A talk on discrete mathematics

The Academy for Discrete Mathematics and Applications is an Indian organisation founded in 2005 to foster and support interest in discrete mathematics. The current president, Ambat Vijayakumar, has started up a Colloquium Lecture seres, and honoured me by asking me to give the first lecture.

I decided that, rather than up-to-the-minute new research, a more reflective talk would be appropriate, and I chose three interactions between discrete mathematics and other parts of mathematics, in all of which I had some involvement:

• Using root systems to prove a stronger version of Hoffman’s conjecture about graphs whose adjacency matrix has smallest eigenvalue −2 (or greater): these must be generalized line graphs (a class invented by Hoffman) with finitely many exceptions);
• The Erdős–Rényi countable random graph and its relationship to Urysohn’s universal and homogeneous Polish space;
• a beautiful graph obtained from the smallest sporadic simple group (the Mathieu group M₁₁).

There were nearly 100 people at the on-line event, and many questions were asked afterwards; time did not permit me to answer all of them, I am afraid.

The slides are here.

One final comment. The organisers had used the Webex platform, and I only realised a day in advance that the Linux version does not permit screen sharing (if you delve into their website you find this is a new feature coming soon). So we had to use the old-fashioned “next slide please” method. So I am not recommending Webex just at the moment!