Geometry/Topology Seminar
Fall 2014
Thursdays (and sometimes Tuesdays) 34pm, in
Eckhart 308

 Thursday October 9 at 34pm in Eck 308
 Darren Long, UCSB
 TBA

Abstract: TBA

 Thursday October 16 at 34pm in Eck 308
 Jeremy Miller, Stanford
 Representation stability for homotopy groups of configuration
spaces

Abstract: In the 1970s, McDuff proved that
configuration spaces of distinct unordered particles in an
open manifold exhibit homological stability. That is,
H_{i}(Conf_{k}(M)) is independent
of k for k>>i. A natural follow
up question is: Do the homotopy groups also stabilize? From
explicit calculations, one can show that this is not the
case. However, in joint work with Alexander Kupers, I have
shown that the rational homotopy groups of configuration
spaces of particles in simply connected manifolds of
dimension at least 3 exhibit representation stability in the
sense of Church and Farb. This follows from a more general
theorem we prove relating the homotopy groups and cohomology
groups of coFIspaces and from the work of Church on
representation stability for the cohomology of ordered
configuration spaces. This result on homotopy groups
suggests that in situations with homological stability, one
should not expect classical stability for homotopy groups.
Instead, one should try to incorporate the fundamental group
into one's definition of stability.

 Tuesday November 4 at 34pm in Eck 308
 Samuel Grushevsky, Stony Brook
 Stable cohomology of the compactifications of the moduli space
of abelian varieties

Abstract: Borel showed that the degree k
cohomology of the moduli space A_{g} of
(complex principally polarized) abelian
gdimensional varieties stabilized as
g grows, that is does not depend on g,
for g>k. Similarly, Madsen and Weiss showed
that the cohomology of the moduli space of curves
M_{g} stabilizes. In this talk we study
the stabilization of the cohomology of compactifications,
observing that the cohomology of the DeligneMumford
compactification of M_{g} does not
stabilize, of the second Voronoi toroidal compactification
of A_{g} likely does not stabilize, while
proving that the cohomology of the perfect cone toroidal
compactification of A_{g} does stabilize,
in degree close to the top. Joint work with Klaus Hulek and
Orsola Tommasi.

 Thursday November 13 at 34pm in Eck 308
 Piotr Przytycki, McGill University
 Balanced walls in random groups

Abstract: This is joint work with John Mackay. We
study a random group G in the Gromov density
model and its Cayley complex X. For density <
5/24 we define walls in X that give rise to a
nontrivial action of G on a CAT(0)
cube complex. This extends a result of Ollivier and Wise,
whose walls could be used only for density < 1/5. The
strategy employed might be potentially extended in future to
all densities < 1/4.
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