**The School of Mathematics and Statistics will be offering a number of individual undergraduate summer research placements in 2021. The list of potential projects is given below. A stipend of €1000 will be paid per student. 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. In 2021, the projects will be run online.**

Applicants should submit:

- their application form online here https://forms.gle/
DFzUo49wL7trqAEN8 , - a cover letter (max 1 page) and CV (max 2 pages) as a single PDF file with filename <lastname_firstname.pdf> to be uploaded here https://www.dropbox.com/
request/iSOcmO7RCGZgZB3Gy7vj Please see guidance on writing a CV suitable for this programme.

The deadline for applications is **Friday 2nd April 2021 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 a minimum of 6 weeks, starting in early 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**

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Three-fold tensors can be used to efficiently store a basis of a subspace of matrices, also known as a rank-metric code. These codes have become a hot topic in recent years, due to applications in the mathematics of communication. This efficiency afforded by tensors raises the possibility of using tensors and matrices to form the basis of a new cryptosystem for Post-Quantum Cryptography (PQC) that is, a system of cryptography which we believe will remain secure even after the development of feasible quantum computers. This is a highly important issue, as much of the cryptography on which our communication currently relies will be rendered obsolete when such a quantum computer is built; experts estimate this will happen in the next 10-15 years.

In this project we will consider tensors over finite fields, and study the tensor rank for three-fold tensors. We will develop algorithms and

The project does not require much background knowledge beyond elementary linear algebra and finite fields.

______________________________

A new Laplace solver computes a fast and accurate solution to the Laplace equation on a polygon or circular polygon with Dirichlet or homogeneous Neumann boundary conditions. The algorithm is based on rational functions. We apply this solver to flows over a range of boundary geometries, such as a step. We take the flow solutions and compute either (a) the resulting hydrodynamic forces acting on different surfaces or (b) transport of small particles through the system.

Tasks:

- Familiarise oneself with theory and code

- Compute a range of flow solutions

- Apply to hydrodynamic forces or transport phenomena

- Consider the physical implications

An application of the project is in industrial flows through connected piping systems or in confined spaces.

References:

Gopal & Trefethen. New Laplace and Helmholtz solvers, Proceedings of the National Academy of Sciences, 116 (2019), 10223

https://people.maths.ox.ac.uk/

Herterich & Dias. Potential flow over a submerged rectangular obstacle: consequences for initiation of boulder motion. European Journal of Applied Mathematics, 31(4) (2020), 646-681.

https://www.cambridge.org/

Herterich & Griffiths. A mathematical model for the erosion process in a channel bend. In preparation.

In this project the student will review the work of The Irish Epidemiological Modelling Advisory Group (IEMAG) on modelling the spread of COVID-19 in Ireland. They will learn about compartmental models for infectious disease modelling (SIR models), and how to analyse them. The student will use daily case data to find the effective reproductive number for the different levels of restrictions. After successfully fitting the model to the data, the student will investigate different strategies for vaccinating the population, with the objective of finding the optimal strategy for minimising further deaths.

______________________________

## Alumni

**2013**

Aideen Costello (TCD; Morgan Stanley and First Derivatives)

Julian Eberley (Theoretical Physics, UCD; Software Developer at Citi)

Daniela Mueller (Mathematical Sciences, UCD; PhD UCD 2020)

Benen Harrington (TCD; PhD University of York 2018)

**2014**

James Fannon (Theoretical Physics, UCD; PhD University of Limerick 2018, Met Eireann)

Andrew Gloster (Theoretical Physics, UCD; MSc Imperial College London; PhD UCD 2018, Arista Networks)

Shane Walsh (Theoretical Physics, UCD; IRC scholar, PhD UCD 2018, Susquehanna International Group)

Adam Keilthy (TCD; PhD University of Oxford 2020; 2-year postdoc at MPIM)

**2015**

Patrick Doohan (Mathematical Studies, UCD; MSc in Applied Mathematics ICL; PhD ICL 2020)

Maria Jacob (ACM, UCD; Statistical Officer Dept of Transport, UK)

Owen Ward (TCD; PhD candidate at Columbia University, New York)

Paul Beirne (Mathematics, UCD; IRC Scholar, PhD UCD 2020; IMVO, Dublin)

**2016**

Christopher Kennedy (ACM, UCD; postgraduate student at UCD)

Emily Lewanowski-Breen (Maths and Science Education, UCD; second-level maths and biology teacher at Wesley College, Dublin)

Michael O’Malley (Stats and Maths, UCD; postgraduate student at STOR-i, University of Lancaster, UK)

**2017**

Adam Ryan (Mathematics, UCD; Analyst for Brown Thomas and Arnotts) - Report

Luke Corcoran (Theoretical Physics, TCD; completed part iii in Cambridge (with distinction), PhD student at Humboldt University Berlin) - Report

Conor McCabe (ACM, UCD; MS Statistical Science at Oxford, Machine Learning Scientist at ASOS) - Report

Joseph Curtis (Statistics, UCD; Core Operations Engineer at Virtu Financial in Dublin) - Report

**2018**

**2020**

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