Many of our seminars are Pure Mathematics Colloquia, so why not check here?

## Previous CIRCA & Algebra Semigroups - 2017 to 2018

Previous Pure Mathematics Colloquia from: 2017/18, 2016/17, 2015/16, 2014/15, 2013/14, 2012/13, 2011/12, 2010/11, 2008/09, 2007/08

Wednesday the 21st of February at 1.00pm in Lecture Theatre D

Rosemary Bailey
University of St Andrews
A substitute for the non-existent affine plane of order 6

A Latin square of order $$n$$ can be used to make an incomplete-block design for $$n^2$$ treatments in $$3n$$ blocks of size $$n$$. The cells are the treatments, and each row, column and letter defines a block. Any pair of treatments concur in 0 or 1 blocks, and it is known that the block design is optimal for these parameters.

If there are mutually orthogonal Latin squares, then the process can be continued, eventually giving an affine plane. But there are no mutually orthogonal Latin squares of order 6, so what should we do if we need a block design for 36 treatments in 30 blocks of size 6?

I will describe how a series of mistakes and wrong turnings in a different research project led to an answer.

Wednesday the 7th of February at 1.00pm in Lecture Theatre D

Peter Cameron
University of St Andrews
Reed--Muller codes and Thomas' conjecture

A countable first-order structure is countably categorical if its automorphism group has only finitely many orbits on n-tuples of points of the structure for all n. (Homogeneous structures over finite relational languages provide examples.) For countably categorical structures, we can regard a reduct of the structure as a closed overgroup of its automorphism group. Simon Thomas showed that the famous countable random graph has just five reducts, and conjectured that any countable homogeneous structure has only finitely many reducts. Many special cases have been worked out but there is no sign of a general proof yet. In order to test the limits of the conjecture, Bertalan Bodor, Csaba Szabo and I showed that a vector space over GF(2) of countable dimension with a distinguished non-zero vector has infinitely many reducts. The proof can most easily be seen using an infinite generalisation of the binary Reed--Muller codes.