We usually think of 2-dimensional manifolds as surfaces
embedded in Euclidean 3-space. Since humans cannot
visualise Euclidean spaces of higher dimensions, it appears
to be impossible to give pictorial representations of
higher-dimensional manifolds. However, one can in fact
draw 1-dimensional pictures truly representing the topology
of surfaces. By analogy, one can draw 2-dimensional
pictures of 3-manifolds (Heegaard diagrams), and 3-dimensional
pictures of 4-manifolds (Kirby diagrams). With a little trick,
one can even draw 2-dimensional (sic!) pictures of at
least some 5-manifolds.


In this talk I shall explain how to draw such pictures
and how to use them for answering topological and
geometric questions. The work on 5-manifolds is joint
with Fan Ding and Otto van Koert.


[Invited by Prof. Ruth Kellerhals]