**The School of Mathematics and Statistics will be offering a number of individual undergraduate summer research placements in 2019. The list of potential projects is given below. A stipend of approximately €700 will be paid per student, with an enhanced stipend for a number of non-local students. However, transport costs to UCD cannot be paid. The programme is aimed specifically at penultimate-year undergraduate students, although students in other years may be admitted in exceptional circumstances. The programme is not restricted to UCD students.**

Applicants should submit the 2019 Undergraduate Summer Research Projects Application Form, a cover letter and CV (max 2 pages) by email to anthony.cronin@ucd.ie with “Summer Projects 2019” in the subject. Please see guidance on writing a CV suitable for this programme.

The deadline for applications is Friday 29th March 2019 at 17:00. Where a mismatch occurs between the number of offers and the demand for projects, candidates will be ranked according to a weighted average of GPA and other factors (e.g. quality of CV, suitability of candidate to a particular project). Successful candidates will be notified in early to mid-April. It is envisaged that the projects will last 6 weeks, starting in June. Details of start dates can be negotiated with individual supervisors.

An undergraduate research project is a great opportunity for students to develop their research skills. See previous year's reports and what some of our alumni have gone on to do! Also, see this recent article on their success.

**Project Titles and Supervisors and Abstracts**

**Complex Networks in Financial Markets** - Dr Michelle Carey

A stock market is considered as one of the highly complex systems, which consists of many components whose prices move up and down without having a clear pattern. The complex nature of a stock market challenges us on making a reliable prediction of its future movements. In this project we aim at building a new method to forecast and understand the future movements of Standard & Poor’s 500 Index (S&P 500) by constructing a complex network from the time-series of the constitutes of the S&P 500. Understanding how the individual companies interact will aid our understanding of the system and help us to provide more accurate predictions of its future values.

**Optimal Bayesian estimators for latent position network models **- Riccardo Rastelli

The latent position model is a widely used statistical model for the analysis of social network interactions. This model postulates that the nodes are embedded as points in a Euclidean social space, and that nodes that are close in this space are more likely to exhibit social interactions. The estimation of the latent positions associated to the nodes is usually performed in a Bayesian setting using Markov chain Monte Carlo methods to sample from a posterior distribution of interest. One of the main difficulties encountered is that the likelihood of a latent position model is unaffected by rotations, translations and reflections of the latent positions. This in turn creates a non-identifiability problem, that hinders the interpretation of the posterior samples. The goal of this project is to adopt a decision theoretic approach to define an optimality criterion that can be used to summarise a posterior sample of latent positions. This would allow one to extract a meaningful point estimate for the model's parameters which would overcome said identifiability issues. Statistical programming in R or C++ will play a crucial role in this project.

Reference: P. D. Hoff, A. E. Raftery, and M. S. Handcock. "Latent space approaches to social network analysis." Journal of the American Statistical Association 97.460 (2002): 1090-1098.

**Interacting Particle Systems, Last Passage Percolation, and Random Matrices** - Elia Bisi and Fabio Deelan Cunden

The theory of interacting particle systems has been developed, from the 1970s on, as a rigorous mathematical setup to model the cooperative behaviour of “particles” or “individuals” distributed throughout space (non-equilibrium gas dynamics, traffic flows, opinion dynamics, spread of epidemics, etc.). Interacting particle systems are random processes where particles, starting from an initial configuration, jump from one to another vertex of a graph according to a random interacting dynamics. The study of these processes revealed a rich structure, disclosing surprising connections with last passage percolation models and the theory of random matrices (Johansson, 2000).

In the so-called oriented swap process (introduced by Angel, Holroyd, and Romik in 2009), particles occupy the vertices of a graph and, if neighbouring, swap after a random time according to given rules. A particle might reach an absorbing state, when it cannot move any further and is trapped in its current location. One basic question in the theory is to determine the absorbing time of a particle, i.e. the time the particle takes to reach an absorbing state. In the oriented swap process on integer intervals, Angel et al. (2009) showed that the absorbing time of a particle is asymptotically described by a random matrix distribution. This connection motivates the study of further observables of the oriented swap process (also on other types of graphs) that have not yet been studied in detail. The aim of this project is to address such problems by studying some concrete low-dimensional examples or carrying out numerical simulations, with a view to finding interesting formulae, and/or developing a systematic theory.

**Resonances and the motion of spinning objects around a Kerr black hole - **Adrian Ottewill, Niels Warburton and Barry Wardell

One of the next great frontiers in gravitational physics will be the construction of a gravitational wave detector in space. To that end, the European Space Agency had committed to building the Laser Interferometer Space Antenna (LISA). One of the key sources for LISA are so-called extreme mass-ratio binaries. These systems consist of a massive black hole orbited by a compact, stellar mass object.

The motion of compact object about a rotating black hole has a very rich structure which evolves as the binary emits gravitational waves. During the binary's lifetime it will evolve through at least one orbital resonance. On a resonance the radial and polar libration times are in a low integer ratio and the usual adiabatic approximation for calculating the orbital motion breaks down and instead the evolution receives a 'kick' from the resonance. These resonances have been studied for test-body geodesic motion but as yet no studies have been made of resonances when the small compact body is itself spinning. Exploring this topic will be the subject of this project.

In particular, the student undertaking this project will study geodesic motion in Kerr spacetime, the Mathisson–Papapetrou–Dixon equations describing a spinning test body in general relativity, and orbital resonances how they are influenced by the spin of the compact object.

References:

Orbital resonances: https://arxiv.org/abs/1009.4923

Spinning body in Kerr spacetime: https://arxiv.org/abs/1512.00376

**Development of a computational framework to bring added colour and motion to the Irish primary school curriculum **- Lennon O'Naraigh

This project will look at the development of mathematical resources to complement the primary school mathematics curriculum in Ireland, from First Class to Sixth Class. At each stage of the curriculum, students are expected to learn new and often abstract mathematical concepts. Great efforts are made to introduce real-world applications and examples.Additional resources, inspired by research-led Mathematics and Statistics may be useful to complement these efforts. In this project, we will look at developing a computational framework to bring more colour and motion to primary school mathematics. We will look at applications that illustrate and complement the primary school mathematics curriculum, including Mechanics, Geometry, Graph Theory, and Probability. We will develop a computational framework in Python and Scratch to give effect to these applications. We will develop a website to make the resources available to primary schools. Finally, we will explore ways in which the efficacy of the programme can be measured, should it be taken up in schools.

Chaoyi Lu - Linear Complementary Dual Matrix Codes (advisor: Dr. Eimear Byrne)

Cian Jameson - Problems in Matrix Theory (advisor: Dr. Anthony Cronin) - Report

Oisin Flynn-Connolly - Random matrices, genus expansions and the symmetric group (advisor: Professor Neil O’Connell) - Report

Khang Ee Pang - Numerical Simulations of the Unsteady Transonic Small Disturbance Equation (advisors: Dr. Lennon O'Naraigh and Andrew Gloster) - Report

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