8. Local Limits of Multi-type Bienaymé-Galton-Watson Trees

Marc Bernard, Mathematics and Statistics, 1-2pm

We consider the random trees commonly known as Galton-Watson trees; a 'branching process' where each individual has a random number of children, based on some random distribution (if the average number of children is less than / equal to / greater than 1, we call the tree sub-critical / critical / super-critical). Sub-critical and critical trees are finite with probability 1, while super-critical trees may be infinite. We "condition" (force) the sub-critical and critical trees to be large in some way (height, number of nodes, etc) and attain a certain result in what we call their local limit. This limiting tree was first discovered by Harry Kesten in 1986, who found that when conditioning a sub-critical or critical tree to have height n, in the limit of n, we attain a tree with only one infinite line of descent. Our research considers other ways to attain the same limiting tree, which we call the Kesten tree, including conditioning based on “types” which are assigned to each node.

12. Black Holes and their Singularities in Quantum Gravity

Sofie Ried, Mathematics and Statistics, 1-2pm

Black holes have diverging spacetime curvature forming a singularity. A positive-mass black hole has a horizon, beyond which nothing can escape. Negative-mass black holes are forbidden, as they do not have horizons around their singularity.

General relativity can’t be consistently applied on small scales. To accurately describe the physics near the singularity of a black hole we need an extension called quantum gravity.

If quantum effects resolve the singularity, negative mass black holes could exist, destabilizing the vacuum and requiring an explanation for their absence.

Changes of space and can be described by a metric. We also include a cosmological constant Λ in our description together with the unimodular time T.

We quantise the metric and impose unitarity with respect to unimodular time T.

A quantum theory constructed this way includes either only positive, negative or zero mass black hole states. Also we find that the black hole singularity is resolved. 

27. Self-Force and the Schwarzschild Star

Abhinove Seenivasan, Mathematics and Statistics, 1-2pm

A charge generates an electric field which also affects the charge itself. This is called the self-force (SF) and been studied since the early 1900s as an interesting theoretical problem. More recently, it has also had experimental motivations, since the discovery of black holes with stars/smaller black holes in orbits around the central black hole. These objects feel a gravitational SF, and this drives them towards the eventual merger. Since the experimental detection of gravitational waves, we have been able to detect and model the orbits of many such two body systems or binaries using SF theory and other approaches to the two-body problem. In this work, we study the SF for a fixed charge outside a constant density star, the Schwarzschild star. We find that as the charge approaches the surface of the star, the SF diverges. We present numerical and analytical approximations for the force in this region and closed form results and numerical data away from the surface. 

34. Reflexive Homology and Operads

Jack Davidson, Mathematics and Statistics, 1-2pm

In algebraic topology, one looks to extract information about topological spaces (which we think of as a generalised version of shapes) via algebraic information associated to these spaces (which we think of as generalising integers). Often, one can use similar techniques to extract information about objects from algebra. A standard algebraic object to study is called an algebra with involution and one such technique to study them is known as the reflexive homology of the algebra. This extracts classical information about how non-commutative the algebra is, along with information about a specified operation that the algebra is equipped with called an involution.

One often wishes to systematically study where theories like reflexive homology really come from, and the answer lies in operads. These are gadgets which slickly encode how many ways there are to multiply together objects. Each operad encodes a certain type of homology, and we determine which one encodes reflexive homology. 

44. Alfv´en Waves in Partially Ionized Plasma with Steady Flows

Nada Alshehri, Mathematics and Statistics, 2-3pm

Plasmas in space objects are not hot enough for full ionisation; thus, ions, electrons, and neutral particles coexist and interact via collisions. Alfvén waves are transversal perturbations in the magnetic field that propagate along field lines and carry large amounts of energy, contributing to plasma heating and acceleration. We consider a two-fluid (charges-neutrals) plasma in the particle collision frequency regime and investigate Alfven wave properties that can propagate in a steady-state plasma when ions and neutrals have different speeds (weak collisions) and directions. Our results show entropy modes become propagating, so flows generate new wave modes. Flows alter wave properties, inducing effects like mode conversion and two-stream instabilities. Wave behaviour is regulated by collisional frequency. Studying plasma waves in astrophysical and laboratory environments, as well as their stability, benefits from correct understanding and interpretation of our results.

54. Mapping Solar Photospheric Flow Fields

Mashael Aldhafeeri, Mathematics and Statistics, 2-3pm

Plasma flows in the solar photosphere can be studied by tracking granules as tracers of the velocity field. These flows have both horizontal and vertical components, influencing solar dynamics. The Balltracking algorithm (Potts et al. 2004) is a highly efficient method for extracting flow fields, offering accuracy and computational speed.

We apply Balltracking to derive velocity fields from solar observations, capturing plasma motion across selected regions. A parametric analysis refines the accuracy of the extracted flow maps. To identify vortical motions, we use the Lagrangian-Averaged Vorticity Deviation (LAVD) method (Haller et al. 2016), which detects coherent vortex structures.

Using SDO/HMI data, we analyze vortex evolution, studying variations in number and size over time. This research enhances our understanding of solar convection, vorticity, and their role in large-scale dynamical processes, contributing to insights into the solar cycle's impact on surface flows. 

63. The Theta Correspndence, Dth Shintani Lifts, and L-Values

Mark Chambers, Mathematics and Statistics, 2-3pm

Wiles’ monumental proof of Fermats last theorem established the equivalence between elliptic curves and modular forms. The Langlands Programme, which is a generalisation of this called the “grand unifying theory of maths”, predicts maps between certain modular forms. One instance is the Theta correspondence - the study of which is closely linked to so-called L-values, which appear in another important problem, the Birch and Swinnerton-Dyer conjecture. 

72. Torsion Growth and Lorentzian Automorphic Forms

Daniel Bassett, Mathematics and Statistics, 2-3pm

Automorphic forms are mysterious objects that play a central role in modern number theory research. One way of accessing them is through calculating the cohomology of certain manifolds. In particular, if these cohomology groups have high rank but low torsion, this is a sign of many automorphic forms associated to these manifolds; conversely low rank but high torsion shows a lack of automorphic forms. We present our results from calculations of torsion cohomology associated to certain hyperbolic 4-manifolds and 5-manifolds, which arise in general relativity and as such are of independent interest in physics. 

86. The Structure and Representation Theory of the Euclidean Algebra in prime Characteristic >𝟐

Faizah Alharbi, Mathematics and Statistics, 3-4pm

We investigate the Euclidean algebra e(3) = sl2 ⋉ V3 in prime characteristic p > 2. While Bavula and Lu provided a comprehensive analysis of the characteristic zero case, this work develops the theory in the prime characteristic.

The study focuses on the universal enveloping algebra U(e(3)) of the Lie algebra e(3). The goal is to classify simple modules and prime ideals of the algebra U(e(3)). Our main results describe the tensor product structure and the centre of the algebra through precise algebraic relationships. These findings extend the understanding of Lie algebras in a prime characteristic, contributing to both pure mathematics and theoretical physics, particularly in areas where discrete mathematical structures are essential.

95. Dark Matter and Dark Energy from Extra Dimensions

Adam Smith, Mathematics and Statistics, 3-4pm

We explored the possibility that dark energy and dark matter could emerge from two fields called dilaton and axion fields. These fields represent the size of extra dimensions and the energy flux wrapping around these compacted dimensions, respectively. We examined how these fields, acting as dark matter and dark energy, influence the evolution of the Universe. In particular, we studied their impact on the formation of galaxies, the development of cosmic structures, and the evolution of the cosmic microwave background (CMB) radiation, which is the radiation left over from the earliest events in the history of the Universe.