This week I’ve been at the rather swish Alan Turing building in Manchester for the MAGIC Postgraduate Conference and LMS Northern Regional Meeting.

I spoke at the former, on the subject of Integer Matrices with Constrained Eigenvalues. Here are my slides: it’s a fairly breezy 15 minute overview of the problem (which integer symmetric matrices have all eigenvalues in [-2,2]?) and its solution, covering Mahler measure, cyclotomic matrices, interlacing, and charged signed graphs. For further reading, here is the paper by McKee and Smyth (my supervisor) with their proof of the presented classification; also by Smyth is a survey on Mahler Measure of 1-variable polynomials.

In my own work I’ve generalised the idea of cyclotomicity (all eigenvalues in [-2,2]) to Hermitian matrices with algebraic integer entries from imaginary quadratic extension fields. I think I have a complete classification of these, with an alternative proof of the above rational integer case as a subcase. The results at least will hoepfully appear here at some point, although for the proofs you’ll probably have to wait for my thesis.

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