9. Geometry of foliated manifolds and singular spaces
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Polish supervisor:
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Robert Wolak
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Cooperating partners:
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Jesus Alvarez Lopez (Univ. Santiago de Compostela, Spain)
Warwick Tucker (Uppsala University)
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Singular spaces like orbifolds, or more generally stratified
spaces are the focus of increasing interest of physicists as more
advanced models find that manifolds are not the right topological
framework. Singular spaces should be taken into consideration.
Likewise foliated spaces also appear in some cosmological models.
The study of geometry and topology, as well as global analysis of
singular space require the development of new tools which take
into account the singularities. There are tools well-known for
manifolds and those developed in the singularity theory. Manifold
tools should be corrected to be able to take into account the
singularities. The other tools are often of local nature and
purely topological or of algebraical nature and pay no attention
to the geometry involved.
We have been working on the problem for several years proposing a
foliated approach, i.e. some important singular spaces, in
particular orbifolds, can be desingularized by a foliated manifold
whose all leaves are compact. Moreover, foliated objects on the
foliated manifold correspond bijectively to orbifold objects.
The main aim of the project will be to develop the theory, to
deepen the understanding of and the properties of singular spaces,
other than orbifolds, which admit foliated desingularization. The
student should develop new foliated tools to obtain important
results of geometry and topology of foliated manifolds in view of
applications to their leaf spaces, the singular spaces, which are
the main interest of the project. The cohomological
characterization of such spaces is also one of the aims.
Possible application of the obtained results to mathematical
models of physics are long term aim of the project.
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