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:

  1. their application form online here https://forms.gle/DFzUo49wL7trqAEN8
  2. 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

 
Dr Anthony Cronin
 
Title: The nonnegative inverse eigenvalue problem (NIEP)
 
As an undergraduate you are used to finding the eigenvalues and eigenvectors for low-dimensional matrices, first by hand and perhaps later, for larger n, numerically with the aid of software such as Matlab, Mathematica, Maple etc. The nonnegative inverse eigenvalue problem (NIEP) is, as the name suggests, an inverse problem to the one of finding eigenvalues. Here the objective is to find the original (entrywise nonnegative) n x n matrix (if it exists!) for which a given list of n complex numbers is the list of eigenvalues. The problem was first posed back in 1938 by Kolmogorov and solved for real lists with just one positive number in 1949 by Suleimanova. The general problem for n = 3 was solved in 1978 by Loewy and London, for n = 4 by Laffey and Meehan in 1997 and for trace zero 5 x 5 matrices by the same authors in 1999. Amazingly the problem is still open for n > 4 despite this being a very active area of research throughout the world. The aim of this project would be to get an appreciation for the techniques used to solve the low dimensional cases and perhaps make progress on related problems such as Monov's conjecture and/or Johnson's Conjecture, which asks which nonnegative properties does the derivative of a characteristic polynomial inherit. The following arxiv link offers an excellent overview to the evolution of results in the problem thus far 
 

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Title: Tensors in Coding and Cryptography
 
Tensors are fundamental objects in mathematics and its applications. In finite dimensions, tensors can be thought of as higher-dimension analogues of matrices; for example, a two-fold tensor corresponds to a matrix, while a three-fold tensor corresponds to a cube of numbers. They have been studied for many years in many contexts, but various foundational problems remaining mysterious. For example, while the rank of a matrix is very well understood, the corresponding notion for a three-fold tensor is much more difficult. 

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 theoretical tools to calculate or estimate the tensor rank, with a view towards demonstrating their potential use in PQC; in particular we would like to show that specific tensors have sufficiently small tensor rank. We will use techniques from linear algebra and coding theory, as well as some ideas from algebraic complexity theory. 

The project does not require much background knowledge beyond elementary linear algebra and finite fields.
 
 
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Title: Patterns of orthogonal matrices
 
We will investigate zero/non-zero patterns of orthogonal matrices in the context of the inverse eigenvalue problem for graphs, applying recent work of Levene, Oblak and Šmigoc. See 
 
 
This project will be partly computational and some programming experience would be an advantage.
 

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Title: Application of potential flows through irregular geometries
 
Fluid flow around an obstacle is a classical problem in applied mathematics. The resulting hydrodynamic forces may cause damage to a structure. Unfortunately, even simplified flows are challenging to solve when the boundary geometry becomes slightly complicated. A 2D steady, inviscid, incompressible, irrotational flow -- a potential flow -- is one example. Here, the flow may be described using Laplace's equation, and may be solved using complex variable approaches. However, it is not always so straightforward. A conformal map theoretically exists, allowing a problem in a complicated geometry to be mapped and solved in a simpler one, but finding the map is the challenge. Analytical solutions exist for polygon boundaries, though corners introduce singularities.

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/trefethen/pnas.pdf

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/core/journals/european-journal-of-applied-mathematics/article/abs/potential-flow-over-a-submerged-rectangular-obstacle-consequences-for-initiation-of-boulder-motion/405240C1E7983BBB43D96B27B8172067

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

 
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Title: Investigating vaccination roll out strategies for COVID-19 in Ireland
 
COVID-19 has had a massive impact in Ireland and across the globe, with over 100 million confirmed cases and over 2 million deaths as of February 2021. Lockdowns and restrictions were imposed to curb the spread of the disease, yet it continues to spread. Fortunately, the end is now in sight, as vaccination administration begins.

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.


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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
 
Cian Jameson (Mathematics, UCD; PhD candidate at UCD) - Report
 
Chaoyi Lu (Statistics, PhD candidate at UCD) - Report
 
Khang Ee Pang, (Applied & Computational Mathematics, UCD; SFI CRT PhD candidate at UCD) - Report
 
Oisin Flynn-Connolly (Masters in Mathematics, Orsay, Paris) - Report

2019
 
Kerry Brooks (Mathematical Science, UCD; BSc stage 4 student) - Report
 
Kevin Cunningham (Theoretical Physics, UCD; BSc stage 4 student) - Report
 
Eoin Delaney (Computer Science, UCD; PhD in the Machine Learning and Statistics at Insight UCD) - Report
 
Shane Gibbons (Mathematics, UCD; Part III at Cambridge) - Report
 
Hou Cheng Lam (Financial Mathematic s, UCD; MSc Data & Computational Science UCD) - Report
 
Jack Lewis (Theoretical Physics, UCD; BSc stage 4 student) - Report
 
2020
 
Hugo Dolan (ACM, UCD; BSc stage 3 student) - Report

Piotr Kedziora (BSc Mathematics, NUIG) - Report

Peter Nee (ACM, UCD; BSc stage 4 student) - Report