmetric multiplicative two-cocycle and we hope that an analysis of potential subobjects of this Hopf algebra will help us to understand THH^MU(HZ).

64. Reduced Gromov-Witten Invariants

Robert Rogers, Mathematics and Statistics, 2-3pm

We develop a theory of counting maps from curves of a fixed genus to a fixed target smooth projective manifold that doesn't involve contributions of maps from curves of a lower genus.

67. Invitation to Quantum Geometry

Kuba Krawczyk, Mathematics and Statistics, 2-3pm

What does geometry look like to a quantum observer? Quantum geometry is an umbrella term for the mathematical frameworks that attempt to answer this question. While the underlying math spans a variety of complex theories, the core concept is remarkably unified: the rigid, exact points of classical geometry dissolve into clouds of quantum probability. This shift occurs because, as dictated by the Heisenberg uncertainty principle, measurements of space are no longer strictly independent -- one observation inherently influences another. In this poster, we aim to give a basic introduction to the subject. Through visual illustrations on familiar surfaces -- like a 2D sphere and a 2D plane -- we bridge the gap between classical intuition and the "fuzzy" reality of quantum space. We also briefly compare the properties of quantum geometries arising in different quantum systems -- from nonrelativistic quantum mechanics to D-branes in string theory.

78. Cellular Structures in Abstract Homotopy Theory and Obstructions to Their Finiteness

Penglin Li, Mathematics and Statistics, 3-4pm

In topology, we build spaces using simple "building blocks" called cells. A natural question is: when can a space be continuously deformed—without tearing or gluing—into one built from finitely many cells? To answer this, mathematicians use tools called "finiteness obstructions." Thomas Athorne, a Sheffield PhD graduate, proved that these cell-gluing steps match a specific algebraic structure. However, his results only apply to classical topology. 

My research extends this to "abstract homotopy theory." This framework applies to any mathematical objects, as long as we can define when two objects are "essentially the same."  Abstract settings are much harder to work with. My work explores how abstract "cells" behave here. My goal is to define clear cellular structures, create valid finiteness obstructions, and extend Athorne’s algebraic results to this broader mathematical world using modern tools like model structures and infinite category theory.

81. Quantifying redshift in chromatic homotopy theory through TR(BP<n>)

Pierre-Louis Guillot, Mathematics and Statistics, 3-4pm

Chromatic homotopy theory has been very successful in helping homotopy theorists understand the category of spectra, a sort of highly structured environment for cohomology theories to live.

It does so by decomposing the category of spectra into various “chromatic heights” and in that context, there is a well-known phenomenon in chromatic homotopy theory that I am interested in : chromatic redshift. It says that applying the complex operation called “algebraic K-theory” on a multiplicative cohomology theory of chromatic height n outputs something of chromatic heights exactly n+1.

Formulations of this phenomenon have been proven using approximations to algebraic K-theory on prototypical “height n” cohomology theories. They use a lot of tricks to do that without actually computing them! Only this month have people managed to do actual computations of one of those approximations (Topological Cyclic Homology). I am trying to compute a different one (Topological Restriction Homology).

94. Lie Algebras and Their Universal Enveloping Algebras

Hadbaa Alqahtani, Mathematics and Statistics, 3-4pm

This research concerns Lie algebras and their universal enveloping algebras, which are fundamental objects in algebra and representation theory. Lie algebras encode infinitesimal symmetries, while universal enveloping algebras provide an associative framework that allows these symmetries to be studied using ring-theoretic methods. 

The work focuses on the universal enveloping algebra of a semi-direct product Lie algebra formed from the simple Lie algebra \mathfrak{sl}_2, acting on a two-dimensional module. By analysing the centraliser of a Cartan element and its relation to the centre of the enveloping algebra, and by identifying connections with Generalised Weyl Algebras, this research contributes to the structural classification of enveloping algebras. Such classifications are central to the study of representations and ideals in non-commutative algebra.