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University of Michigan
Department of Mathematics
Fall 2005 Topology Seminar
Thursdays 3-4, 4088 East Hall
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For more information, contact Emina Alibegovic.
If you want to check the list of seminars held in previous terms, click on
the appropriate link below.
Winter 2005, Fall 2004, Winter 2004, Fall 2003, Winter 2003, Fall 2002, the year 2000-1.
Schedule of Talks
Abstracts
September 22
Motivated by conformal field theory we introduce the rigged
moduli space of Riemann surfaces with parametrized boundary
components. By generalizing the class of parametrizations to
quasisymmetric maps we can connect to the standard Teichmueller theory
of surfaces with boundary. In particular, we show that the rigged
moduli space can be obtained from the usual Teichmueller space of a surface
with boundary by quotienting with a subgroup of the mapping class
group. By sewing caps onto the boundary
components, a connection can be made with the Teichmueller space of a
punctured surface. A surprising upshot is a possibly new way of
understanding the usual Teichmueller space of a surface with boundary.
No technicalities will be discussed.
This is joint work with Eric Schippers.
October 27
We will go straight to applications of layered-triangulations. These
include using layered-triangulations of lens spaces to capture some
classical results about the topology of lens spaces; for example, after
classifying normal and almost normal surfaces in layered-triangulations of lens
spaces (the good point here is that there just aren't many), we obtain earlier
results of Bredon and Wood classifying those lens spaces that admit embedded
nonorientable surfaces and classifying the embedded nonorientable surfaces in
each such lens space. We characterize efficiency of layered triangulations of
lens spaces and apply layered-triangulations to obtain the results of
Waldhausen and those of Bonahon and Otal to classify Heegaard splitings of
$S3$ and $S2\times S1$ and all (other) lens spaces, respectively. We also
use layered-triangulations of the solid torus to construct canonical
triangulations for Dehn-fillings of knot exteriors; as an example we apply
these triangulations to classify the Heegaard splittings of Dehn-fillings of
the Figure-eight knot exterior. Time permitting (which we can assume it will
not), we will define layered triangulations of higher genus handlebodies and
general $3$--manifolds.
November 18
In Euclidean space, the intersection of two balls can be very long and
thin (for instance two balls with very large radius and with small
intersection). In the hyperbolic plane this is not the case, the
intersection of any
two balls is always very round. More generally, I will explain why a
geodesic metric space is delta-hyperbolic in the sense of Gromov if and
only if the intersection of any two metric balls is almost a ball.
This is joint work with Graham Niblo.
December 2
Relative property (T) has recently been used to construct a variety of new
rigidity phenomena, for example in von Neumann algebras and the study of
orbit-equivalence relations. However, until recently there were few
examples of group pairs with relative property (T) available through the
literature. Which motivated the main theorem I will discuss. It gives a
necessary and sufficient condition for contructing group pairs with
relative property (T) from Abelian groups of finite and nonzero Q-rank.
The proof is constructive. The main ingredients are Furstenberg's
celebrated lemma about invariant measures on projective spaces and the
spectral theorem for the decomposition of unitary representations of
Abelian groups. Methods from algebraic group theory, such as the
restriction of scalars functor, are also employed.
In this talk, I will demonstrate the proof of a special case of the
theorem which embodies all of the essential ingredients.
December 8
In this talk, I will review the theory of topological
modular forms, which is a generalized cohomology theory defined
by Mike Hopkins and Haynes Miller. I will also mention my joint
construction with Hao Xing involving the topological modular
form cohomology of a 10-manifold, which is relevant in physics
(hep-th/0511011).
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