Mathematical and Statistical Sciences

Geometric Rigidity Theory
This is concerned with the geometric and combinatorial mathematics associated with bar and joint frameworks and other related structures. Recently I am interested in connections with polyhedral combinatorics and more general simplicial complexes.

Linear and multiilinear algebra over local rings
Local rings arise naturally in several mathematical contexts. We have studied hermitian and skew hermitian forms over such rings and proved several new classification theorems for such forms. We have also investigated analogues of the classical matrix groups in this context. 

Problem Solving in Mathematics
I am interested in the role of problem solving as a driver of learning in mathematics. Undergraduate mathematics faces special challenges in implementing a research-led approach. Due to the highly abstract and technical nature of contemporary research in many (but not all!) mathematical disciplines it can often be difficult to introduce undergrad students to open problems in contemporary mathematics. However there is a rich tradition of mathematical problem solving that can provide a bridge. Throughout my career at NUI Galway I have tried to promote student interest in problem solving contest at all levels. In particular 

• The mathematical enrichment programme seeks to engage preuniversity students in extracurricular mathematics with a particular focus on preparing for so-called Olympiad contests such as the Irish Mathematical Olympiad and the International Mathematical Olympiad. I been heavily involved in the administration and teaching of this course for the last 15 years.
• The PRISM contest is also for preuniversity students and is one of the largest one day mathematical contests in Ireland with typically 4000-5000 students participating. I was one of the founders of this contest and have been responsible for maintaining and upgrading its website for the last 10 years.

E-assessment in mathematics 
I am very interested in using new technologies to support learning. Mathematics is a subject that offers particular opportunity to support student learning with automated assessment technologies. In recent years, several new developments contributed to an rapid growth in the area of mathematical e-assessment. In particular 

• MathJax essentially solved the problem of presenting complex mathematical notation on the web
• Open source computer algebra systems such as SageMath and Maxima have advanced to a high level of sophistication - comparable to state of the art systems such as Mathematics/Maple.
• Widespread student access to the web through a variety of devices and platforms
• Web development in general has changed dramatically in the last decade. The latest web development frameworks have made it much easier to develop sophisticated/responsive applications without requiring a large team of developers

I have developed an online e-assessment system for mathematics called Ubbr. This exploits the power of SageMath and Django to provide an easy methid of creating web assessments for all levels of undergraduate mathematics. It is, to my knowledge, the first such system to be based on SageMath, which is the leading open source computer algebra system. Because of this it allows the user to easily create problems in topics such as abstract algebra/graph theory/set theory that have not traditionally been adressed by other e-assessment systems. 

I have used this system successfully in the last two academic years, both for large group teaching to first year service courses and for more specialised courses in actuarial mathematics for third year students. I plan to roll out the system to several other courses in the 2017-18 academic year. 

See https://eams.ncl.ac.uk/session/ubbr-a-python-sagemath-package-for-mathematical-e-assessment/
for a talk that I gave about this system at the E-Assessment in Mathematical Sciences conference at Newcastle University in September 2016.