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Category Archives: exposition
Deep isoclinism
Graphs and groups, in my view, are two subjects engaged in a wide-ranging dialogue at present. Graphs can be used to describe interesting classes of groups, and groups to construct interesting graphs. But I am delighted that recently, in a … Continue reading
Digraphs on groups
I have spent a lot of time recently thinking about graphs on groups. To recall the rules: the vertex set must be the group (in general, but not here, I allow an automorphism-invariant subset or the quotient by an automorphism-invariant … Continue reading
The rational world and the rational Urysohn space
The set Q of rational numbers is obviously an interesting topological space. In 1920, Waclaw Sierpiński gave a lovely characterisation of it. The simplest way to state it is to say that a countable, metrisable, space without isolated points is … Continue reading
Posted in exposition, open problems
Tagged Urysohn space, Sierpinski's theorem, cycle structure
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A birthday discovery
Some of the standout results about graphs on groups are characterisations of the groups G for which two types of graph (for example, the power graph and the commuting graph) coincide on G. Sometimes the proofs are long and difficult, … Continue reading
JLMS centenary
This year is the 100th anniversary of the Journal of the London Mathematical Society. They have celebrated the centenary by an issue of the journal containing ten papers, each starting from an important paper published in the Journal. The entire … Continue reading
A remarkable computation
Today I watched, on Natalia Maslova’s on-line seminar from Yekaterinburg, a talk by Sergey Shpectorov from Birmingham, on the non-existence of a strongly regular graph with parameters (85,14,3,2): this is a graph with 85 vertices, regular with valency 14, and … Continue reading
The Universe according to Leibniz?
I have just been reading Lee Smolin’s recent book Einstein’s Unfinished Revolution. My sources for what is going on deep in theoretical physics are Carlo Rovelli (whom I met at How the Light Gets In some years ago), Lee Smolin, … Continue reading
Posted in exposition, Uncategorized
Tagged Lee Smolin, Leibniz, non-locality, quantum mechanics, realism
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Ramsey, Fraïssé, and orders
In the 1980s, Jarik Nešetřil investigated Ramsey classes: these are classes of structures over a fixed relational language satisfying the condition that, for any structures A,B, there exists C such that, if the embeddings A→C are coloured red and blue, … Continue reading
Posted in exposition, mathematics
Tagged Fraïssé classes, Kechris, KPT theorem, Pestov, Ramsey classes, Todorčević
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Why transformation monoids are harder than permutation groups
For permutation groups (or transformation monoids), we don’t need to assume the associative law, since composition of functions is always associative. So a permtation group is a set of mappings satisfying the identity, inverse and closure axioms. This implies that … Continue reading
Graphs defined on algebras
Next February, I will be speaking at AAA108 (Arbeitstagung Allgemeine Algebra) in Vienna. I thought this might be a chance to take some of the work about graphs defined on groups, and see whether it can be extended to arbitrary … Continue reading
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