### Undergraduate Research Projects 2017

### The School of Mathematics and Statistics will be offering a number of individual summer research placements in 2017. 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 the2017 Research Projects Application Form, a cover letter and CV (max 4 pages) by email to robert.osburn@ucd.ie with “Projects 2017” in the subject. Please see attached a guidance on writing a CV suitable for this programme.

The deadline for applications is **Friday 31st March 2017 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 at the 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 Lennon Ó Náraigh

### Title: Parallel I/O for Computational Fluid Dynamics

High-performance Computational Fluid Dynamics (CFD) exploits parallel algorithms to boost the performance of CFD calculations, leading to ultra-high-resolution simulations that can be achieved in a timely fashion. However, Input and Output (I/O) – especially for the generation of output data to be used for postprocessing, visualization, and characterization of the fluid system – is sometimes a forgotten aspect of such high-performance computing.

In this project the student will work on a computational model for turbulent flows (S-TPLS) and implement a parallel data I/O framework to improve the performance and scalability of the code. The framework will be based on the NetCDF standard. The student will implement the new framework and test the performance of the code when run in parallel on many CPUs. Time permitting, there will be opportunities to work on other more fluids-based aspects of the computational model, including improvement of the large-eddy simulation technique used in the code for the resolution of sub-grid-scale turbulence features

### Dr Kevin Hutchinson

### Title: Bloch Groups and Dilogarithms over finite fields.

Bloch groups and scissors congruence groups of fields arose first in the 1980s out of connections between 3-dimensional hyperbolic geometry, algebraic K-theory

and homology calculations of linear groups such as SL_2. They are also closely related to the properties of the dilogarithm function which plays a role in geometry and number theory.

This project concerns scissors congruence groups and dilogarithms associated to finite fields. There are many unanswered questions in this arena of interest

to current mathematical research. For example, the scissors congruence group of finite field has a simple explicit description. It's size and structure are known, but only by invoking deep theorems from K-theory and homology. A direct and more elementary proof of these results is very desirable.

### Dr Mark Dukes

### Title: Permutation patterns

Permutation patterns is an area of combinatorics that studies ordered patterns contained in permutations. The area has been studied implicitly for a long time, and explicitly for the last few decades, owing to their intrinsic interest and to the fact that they often turn out to be the right `language' to express important properties of structures studied in theoretical computer science, mathematics and physics. In particular, characterisations of other structures can sometimes be translated into characterising permutations avoiding or having a prescribed number of occurrences of a given pattern.

The origin of the field of permutation patterns can be traced back to the study of sortability by certain devices studied in computer science, namely stacks and queues. This development started with Knuth, who showed, in The Art of Computer Programming that permutations avoiding the pattern 231 are precisely those that can be sorted to increasing order with the help of a stack. The aim of this project is to look at some outstanding problems in the field, with an emphasis on enumeration and connections to other parts of discrete mathematics.

### Dr Marius Ghergu

### Title: Subharmonic Functions in Real Analysis.

In mathematics, subharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.

Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a line intersect at two points, then the graph of the convex function is below the line between those points. In the same way, if the values of a subharmonic function are no larger than the values of a harmonic function on the boundary of a ball, then the values of the subharmonic function are no larger than the values of the harmonic function also inside the ball.

During the project the student will learn about most important properties of subharmonic functions, their integral representations and how they are used in other applications of real life.

## 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)

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