Geometry/Topology Seminar
Winter 2017
Thursdays (and sometimes Tuesdays) 34pm, in
Eckhart 308

 Thursday January 19 at 34pm in Eck 308
 Christin Bibby, University of Western Ontario
 Representation stability for the cohomology of arrangements associated to root systems

Abstract: From a root system, one may consider the
arrangement of reflecting hyperplanes, as well as its toric
and elliptic analogues. The corresponding Weyl group acts on
the complement of the arrangement and hence on its
cohomology. We consider a sequence of linear, toric, or
elliptic arrangements which arise from a family of root
systems of type A, B, C, or D, and we study the rational
cohomology as a sequence of Weyl group representations. Our
techniques combine a Leray spectral sequence argument
similar to that of Church in the type A case along with
FI_{W}module theory which Wilson
developed and used in the linear case.

 Thursday January 26 at 34pm in Eck 308
 Sam Nariman, Northwestern
 FriedlanderMilnor's problem for diffeomorphism groups

Abstract: Let G be a finite dimensional Lie group
and G^{d} be the same group with discrete topology.
The natural homomorphism from G^{d} to G induces a
continuous map from BG^{d} to BG. Milnor conjectured
that this map induces a padic equivalence. In this talk, we
discuss the same map for infinite dimensional Lie groups, in
particular for diffeomorphism groups and symplectomorphisms.
In these cases, we show that the map from BG^{d} to
BG induces split surjection on cohomology with finite
coefficients in "the stable range". If time permits, I will
discuss applications of these results in foliation theory,
in particular flat surface bundles.

 Tuesday February 7 at 34pm in TBA
 Dan Petersen, University of Copenhagen
 The gravity operad and Francis Brown's partial compactification of M_{0,n}

Abstract: The gravity operad is a certain operad
built out of the cohomology of the moduli space
M_{0,n} of npointed smooth genus zero
curves. We show that as a nonsymmetric operad, it can be
described combinatorially in terms of gluing together
certain polygons with marked chords. In particular, this
description implies that the nonsymmetric gravity operad is
free. The result can be interpreted geometrically in terms
of a partial compactification of M_{0,n}
denoted M_{0,n}^{\}delta, which
was introduced by Francis Brown and whose cohomology plays a
role in the study of MZVs. We see that the cohomology of
M_{0,n}^{\}delta gets identified
with the generators of the gravity operad. The calculation
of the cohomology of
M_{0,n}^{\}delta is new. (joint
with Johan Alm) (Note that Geometry/Topology seminar meets
on Tuesday this week.)

 Tuesday February 14 at 34pm in Eck 308
 Igor Dolgachev, Michigan
 Decomposition and Inertia subgroups of birational automorphism groups of algebraic surfaces

Abstract: I will discuss a problem of whether an
automorphism of an algebraic surface of positive entropy can
fix a curve on the surface pointwisely. This problem was
raised almost a hundred years ago and the first such
examples have been found only recently by Jeremy Blanc and
John Lesieutre. I will discuss an example of a group of
automorphisms of a rational or a K3 surface that is
isomorphic to a nonelementary discrete group of isometries
of a hyperbolic space that leaves a smooth rational curve
invariant and hence admits a nontrivial homomorphism to the
group of Moebius transformations. The problem of injectivity
of this map is related to the problem of freeness of some
rotation group associated with a regular tetrahedron.

 Wednesday February 22 at 34pm in Eck 308
 Chris Leininger, UIUC
 Wordhyperbolic surface bundles

Abstract: The work of FarbMosher and Hamenstadt
provides a necessary and sufficient condition for the
fundamental group of a closed surface bundle over any
compact space to be wordhyperbolic. The condition is
geometric in nature, involving the monodromy homomorphism
and the action on Teichmuller space. Gromov's
hyperbolization question, in the special case of surface
bundles, asks whether the condition on the action can be
relaxed to an algebraic one. In this talk I will discuss
this problem, and some joint work with Bestvina, Bromberg,
and Kent providing results in this direction. (This talk is
on Wednesday, and there is no talk on Thursday.)

 Tuesday March 28 at 34pm in Eck 308
 Jenny Wilson, Stanford
 Stability in the second homology of Torelli groups

Abstract: In this talk, I will describe stability
results for two families of groups, the Torelli groups of
automorphisms of free groups, and the Torelli groups of
mapping class groups of surfaces with one boundary
component. Specifically, I will explain the following
statement: the degree2 integer homology groups of these
Torelli groups are centrally stable when viewed as
representations of GL_{n}(Z) or (respectively)
Sp_{2n}(Z). This project uses a framework developed
by Putnam, ChurchEllenbergFarb, and PutnamSam. It is
joint work with Jeremy Miller and Peter Patzt. (This talk is
on a Tuesday.)

 Thursday April 20 at 34pm in Eck 308
 Fedor Manin, University of Toronto
 TBA

Abstract: TBA

 Thursday April 27 at 34pm in Eck 308
 Tam NguyenPhan, SUNY Binghamton
 An analog of the Tits "building" in nonpositive curvature

Abstract: Locally symmetric manifolds (of noncompact
type) form an interesting class of nonpositively curved
manifolds. By BorelSerre, the thin part of the universal
cover of an arithmetic locally symmetric space is
homotopically equivalent to the rational Tits building,
which is homotopically a wedge of spheres of dimension q1,
where Q is the Qrank of the locally symmetric space. In
general, q is less than or equal to n/2. We show that this
is not an arithmetic coincidence in a weaker sense, which is
that if M is a noncompact, bounded nonpositively curved
manifold with finite volume and no arbitrarily small
geodesic loops (so that M is tame), then any nontrivial
homology cycle in the thin part of \tilde{M} must have
dimension less than or equal to n/2  1. For each such
cycle, we construct a complex at infinity of dimension less
than n/2 that is an analog of the Tits building which we
collapse the cycle onto. I will describe how this is done.
Due to the high number of requests, we are no longer accepting speakers via selfinvitations.
For questions, contact