Just a brief note that my second paper on my thesis topic has been accepted by the Journal of Number Theory. The full citation is: Graeme Taylor, Lehmer’s conjecture for matrices over the ring of integers of some imaginary quadratic fields, Journal of Number Theory, Volume 132, Issue 4, April 2012, Pages 590-607, ISSN 0022-314X, [...]
Archive of posts filed under the cyclotomic category.
Cyclotomic Matrices and Graphs: Waterloo
I gave the Number Theory Seminar at the Department of Pure Mathematics, University of Waterloo on Thursday, September 8th. My slides are available in presentation or handout form (except the latter is missing the interlacing demo which didn’t render into individual slides correctly). I used this talk as an opportunity to present some results that [...]
Cyclotomic Matrices and Graphs over the ring of integers of some imaginary quadratic fields
Online access to my first paper.
Cyclotomic Matrices and Graphs: Warwick
I’m continuing to tour my Cyclotomic Matrices and Graphs talk; today I presented it at the University of Warwick. Here’s the latest and greatest iteration of the slides, mostly unchanged except for the current state of the computer search for minimal noncyclotomics of at most ten vertices. I’d hoped to finish that this month, but [...]
Talks on Cyclotomic Matrices and Graphs
I’m speaking this afternoon at the Heilbronn Seminar in Bristol: my slides are available here. This is essentially (up to permutation, and modulo errors!) the talk I gave at Royal Holloway in October, although the last few slides have been replaced with a result I’ve found since then. I try to avoid technical details of [...]
Thesis available
Downloadable copy of my thesis.
Viva
Today I successfully defended my PhD thesis, Cyclotomic Matrices and Graphs. There are of course numerous corrections to be made, but I hope to have those done within the next couple of weeks and to make the final version available online. Until then, here is the abstract: We generalise the study of cyclotomic matrices – [...]
Workshop on Discovery and Experimentation in Number Theory
I’m back from the Fields Institute in Toronto, where I spoke at the above workshop, on my usual topic of cyclotomic and 4-cyclotomic matrices/graphs. During the talk I described my conjecture that a graph is maximal cyclotomic if-and-only if it’s 4-cyclotomic, and after an hour at the blackboards with James McKee I now have a [...]
An infinite family of n-hypercubes
Having hit a bit of a wall trying to prove that a maximal cyclotomic matrix necessarily squares to 4I, I’ve been exploring related questions instead. For instance, it only took a couple of tweaks to my code to search for matrices that square to 3I instead of 4I; there turn out to be only finitely [...]
A dangerous assumption
I’ve often said that trying to explain your work to others is the best way to check you understand it, and whilst preparing for this week’s talk at Cambridge (which was apparently well received!) I started to have some doubts about an algorithm I’d been using. In the end I didn’t go into enough technical [...]
