MA3823 Group Theory
This course, typically taken by third year students at WPI is an introduction to abstract algebra and to axiomatic methods. I have taught the course since 2016. While the required syllabus finishes with the first isomorphism theorem, I generally finish the course with discussion of the Sylow theorems and (unexamined) discussion of free groups, paradoxical decompositions and the Banach-Tarski paradox. My lecture notes follow, with the weekly assessment.
M1_AxiomsEtc.pdf (Assignment1.pdf)
M2_Subgroups.pdf (Assignment2.pdf)
M3_PermGroups.pdf (Assignment3.pdf)
M6_Sylow.pdf (Assignment5.pdf)
Since 2017, I have developed a series of interactive learning sessions which take place in parallel to traditional lectures. Most of the sessions focus on using techniques developed in class to analyze the symmetries of regular n-gons in two dimensions and the cube in three dimensions. Students seems to benefit from the sessions, and report that they make some abstract concepts much more concrete.
Lecturing prior to Worcester Polytechnic Institute
Coding theory
I taught the coding theory part of MATH3302 `Coding and cryptography' in Semester 1 of 2013 in the University of Queensland.
Permutation groups
In Semester 1 of 2014 I offered an Honours level reading course on permutation groups. We worked through the first several chapters of Isaacs' Algebra (as far as Wielandt's proof of the Sylow theorems). Then we worked through the first chapter of Wielandt, and finished with the proof of the O'Nan-Scott theorem as contained in Dixon & Mortimer's Permutation Groups.
I designed, wrote and delivered the following lecture courses in the National University of Ireland, Galway in the second semester of the 2011/2012 academic year (January-March 2012). I also set and corrected the exams for these courses, and dealt with all aspects of course administration.
MA313: Linear Algebra II: This was a capstone algebra course for final year Arts students studying (honours) Mathematics or (pass) Mathematical Studies. It was also taken by a number of pass Science students and some taught masters students. There were approximately 70 students in total. The topics covered were:
Inner products, orthogonality and Fourier series
Orthogonal and symmetric matrices
An introduction to linear programming.I prepared a substantial set of notes, including problems and selected solutions. These are available here..
Introduction to Mathematics
This was a course aimed at disadvantaged and mature students hoping to studying Science or Engineering. There were approximately 35 students in total, from a wide range of backgrounds and with varying levels of ability. I taught the half of the course devoted to algebra. The topics covered were
Numbers, functions and polynomials
Complex numbers, roots of equations
Trigonometry, lengths, angles, areas and volumes in two and three dimensions
Systems of two linear equations in two unknowns, solution via matrix inversion
Tutoring Experience
I worked as a tutor (teaching assistant) throughout my time as a graduate student at the National University of Ireland, Galway from October 2007 to March 2012. I worked on most of these courses multiple times.
Masters: Representation theory and character theory of finite groups. This was an advanced course offered to taught Masters and research PhD students. The course was a standard introduction to representation theory as far as Frobenius reciprocity. (The material covered was more or less Chapters 1-21 of Liebeck and James.)
MA387/533: Probability. This was the most advanced theoretical probability course taught in Galway, covering discrete and continuous probability theory as far as a proof of the central limit theorem.
CS402: Cryptography and number theory. This was a final year/masters level course for students in mathematics and computer science. It covered affine cryptosystems, RSA, factorisation techniques and an introduction to cryptography on elliptic curves over finite fields.
MA344: Group Theory II. This was a third year course in group and monoid actions, covering basic permutation group theory and Sylow theory.
MA180: First Year Honours mathematics. This course contained an introduction to number theory, linear algebra, introductory analysis (including epsilon-delta) and techniques of integration.
MA180 (Gaeilge): Céad Onóireacha. As the only graduate student fluent in the Irish language, I taught the Irish language version of the First Year Honours syllabus too.