# Category Archives: exposition

a post aimed to teach something

## Mathematical Structures

My last major job at Queen Mary University of London more than ten years ago was designing and presenting a new first-semester first-year module to be taken by all students on mathematics programmes or joint programmes involving mathematics. I discussed … Continue reading

Posted in exposition, teaching | | 4 Comments

## Two things that should be related

Here are two things that look as if there should be a relation between them. A graph duality In my paper on graphs defined on groups, I invented an ad hoc duality relation between pairs of graphs. The context is … Continue reading

## A theorem on polytopes

You know what polygons and polyhedra are. How do we extend their study to higher dimensions? There are two parts to this question. The first involves incidence geometry: vertices, edges, faces, etc. Here the generalisation is fairly straightforward. A polygon … Continue reading

Posted in doing mathematics, exposition | | 3 Comments

My work with João Araújo and other semigroup theorists has produced a number of permutation group properties which lie between primitivity and 2-homogeneity, especially the synchronization family. Another of these is the road closure propery, which I have discussed here … Continue reading

## BCC29 at Lancaster

Last week we celebrated the 29th British Combinatorial Conference in Lancaster, face to face. (As a side observation, this was by far the largest social gathering I have been at since the start of the pandemic; I found it both … Continue reading

## Peter Neumann’s 3p paper

In 1955, Helmut Wielandt published a paper proving the following theorem: Let G be a primitive permutation group of degree 2p, where p is a prime greater than 3, which is not doubly transitive. Then p = 2a2+2a+1 for some positive integer … Continue reading

## An O’Nan-Scott note

My old friend Leonard Soicher is visiting St Andrews this week, having made a better job of retiring than I did. Today he gently took me to task for using the expression “simple diagonal type” for one of the types … Continue reading

## Graphs on groups, 13

There are many results about the universality, or otherwise, of various graphs defined on groups: answers to questions of the form “for which graphs Γ is there a group G such that Γ is isomorphic to an induced subgraph of … Continue reading

## Graphs on groups, 12

One thing I have learned from the project is that the most interesting question about graphs defined on groups is this: given two types of graph defined on a group G, what is the class of groups for which the … Continue reading

Posted in doing mathematics, exposition | | 1 Comment

## Graphs on groups, 11

A brief interlude to describe another recent preprint, and as in the preceding post I will concentrate on one result in the paper. I don’t know why it happens, but in this project one of the most interesting graph parameters … Continue reading