Talk #1 (1:00pm–2:30pm)
Paul Cusson
"When is a closed parallelizable manifold a homogeneous space?"
Given a compact parallelizable real n-manifold M, the following theorem is known: M ∼= G/H for G a simply connected Lie group and H a discrete subgroup of G if and only if there exists n everywhere linearly independent vector fields X1, . . . ,Xn and constants αk ij such that for each 1 ≤ i < j ≤ n, [Xi,Xj ] = αkijXk. This is in contrast to the holomorphic case, where a compact parallelizable complex manifold is guaranteed to have such structure constants. After proving this theorem, we will look at how this result could possibly be used as a tool for an alternative proof of the Poincar´e conjecture, if one were so bold.
Talk #2 (2:30pm–4:00pm)
Henry Li
"Riemann
surfaces
and
Fuchsian
groups"
In
this
talk,
we
explain
how
Riemann
surfaces
of
genus
greater
than
one
can
be
constructed
as
the
quotient
of
the
upper-half
plane
by
a
Fuchsian
group.
MC 5403