# Keynote Speakers

**Prof.
Peter Cameron
University of St Andrews**

Peter Cameron is professor of Mathematics at the University of St Andrews, and professor emeritus at Queen Mary University of London, having previously held a position at Oxford University. He was awarded both the Junior and Senior Whitehead Prizes of the London Mathematical Society and the Euler Medal of the Institute of Combinatorics and its Applications. He has worked in various parts of combinatorics, group theory, logic and statistics, and has written more than 350 papers with over 200 coauthors (including Paul Erdős). He has recently been involved with a large project with mainly Indian mathematicians studying graphs defined on groups.

Speech Title: "The ADE Affair"

Abstract: The Coxeter–Dynkin diagrams of type ADE (shown above) are almost ubiquitous in mathematics, arising in areas from Lie algebras to general relativity, representation theory to finite graph theory, regular polyhedra to singularity theory. In 1978, Vladimir Arnold proposed the problem of explaining their extremely wide range as a “moden Hilbert problem”. This problem is by no means solved, and recent developments have added more occurrences such as cluster algebras. In my talk I will explain some of the many occurrences of these diagrams, and some of the connections between them. A highlight is the observation of John McKay (the person who originated “monstrous moonshine”) on a direct connection between quivers for the binary polyhedral groups and the ADE root systems. It must be said, though, that there is much still to do!

**Prof.
R. A. Bailey
University of St Andrews**

R. A. Bailey is Professor of Statistics at the University of St Andrews, and a Fellow of the Royal Society of Edinburgh. She worked for the Medical Research Council's Air Pollution Research Unit before studying at the University of Oxford, where she obtained a BA in Mathematics and a DPhil in Group Theory. As a post-doctoral fellow at the University of Edinburgh she learnt how to apply group theory to problems in design of experiments. She spent ten years applying this knowledge in the Statistics Department at Rothamsted Experimental Station, before moving to academia, being Head of Department or School at Goldsmiths College and at Queen Mary College, both in the University of London. She was President of the then-British Region of the International Biometric Society from 2000 to 2002, and has also served on various committees of the London Mathematical Society, the Royal Statistical Society and the Institute of Mathematical Statistics.

Speech Title: "Designs on Strongly-regular Graphs"

Abstract: Some
particularly nice graphs
are the strongly-regular
graphs. Their edges and
non-edges form the
associate classes of an
association scheme. The
corresponding
Bose–Mesner algebra
(linear combinations of
the adjacency matrices)
has three common
eigenspaces, one of
which is V_{0},
which consists of the
constant vectors. In
classical work on design
of experiments, the
experimental units are
grouped into b blocks of
size k. This corresponds
to the strongly-regular
graph consisting of b
complete graphs of size
k, with no edges between
them. In some other
experiments, the
experimental units are
all pairs of individuals
who have to undertake a
given task together. If
all such pairs are used
exactly once each, then
the set of pairs forms a
triangular association
scheme. Two types of
design are particularly
important. In *
balanced* designs,
the variance of the
estimated difference
between any two
treatments is the same.
In *orthogonal*
designs, the linear
combination of responses
which gives the best
unbiased estimator of
any difference between
treatments is obtained
by simple averaging.
Such designs are often
said to have *
commutative orthogonal
block structure*. I
will give some
constructions for
balanced designs and
some for designs which
have commutative
orthogonal block
structure, in each
scenario.