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I have this information from Vladimir Guletskii. Maybe somebody could post/advertise it on nCafe as well. The deadlines are really soon so it is an urgent information.
The Department of Mathematical Sciences of the University of Liverpool has one EPSRC DTG studentship award for research leading to a PhD starting on 1 October 2012. Dr Jon Woolf proposes the following PhD research programme for this vacancy to work under his supervision:
Directed homotopy theory and (infinity,1)-categories
Directed homotopy theory studies spaces equipped with some notion of direction or ordering of points; there are several variants but the most relevant one for this project is locally pre-ordered spaces. These are spaces such that the points of each open subset are pre-ordered (suitably compatibly with the topology, and with the other local pre-orders). There are many natural examples: stratified spaces (points in higher codimension strata are deeper), space-time (ordered by causality), geometric simplices (points have the same ‘level’ as the highest vertex in the closure of the face in which they lie), and more generally geometric realisations of simplicial sets.
The initial aim of this project is to study an analogue of Grothendieck’s homotopy hypothesis
‘homotopy-types are infinity-groupoids, i.e. (infinity,0)-categories’;
namely that ‘directed homotopy types are (infinity,1)-categories’.
(Here, by a directed homotopy type we mean a directed space up to equivalence under an undirected notion of homotopy. Working with directed homotopies is also interesting, but leads to a different theory.) To be more precise, the aim is to set up a model structure on the category of locally pre-ordered spaces which is Quillen equivalent, via directed analogues of the singular simplicial set and geometric realisation functors, to the Joyal model structure on simplicial sets. The fibrant objects of the latter are precisely the quasi-categories, i.e. the simplicial models of (infinity,1)-categories.
The conjectural model structure on locally pre-ordered spaces should have a close connection with well-studied notions in the homotopy theory of stratified spaces (and from this it would derive much of its interest). In particular, the fibrant objects should include Quinn’s homotopically stratified sets, and for these weak equivalence should specialise to stratum-preserving homotopy equivalence.
The Initial applications (CV and Personal Statement on Research) can be emailed to jonwoolf@liverpool.ac.uk
Let me remind you of this urgent entry if you know somebody interested.
I posted it at the Cafe. I wonder if there’s an effect working here that I’ve seen in philosophy. It’s much easier to fill a postdoc with prescribed content than a PhD position.
Great, thanx.
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