Low dimensional objects such as graphs, surfaces, and 3-manifolds
have the advantage that we can visualise them. We will study these
spaces and the groups that relate to them, in particular the fundamental
group and their groups of symmetries.
Possible topics include:
Dynamics of graph and surface automorphisms: Consider a homeomorphism
from a surface to itself. What can we say about the dynamics of iterated
applications of this map? For instance, the homemorphism takes curves
to curves, and with each iteration, they might become more and more
complicated. Do we still obtain something reasonable in the limit?
JSJ-decompositions of 3-manifolds and finitely presented groups:
What happens in the fundamental group of a manifold, when you chop this
space up into simpler pieces? You would hope that the fundamental group
decomposes, in some sense, into simpler pieces too. Conversely, if
you can decompose the fundamental group, can you realise this splitting
Algorithmic problems in low dimensional topology: Someone gives you
a knot. Can you decide if that gadget is really knotted? Or someone
gives you a surface, can you tell whether it is the torus? Which
invariants of topological spaces can be computed?
This class will be accessible to first year graduate students.
Some familiarity with topological spaces and groups, as provided
by a undergraduate level topology or algebra class, is welcome.