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 14th of March at 1.30pm in Lecture Theatre D

Wilf Wilson
University of St Andrews
Computing direct products of semigroups

Direct products of semigroups are completely straightforward to define. However, to perform many algorithms from computational semigroups theory, we require a generating set for a given semigroup - and direct products are not defined in terms of generating sets. I will talk about some aspects of practically constructing reasonably small generating sets for direct products of semigroups.

Wednesday the 14th of March at 1.00pm in Lecture Theatre D

Fernando Flores Brito
University of St Andrews
More on congruences of EndF_n(G)

I will discuss the outcomes of the methodology of my first approach on classifying the congruences of this monoid, the results I have about congruences of its minimal ideal, and some other results about congruences of higher rank.

Wednesday the 28th of February at 1.00pm in Lecture Theatre D

Craig Miller
University of St Andrews
Right noetherian semigroups

A semigroup $$S$$ is right noetherian if every right congruence on $$S$$ is finitely generated. In this talk, we will begin by looking at some known examples of right noetherian semigroups. We will then consider whether the property of being right noetherian is preserved by various semigroup constructions.

Wednesday the 28th of February at 1.30pm in Lecture Theatre D

Ashley Clayton
University of St Andrews
Finitary conditions for fiber products of free objects

If $$A$$ and $$B$$ are two algebras of the same type, then a subdirect product of $$A$$ and $$B$$ is a subalgebra $$C$$ of $$A\times B$$, such that the projections from $$C$$ onto $$A$$ and $$B$$ are surjective. An important tool for considering subdirect products is via fiber products of algebras, which can be constructed via homomorphisms from two algebras onto a common image. In this talk, we give some results to the typical finitary questions for fiber products of free semigroups and monoids, and consider some further related construction questions.

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.