Between 2006 and 2010 I held a scholarship from the Maxwell Insitute for postgraduate study at the University of Edinburgh‘s Mathematics Department. My thesis was supervised by Chris Smyth, and generalises the study of Mahler measure via integer symmetric matrices and charged signed graphs to Hermitian matrices / labelled graphs over rings of integers of imaginary quadratic extensions. The abstract can be found here, or you download the full text.

During my time in Edinburgh, I was involved with the following activities.

## Conferences attended

- Maths 2010 (Joint BMC/BAMC Meeting) Edinburgh, UK April 6-9 2010.
- Young Researchers in Mathematics Cambridge, UK March 25-27 2010.
- The Diverse Faces of Arithmetic Norwich, UK December 14-16 2009.
- Workshop on Discovery and Experimentation in Number Theory Toronto, CA September 22-26 2009. (Invited Talk)
- Beyond Part III Cambridge, UK April 16-18 2009. (Contributed Talk)
- MAGIC Postgraduate Student Conference 2009 Manchester, UK January 12-14 2009. (Contributed Talk)
- 12th Workshop on Elliptic Curve Cryptography (ECC 2008) Utrecht, NL September 22-24 2008.
- Eighth Algorithmic Number Theory Symposium ANTS-VIII Banff, CA May 17-22 2008
- SAGE Days 6: Cryptology, Number theory, and Arithmetic Geometry Bristol, UK November 10/11 2007.
- ECC2007 11th Workshop on Elliptic Curve Cryptography Dublin, IE September 5-7 2007.
- Topics in Algebra, Geometry and Number Theory Utrecht, NL August 13-24 2007.
- 25th Journées Arithmétiques Edinburgh, UK July 2-6 2007.

## Talks Given

- Integer Matrices with constrained Eigenvalues
*Workshop on Discovery and Experimentation in Number Theory, September 2009*

30 minute presentation of the classification of maximal cyclotomic graphs in the rational integer case, and of 4-cyclotomic graphs in the imaginary quadratic extension case; plus evidence for the latter being all maximal cyclotomic graphs over such rings. - Why Cryptography Doesn’t Guarantee Security
*Geometry Club, April 29, 2009*

Brief overview of Public key cryptography, DLP, Diffie-Hellman key exchange and the man-in-the-middle attack; cryptographic protocol design- authentication, replay attacks, mutual authentication, reflection attacks, the undecidability of secrecy via reduction to Post’s Correspondence Problem; the challenge of post-quantum computing cryptography, lattices and the shortest/closest vector problems.*(Talk and chalk, so no slides; some related writings are available.)* - Cyclotomic Matrices and Graphs
*Beyond Part III conference, April 2009*

45 minute overview of the classification of cyclotomic matrices corresponding first to graphs, then charged signed graphs; plus some observations on my generalisations to cyclotomic matrices over rings of integers. (*Sorry, no slides: this was a traditional ‘talk and chalk’!*) - Integer Matrices with constrained eigenvalues
*MAGIC Postgraduate conference, January 2009*

15 minute introduction to a special case of my research problem, the classification of integer symmetric cyclotomic matrices: covers Mahler measure, cyclotomic matrices, interlacing and charged signed graphs. - Public Key Cryptography and Secret-sharing
*Postgraduate Colloquium, October 30, 2008*

A brief discussion of the need for public key cryptography, an illustration of Diffie-Hellman by mixing paint, scalar multiplication/DLP as a potential one-way function, and some open problems. - Computational Aspects of Elliptic Curve Cryptography
*Geometry Club, April 18, 2008*

A computer-science view of ECC- Diffie-Hellman and one-way functions for key exchange; the generic Discrete Logarithm Problem and BSGS algorithm; scalar multiplication- addition chains, fast exponentiation, m-ary methods and windowing; group law implementations, Side-channel attacks and the Edwards form.

For a recent example of a side-channel attack, see this article from the Register. - The Mathematics of Being Nice
*Postgraduate Colloquium, October 11, 2007*

A brief discussion of the Prisoner’s Dilemma and the emergence of cooperation from a non-cooperative setting. **First Year Report**: Full report / OHP slides. (N.B. report does not include the source code appendix).- Hyperelliptic Curves over finite fields
*Geometry Club, April 27, 2007*

An overview of the geometry of hyperelliptic curves as used in cryptography: the discrete logarithm problem and ElGamal; explicit construction of the Jacobian of the curve; Frobenius endomorphism, zeta functions and the point counting problem.

## Seminars and courses

I often attended the Postgrad colloquia; and could sometimes be found at the Geometry club or Informatics seminars from the LFCS, IPAB, or Interdisciplinary Tea series. I also participated in various Transkills courses, in particular the *Science Communication in Action program* (see the Outreach page for more on that.)

**Extended courses**

- 2009/10 Semester 1 Information Security: Design Fundamentals and Selected Research Topics.
- 2009/10 Term 1 Languages for all: Norwegian 1.1.
- 2006/07 Semester 2 Informatics: Formal analysis of cryptographic protocols. (some notes.)
- 2006/07 Semester 1 Informatics: Gröbner bases (informatics).
- 2006/07 Semester 1 Mathematics Course: Complex Algebraic Surfaces

## Tutoring

- 2009/10 Semester 1: Honours 1 Practical calculus (MAT-1-PCa) / Solving equations (MAT-1-SEq) (
**Feedback**) - 2008/09 Semester 2: Honours 1 Geometry and Convergence (MAT-1-GCo) / Group theory (MAT-1-GTh) (
**Feedback**) (*Nominated for a EUSA Teaching Award*.) - 2008/09 Semester 1: Honours 1 Practical calculus (MAT-1-PCa) / Solving equations (MAT-1-SEq); Mathsbase
- 2007/08 Semester 2: Honours 1 Geometry and Convergence (MAT-1-GCo) / Group theory (MAT-1-GTh); Honours 2 Discrete Mathematics (MAT-2-DiM)
- 2007/08 Semester 1: Honours 1 Practical calculus (MAT-1-PCa) / Solving equations (MAT-1-SEq); Mathsbase
- 2006/07 Semester 2: Honours 1 Geometry and Convergence (MAT-1-GCo) / Group theory (MAT-1-GTh)
- 2006/07 Semester 1: Honours 1 Practical calculus (MAT-1-PCa) / Solving equations (MAT-1-SEq); Mathsbase