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

 Thursday October 9 at 34pm in Eck 308
 Alden Walker, University of Chicago
 Roots, Schottky Semigroups, and a proof of Bandt's Conjecture

Abstract: In 1985, Barnsley and Harrington defined a
"Mandlebrot set" M for pairs of complex dilations. This is
the set of complex numbers c such that the limit set
generated by the pair of dilations x> cx+1 and x>
cx1 is connected. The set M is also the closure of the set
of roots of polynomials with coefficients in {1,0,1}. As
with the usual Mandlebrot set, M has strong connections to
dynamics and algebra, and it has been studied by Bousch,
Bandt, Solomyak, Xu, Thurston, and Tiozzo. For a geometric
group theorist, the study of M is qualitatively similar to
the study of Kleinian groups acting on their limit sets or
on universal circles. Barnsley and Harrington noted the
(numerically apparent) existence of infinitely many "holes"
in M, which correspond to exotic components of the space of
Schottky semigroups. Bandt rigorously confirmed a single
hole in 2002 and conjectured that the interior of M is dense
in M away from the real axis. We give the new technique of
"traps" to certify an interior point of M, and we use these
traps to prove Bandt's conjecture and certify the existence
of infinitely many holes in M. The only prerequisite for
this talk is pointset topology. Fun pictures will be
provided. This is joint work with Danny Calegari and Sarah
Koch.

 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.

 Thursday October 23 at 34pm in Eck 308
 Darren Long, UCSB
 Geometric Configurations of Primes

Abstract: A number field that contains a
sufficiently large set of integers whose pairwise
differences are all either units or generate prime ideals
with a suitable property, has class number 1. Such sets of
integers  which we call "LenstraHurwitz cliques"  are
surprisingly easy to find, and let us show that many number
fields have class number 1.

 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.

 Tuesday November 18 at 34pm in Eck 308
 Patricia Hersh, NC State
 Regular cell complexes in total positivity

Abstract: This talk will focus upon stratified
spaces arising from total positivity theory and
combinatorial representation theory, with background and
motivations provided along the way. We prove that certain
such stratified spaces having the Bruhat orders as their
posets of closure relations are regular CW complexes
homeomorphic to closed balls. A special case is the link of
the identity in the space of upper triangular, totally
nonnegative matrices with 1's on the diagonal, stratified
according to which minors are positive and which are 0. This
confirms a conjecture of Sergey Fomin and Michael Shapiro,
completing the solution of a question of Joseph Bernstein. I
will briefly discuss some ingredients that went into the
proof, including the role of the 0Hecke algebra of a finite
Coxeter group, combinatorics of reduced and nonreduced
words, and a new criterion for deciding if a finite CW
complex is regular (with respect to a choice of
characteristic maps) based on an interplay of combinatorics
with topology. As time permits, I will also indicate ways in
which this story could perhaps be generalized.

 Thursday November 20 at 34pm in Eck 308
 Christian Rosendal, UIC
 Large scale geometry of topological groups

Abstract: Geometric group theory is based on the
fundamental observation that the word metrics on a f.g.
group given by distinct finite generating sets are
biLipschitz equivalent, i.e., differ at most by a
multiplicative constant. This observation makes it possible
to treat finitely generated groups as geometric objects as
long as the methods employed are insensitive to the
multiplicative error. This study carries quite easily over
to compactly generated locally compact groups, but so far
more general topological transformation groups, e.g.,
homeomorphism and diffeomorphism groups, have resisted
treatment from this perspective due to the presumed absence
of canonical generating sets. We shall present some newly
developed tools for overcoming this and consider the ensuing
large scale geometric theory of these groups.

 Thursday December 11 at 34pm in Eck 308
 Dan Margalit, Georgia Institute of Technology
 Combinatorial models for mapping class groups

Abstract: In the 1990s Ivanov proved that every
automorphism of the complex of curves is induced by an
element of the mapping class group of the corresponding
surface. This theorem has found many applications to the
theory of mapping class groups and has also inspired many
similar results of the form that the automorphism group of
some specific simplicial complex is isomorphic to the
mapping class group. As a result, Ivanov made a
metaconjecture that all "sufficiently rich" simplicial
complexes associated to a surface should have the mapping
class group as its group of automorphisms. Tara Brendle and
I give a wide class of simplicial complexes satisfying
Ivanov's metaconjecture, and moreover we give necessary and
sufficient conditions for certain types of simplicial
complexes to have automorphism group the mapping class
group.
For questions, contact