Bohemian Matrices in Manchester

Bohemian matrices are families of matrices in which the entries are drawn from a fixed discrete set of small integers (or some other discrete set). The term is a contraction of BOunded HEight Matrix of Integers and was coined by Rob Corless and Steven Thornton of the University of Western Ontario. Such matrices arise in many situations:

• adjacency matrices of graphs have entries from ;
• Bernoulli matrices, which occur in compressed sensing, have entries from ;
• Hadamard matrices have entries from and orthogonal columns; and
• matrices with elements from provide worst case growth factors for Gaussian elimination with partial pivoting and yield the most ill conditioned triangular matrices with elements bounded by .

Rob’s group have done extensive computations of eigenvalues and characteristic polynomials of Bohemian matrices, which have led to interesting observations and conjectures. Many beautiful visualizations are collected on the website http://www.bohemianmatrices.com as well as on the Bohemian Matrices Twitter feed.

In June 2018, Rob and I organized a 3-day workshop Bohemian Matrices and Applications, bringing together 16 people with an interest in the subject from a variety of backgrounds. The introductory talks by Rob, Steven, and I were videod (and are embedded below), and the slides from most of the talks are available on the conference website.

We scheduled plenty of time for discussion and working together. New collaborations were started, several open problems were solved and numerous new questions were posed.

The workshop has led to various strands of ongoing work. Steven has created the Characteristic Polynomial Database, which contains more than polynomials from more than Bohemian matrices and has led to a number of conjectures concerning matches of properties to sequences at the On-Line Encyclopedia of Integer Sequences. Three recent outputs are

• E. Y. S. Chan, R. M. Corless, L. Gonzalez-Vega, J. R. Sendra, J. Sendra and S. E. Thornton, Bohemian Upper Hessenberg Matrices, ArXiv preprint arXiv:1809.10653, September 2018.
• E. Y. S. Chan, R. M. Corless, L. Gonzalez-Vega, J. R. Sendra, J. Sendra and S. E. Thornton, Bohemian Upper Hessenberg Toeplitz Matrices, ArXiv preprint arXiv:1809.10664, September 2018.
• S. L. Hill, M. N. Huxley, M. C. Lettington and K. M. Schmidt, Some Properties and Applications of Non-trivial Divisor Functions, ArXiv preprint arXiv:1806.00651, June 2018; to appear in The Ramanujan Journal [added to this post October 26, 2018].

Sponsorship of the workshop by the Heilbronn Institute for Mathematical Research, the Royal Society and the School of Mathematics at the University of Manchester, as well as support from Canada for some of the Canadian participants, is gratefully acknowledged.