Logic List Mailing Archive

PhD student position on "Homogeneous structures", Leeds (U.K.)

Homogeneous structures, 
homomorphism-homogeneity and automorphism groups

As part of an EPSRC grant at the University of Leeds with the title above, 
there will be a fully funded PhD studentship, and we seek suitably 
qualified candidates to begin their PhD in the autumn of 2009.


The PhD student appointed will be jointly supervised by Professors 
Macpherson and Truss, and will work on an aspect of the EPSRC research 
grant.  He or she will interact not only with the two investigators, but 
also with the postdoctoral research assistant, Dr Deborah Lockett. There 
will be regular meetings between the RA, the PhD student, and the 
investigators, and other interested research students, and probably a 
seminar with this group. We plan to hold a short meeting in Leeds on 
homogeneous structures.

The project concerns aspects of homogeneous structures. A countable 
structure is said to be 'homogeneous' if any isomorphism between finite 
substructures extends to an automorphism. The initial theory of 
homogeneous structures was developed as part of model theory. One of the 
key achievements was a classification by Cherlin (1998), of the 
homogeneous digraphs. The class of examples has great complexity but the 
description is clean and beautiful. However, the classification sheds 
little light on what homogeneous (even binary) structures look like in 
general.

The very general framework of homogeneity means that the subject touches 
many parts of mathematics, such as model theory, connections of finite 
model theory with computer science, group theory, descriptive set theory, 
and, in particular, combinatorics. Much of this has developed since 
Cherlin's memoir. For example, there is now wide interest in homogeneous 
metric spaces, in connections with structural Ramsey theory in 
combinatorics, and with topological dynamics. It has become urgent to 
revisit classification in homogeneous structures, to identify how far it 
can reasonably be taken, and whether, if one requires less than full 
classification, meaningful descriptions remain.

This is a project in combinatorics, but it has strong connections with 
model theory, group theory, and theoretical computer science. Candidates 
with interests in any or all of these areas may apply.

The studentship is advertised on 
http://www.findaphd.com/search/showproject.asp?projectid=22740&searchtype=n&page=1 
and anyone interested is encouraged to contact one or both of the 
investigators on pmtjkt@leeds.ac.uk<mailto:pmtjkt@leeds.ac.uk> or 
pmthdm@leeds.ac.uk.