Student: Cameron Heather

Supervisor: Richard Daniel (PGR)

One of the biggest remaining mysteries in physics is dark energy, which we know is the cause of the expansion of the universe, but still eludes our understanding. In this project, we looked at a very general model with a scalar field which mediates a fifth force. Via the chameleon mechanism this fifth force is dependent on the density of interacting bodies. As such, Newtonian mechanics can be recovered despite this ‘hidden force’.

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Student: Benjamin Andrews

Supervisor: Dr Jonathan Jordan

Preferential attachment graphs are a model of random network, where nodes (which have one of several types) are added to a network and connected to others already in it at random. This is a good model for a lot of situations in the real world, such as people’s brand preferences (where the types are the brands). Preferential attachment means that nodes are more likely to be connected to others that already have more connections, and choose their type based on them. This can be likened to people being more likely to choose their brand preference based on that of influential people, for example. It was conjectured that with three different types of nodes, the proportions of each type in the network would always converge. However, this is not the case in at least one setup, where new types are selected using a "rock-paper-scissors" style process. We explore similar processes to these, which do end up converging, and discuss the idea that cases where convergence does not happen are probably rare and special.

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Student: Elisabetta Dixon

Supervisors: Prof Caitlin Buck, Bryony Moody (PGR)

Accuracy in Archaeological dating is important. There are different models we can use that gives us a calendar date range from a radiocarbon date to estimate the age of an archaeological find. Differing models can produce contradictary date ranges, and with multiple phases or multiple dates within a phase, this 'difference' can multiply significantly. We wanted to explore and compare these differences to see just how significant model choice might be to an archaeologist wanting to construct a chronology from a particular site.

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Student: Sam Felton-Dobbs

Supervisor: Dr Alison Poulston

When performing Bayesian inference, often the likelihood cannot be evaluated. In this case, methods have been developed to approximate the target posterior distribution. This presentation will introduce rejection ABC as such a method and show how it can be used in a toy example. I will also present an example animal movement model to demonstrate the real world impact of these likelihood-free inference methods

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Student: Kacper Mytnik

Supervisor: Prof Sarah Whitehouse

The project is focused on a certain object in algebraic topology; the fundamental group of a space X with base point x. It is denoted π1(X,x) which is a set of equivalence classes of loops based at the point x, endowed with the group structure under the operation of concatenation of loops. This object is very useful in classifying connected spaces up to homotopy equivalence however it can be notoriously hard to compute. A very useful tool for computing the fundamental group of a space is the Van Kampen theorem. Briefly, it says that for sufficiently nice spaces that can be broken up into two pieces U and V, such that the intersection is non-empty and the union of the two pieces is the whole space, we have π1(X,x)=π1(U,x)*π1(V,x), which is the amalgamated free product of the fundamental groups of the two pieces. In order to prove this theorem, one can proceed by directly examining the loops or by employing an equivalence of categories amongst other methods. In this project I did the latter and here I will discuss what went into establishing the equivalence of the category of covers and the category of π-sets.

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Student: Louisa McKenna

Supervisor: Ines Krissaane (PGR)

The aim of the project was to familiarise myself with Bayesian statistics, as I did not take Bayesian Statistics as a third year module, and to use this field of statistics to solve problems, such as the Birthday problem and the Monty Hall problem, in R. I then adapted this to explore a range of sampling methods including Monte Carlo, Gibbs and Metropolis-Hastings, meaning that when we cannot infer exactly, we approximate the true distribution by taking samples. I was able to use these sampling methods alongside the Ising model, and link this further to images created by MRI scans. Finally, I was able to use these imaging techniques to show how to make an image less blurry through the R Shiny App.

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Student: Calum Hughes

Supervisor: James Brotherson (PGR)

Simplicial sets are an extension of the notion of the simplicial complex, which is (roughly) a way to split up a topological shape into n-dimensional triangles in order to learn information about the number of n-dimensional holes and connectedness of the shape. It has uses in algebraic topology, and especially in homotopy theory, but my project favoured building geometric intuition for what a simplicial set is. The terminology for a lot of algebraic topology is written in the language of category theory, which provides shortcuts to understanding and proving related ideas in different mathematical areas. A few of the main ideas of category theory were first introduced before moving onto a relatively intuitive introduction to simplicial sets.

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Student: Callum Murton

Supervisor: Valentin Breaz (PGR)

Machine Learning is an extensive field that lies at the intersection between Statistics and Computer Science, with several exciting applications. One key algorithm of Machine Learning is Gaussian Processes, which is a very popular method of predicting data. Gaussian Processes come with many great advantages which makes predicting data very flexible. Especially when compared to other methods such as Linear Regression, which has its limitations. From the research, we wanted to motivate the use of Gaussian Processes and investigate why they are so commonly used. Also, we wanted to understand how they work and the theory behind them; enabling us to apply them for ourselves. This led us to the use of Bayesian Inference and Covariance Functions (Kernels), which are instrumental aspects of Gaussian Processes. Research methods included, reading ‘Gaussian Processes for Machine Learning’, and attending an online Gaussian Processes Summer School. Finally, we applied our research to a practical problem. Where given some training data, we fitted a Gaussian Process such that it gave us accurate predictions with uncertainty.

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Student: Murat Delibas

Supervisor: Dr Ashley Willis

To trigger chaotic dynamics, sometimes we need to apply a small ‘kick’ to get things started. We tend to be interested in equilibrium solutions to a dynamical system, when in practice, even if a solution is stable, a small perturbation from it could lead to chaos. The onset of turbulence in flow through a pipe (or between flat plates) is a good example – simple flow straight through the pipe (‘laminar’ flow) is possible, but large enough perturbations trigger a chaotic turbulent flow. A nice system that mimics this with only 9 coefficients was derived by Moehlis, Faisst and Eckhardt (2004). A perturbation of special interest is the smallest perturbation that will trigger chaos, but how can we find what this perturbation is? Generically the basin of attraction of a chaotic attractor is fractal, but is that the case in the vicinity of this ’optimal’ perturbation?

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Student: Shengzhi Luo

Supervisors: Dr James Cranch, Dr Fionntan Roukema

With the knowledge that there are two knots in different shapes with the same invariant the approach of this project is to try to use different invariants to tell knots apart with various crossings and strings and compare each invariant to get a relatively reasonable invariant. To learn about invariants, the project started with braids theory in two dimensions which was a relation to knots in three dimensions. After done with colorings, Python was used to count all braids in the list by 3-coloring to see if they had the same braid closure and thus there would be 1, 9, 27 three-colorings to sort knots. Therefore, on the basis of coloring, quandles were introduced to involve more colorings and get programming digitally. For the second goal, entropies were needed to measure how good a knot invariant was and by running Python through the list of quandles which was up to 14 numbers entropy showed some regularity and similarity. There were symmetries on knots and quandles for the same entropies with different invariants and those usually wouldn’t be strong invariants. Moreover, quandles with more elements normally would have lower entropy among all knots.

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Student: Philip Kmentt

Supervisor: Farhad Allian (PGR)

The project revolved around Fourier transforms and their importance in signal processing and solar physics. Briefly, a Fourier transform is a mathematical transformation that transforms a signal in time into its dominant frequencies and portrays these in a frequency domain. Fourier transforms have vast applications in data analysis and especially physics, as they help to describe the behaviour of waves, which I investigated in my summer project, both theoretically and by use of code. The aim of my project was to enhance my coding skills, to develop a better understanding of the Fourier transform and also to appreciate how the Fourier transform can be used to model the sun’s corona, an area in which my project supervisor is actively engaged in. I achieved this successfully by dedicating my time towards creating and improving code in Python to model explicit data from NASA I was sent.

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Student: Barry Zhang

Supervisor: Dr Rekha Jain

Lagrangian points L1 to L5, discovered by French mathematician Joseph Louis Lagrange, are locations in space where gravitational forces and the orbital motion of a body balance each other. Therefore, these points are used by spacecrafts to float. In 1994, NASA launched the Wind spacecraft, and it was placed, with nudge, in the L1 Lagrangian point of the Sun-Earth system. The Solar and Heliospheric Observatory (SOHO) also slowly orbited around the L1 Lagrangian point. WMAP, Planck and the James Webb Space Telescope are all using L2. The L3 point is hidden behind the Sun at all times and so not used but is a popular topic for science fiction movies. The L4 and L5 points could provide stable orbits under certain conditions.

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Student: Jai Schrem

Supervisor: Dr Moty Katzman

We first construct a direct system of groups and its direct limit in abstract terms, beginning with some basic definitions and notations that are required, such as a partially ordered set and some new notation for homomorphisms. Next, we looked at an example of a direct system of prime power order cyclic groups and construct the Prüfer group of type p∞ as its direct limit.

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