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!
About Peter Cameron
I count all the things that need to be counted.
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