# Category Archives: doing mathematics

how is it done?

## 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

## The road closure property

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

## COBS and equitable partitions

It happens sometimes that researchers working in different fields study the same thing, give it different names, and don’t realise that there is further work on the subject somewhere else. Here is a story of such a situation, which arose … Continue reading

## An apology

What would life be like if I could remember all the things I ever knew? Yesterday I was led to something I posted here twelve years ago. This was based on a talk to the London Algebra Colloquium by Mark … Continue reading

Posted in doing mathematics | | 4 Comments

## The enhanced power graph is weakly perfect

Earlier this year, I posed a combinatorial problem, a solution to which would imply that, for any finite group G, the enhanced power graph of G is weakly perfect, that is, has clique number equal to chromatic number. Recall that … Continue reading

Posted in doing mathematics | | 19 Comments

## More on the 3p paper

I wrote here about Peter Neumann’s paper on primitive permutation groups of degree 3p, where p is a prime number. Well, summer is almost over, but my undergraduate research intern Marina Anagnostopoulou-Merkouri and I have done our work and produced … Continue reading

## Why I’d like to see this solved

I am aware that quite a number of people have been captivated by the problem I posed. So here is the motivation for it, with some additional remarks and commennts. First, to repeat the problem: Problem: Let n be a … Continue reading

Posted in doing mathematics | | 3 Comments

## I’d like to see this solved

Here is a problem that I would really like to see solved. I have spent quite a bit of time on it myself, and have suggested it to a few other people, but it still resists all attacks, though it … Continue reading

Posted in doing mathematics, open problems | 9 Comments

## 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