### Undergraduate Research Projects 2018

### The School of Mathematics and Statistics will be offering a number of individual summer research placements in 2018. The list of potential projects is given below. A stipend of approximately 600 Euro 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 2018 Research Projects Application Form, a cover letter and CV (max 4 pages) by email to robert.osburn@ucd.ie with “Projects 2018” in the subject. Please see attached a guidance on writing a CV suitable for this programme.

The deadline for applications is **Friday 30th March 2018 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 what some of our alumni have gone on to do! Also, see this recent article on their success.

### Project Titles and Supervisors

### Dr. Eimear Byrne

#### Title: Linear Complementary Dual Matrix Codes

### Dr. Anthony Cronin

#### Title: Problems in Matrix Theory

### Professor Neil O’Connell

#### Title: Random matrices, genus expansions and the symmetric group

The study of random matrices has its origins in the pioneering works of John Wishart in mathematical statistics in the 1930’s, and of Eugene Wigner and Freeman Dyson in nuclear physics in the 1950’s and 1960’s. It has since emerged as very active area of modern research, mainly due to its remarkable connections across various fields, including number theory, enumerative combinatorics, representation theory, statistical physics and quantum chaos. One such connection was discovered by Harer and Zagier (1986), who found that `moments’ of random matrices can be written as polynomials (in the dimension) with positive integer coefficients and, moreover, these coefficients have a combinatorial interpretation: a(k,g) is the number of ways to match and glue the edges of a regular 2k-gon to obtain an orientable surface of genus g. Another connection, which was discovered more recently (Ph. Biane, Proceedings of the International Congress of Mathematicians, 2002), relates moments of random matrices to the asymptotic (representation) theory of the symmetric group S_n, when n is large. This connection motivates the study of some new classes of random matrices, which have not yet been studied in detail and as such are not so well understood. The aim of this project is to consider and study some concrete low-dimensional examples, with a view to finding interesting formulae, combinatorial interpretations and/or developing a systematic theory.

### Dr. Lennon O Naraigh

#### Title: Numerical Simulations of the Unsteady Transonic Small Disturbance Equation

The Unsteady Transonic Small Disturbance (UTSD) Equation is used to describe shock structure when a shock reflects off a wedge. While much work has been completed regarding the global structure of solutions to the UTSD equation open questions remain about the behaviour of the solution under conditions where the incident and reflected shock detach from the wedge and produce a Mach stem. This project will numerically examine the structures associated with this phenomena.

Although the UTSD equation is an apparently routine quasilinear partial differential equation in its dependence on the spatial dimensions x and y and time, it nonetheless possesses almost unique aspects which make its numerical solution non-trivial. In particular, the structure of the imposed boundary conditions is such that at each point in time the solution is obtained from boundary information effectively by propagating information from the right-hand boundary along the x-direction. As such, the necessary solution approach is equivalent to solving a heat equation in the timelike x-variable and the spacelike y-variable, at each point in time -- hence, a heat equation in effectively one spatial dimension.

In this project the student will produce a code to numerically solve the UTSD equation based on algorithms available in the literature. Focus will be put on learning the C programming language required to achieve good computational speed, the use of numerical flux functions to capture shocks accurately and the use of grid stretching methods to capture a high resolution solution in areas of interest . After implementing the base code, time permitting, the student will extend the code to capture further features of interest.

## Alumni

**2013**

Aideen Costello (TCD; now at Morgan Stanley and First Derivatives)

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

Daniela Mueller (Mathematical Science UCD; now PhD candidate in UCD)

Benen Harrington (TCD, now a PhD candidate at University of York)

**2014**

James Fannon (Theoretical Physics, UCD; now PhD candidate in UL)

Andrew Gloster (Theoretical Physics, UCD; MSc at Imperial College London; now PhD candidate in UCD)

Shane Walsh (Theoretical Physics, UCD; now PhD Candidate in UCD and IRC scholar)

Adam Keilthy (TCD; now a PhD candidate in Oxford University)

**2015**

Patrick Doohan (Mathematical Studies, UCD; H Dip in Mathematical Science, UCD; MSc in Applied Mathematics at Imperial; now PhD candidate at Imperial College London)

Maria Jacob (ACM, UCD; now PhD candidate at University of Reading/Imperial College London)

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

Paul Beirne (Mathematics, UCD; now a PhD at candidate in UCD)

Daniel Camazon Portela

**2016**

Christopher Kennedy (ACM, UCD)

Emily Lewanowski-Breen (Maths and Science Education, UCD)

Michael O’Malley (Stats and Maths, UCD)

**2017**

Adam Ryan (Mathematics, UCD)

Luke Corcoran (Theoretical Physics, TCD)

Conor McCabe (ACM, UCD)

Joseph Curtis (Statistics, UCD)

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