Geometry/Topology Seminar
Fall 2023
Thursdays 3:30-4:30pm, in
E308
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- Thursday October 5 at 3:30-4:30pm in E308
- Nicholas Wawrykow, UChicago
- Representation Stability and Disk Configuration Spaces
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Abstract: Church--Ellenberg--Farb and Miller--Wilson
proved that for a nice enough manifold X and fixed k, the
k-th homology group of the ordered configuration space of
points in X stabilizes in a representation-theoretic sense
as the number of points in the configuration space
increases. By fixing a metric on X and replacing points with
open unit-diameter disks, we get a new family of
configuration spaces where the geometry of X comes to the
forefront. One of the simplest of these disk configuration
spaces is conf(n,w), the ordered configuration space of
unit-diameter disks in the infinite strip of width w. The
homology groups of conf(*,w) do not stabilize in the sense
of Church--Ellenberg--Farb, Miller--Wilson; however, Alpert
proved that when the width of the strip is 2 they stabilize
in a related sense. In this talk I use twisted algebras to
describe various notions of representation stability, and
expand on the work of Alpert--Manin to show that for all
widths w at least 2, the rational homology of conf(*,w) is a
finitely presented twisted algebra that exhibits notions of
1st and higher order representation stability.
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- Thursday October 12 at 3:30-4:30pm in E308
- Dan Minahan, Georgia Tech
- Finiteness in the homology of the Torelli group
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Abstract: The Torelli group of a closed, oriented
surface is the kernel of the action of the mapping class
group on the first homology of the surface. Birman asked if
the Torelli group is finitely presented for all surfaces of
sufficiently large genus. We will a discuss our recent
theorem, which says that the second rational homology of the
Torelli group is finite dimensional for all closed, oriented
surfaces of sufficiently large genus. This result rules out
the simplest obstruction to the finite presentability of the
Torelli group.
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- Thursday October 19 at 3:30-4:30pm in Ry177
- Nick Salter, Notre Dame
- The equicritical stratification and stratified braid groups
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Abstract: One of the many guises of the braid group
is as the fundamental group of the space of monic squarefree
polynomials. From this point of view, there is a natural
"equicritical stratification" according to the
multiplicities of the critical points. These equicritical
strata form a natural and rich class of spaces at the
intersection of algebraic geometry, topology, and geometric
group theory, and can be studied from many different points
of view; their fundamental groups ("stratified braid
groups") look to be interesting cousins of the classical
braid groups. I will describe some of my work on this topic
thus far, which includes a partial description of the
relationship between stratified and classical braid groups,
and some progress towards showing that the equicritical
strata are K(\pi,1) spaces.
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- Thursday October 26 at 3:30-4:30pm in Ry 177
- Jacob Russell, Rice
- Geometric finiteness and the geometry of surface group extensions
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Abstract: The theory of convex cocompact subgroups
of the mapping class group contains two intertwining
threads. One is the rich analogy between these subgroups of
the mapping class group and the convex cocompact Kleinian
group that inspired them. The other is work of Farb and
Mosher plus Hamenstaedt that shows convex cocompactness is
precisely the property that characterizes when an extension
of a surface group is Gromov hyperbolic. Both of these
threads have natural generalizations that are unresolved.
Among Kleinian groups, convex cocompactness is a special
case of geometric finiteness, yet no robust notion of
geometric finiteness has emerged for the mapping class
group. In geometric group theory, there are a variety of
generalizations of Gromov hyperbolicity, but there is no
characterization of these geometries for surface group
extensions. Mosher suggested these threads should continue
to intertwine with geometric finiteness in the mapping class
group (however it is eventually defined) being equivalent to
some generalization of Gromov hyperbolicity of the
corresponding surface group extension. Inspired by their
work on Veech groups, Dowdall, Durham, Leininger, and Sisto
conjectured that this generalized hyperbolicity could be the
hierarchical hyperbolicity of Behrstock, Hagen, and Sisto.
We provide evidence for this conjecture by showing that
several classes of subgroups that should be considered
geometrically finite (stabilizers of multicurves, twist
subgroups, cyclic subgroups) correspond to surface group
extensions that are hierarchically hyperbolic.
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- Thursday November 2 at 3:30-4:30pm in E308
- Andreas Stavrou, UChicago
- Configuration spaces of surfaces and the Johnson filtration
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Abstract: The mapping class group of a surface
naturally acts on the homology of the configuration spaces
of the surface. The arising representations vary in
complexity with the number of configuration points and with
the flavor of configurations, and this complexity can be
measured by the triviality of the action of the Johnson
filtration. In this talk, based on joint work of Bianchi and
myself, I will show how to construct explicit classes in the
homology of ordered configurations of surfaces acted on
non-trivially by deeper Johnson elements.
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- Thursday November 9 at 3:30-4:30pm in E308
- Matthew Stover, Temple
- Residual finiteness of discrete subgroups of Lie groups
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Abstract: A famous theorem of Malcev implies that
finitely generated subgroups of linear Lie groups are
residually finite (RF), but Deligne showed that lattices in
nonlinear Lie groups need not be RF. I will start by
describing what is known/expected in the more general
setting of central extensions of lattices in adjoint simple
Lie groups. It ends up that, modulo the congruence subgroup
problem, real rank one lattices are the most mysterious with
PU(n,1) as arguably the key case. Then I will explain joint
work with Domingo Toledo proving that a certain perspective
on showing lattices in finite covers of PSL(2,R) are RF
generalizes to central extensions of the `simplest' lattices
in PU(n,1) (with some Hodge-theoretic hypotheses on the
characteristic class of the extension in low dimensions). If
I have time, I'll explain how our results can be used to
construct smooth projective varieties admitting a negatively
curved Kahler metric that are not homeomorphic to a locally
symmetric space, which was new for all dimensions greater
than four.
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- Thursday November 16 at 3:30-4:30pm in E308
- Bradley Zykoski, Northwestern
- L infinity-Delaunay Triangulations in Teichmuller Dynamics
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Abstract: In this talk, I will discuss how the L
infinity-isodelaunay decomposition of a stratum H gives rise
to a computable simplicial complex homotopy equivalent to H,
and does the same for every closed GL(2,R) orbit in H. I
will also discuss the progress towards proving the analogous
conjecture for all GL(2,R) orbit closures. The L
infinity-Delaunay triangulation of a translation surface
generalizes the classical Delaunay triangulation of a planar
pointset, which is dual to the Voronoi diagram. Moduli
spaces of translation surfaces ("strata") are covered by
finitely many regions (the "L infinity-isodelaunay
decomposition"), such that all surfaces in a given region
admit isomorphic L infinity-Delaunay triangulations. The
nerve of this covering is locally infinite, and our central
technical result is that we may identify a finite subcomplex
homotopy equivalent to H by analyzing translation surfaces
with "large" embedded cylinders.
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- Thursday November 30 at 3:30-4:30pm in E308
- Mike Wolf, Georgia Tech
- Ray structures on Teichmuller space
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Abstract: We describe some rays in Teichmuller space
related to harmonic maps and explain how these rays
interpolate between Teichmuller metric geodesics and some
Thurston metric geodesics. A nuance of the Thurston metric
is that there need not be unique geodesics between points,
but using the special Thurston geodesics picked out by this
additional energy minimization problem, we find some new
results on the Thurston metric: (i) a topological
exponential map from each internal point of Teichmuller
space to the Thurston boundary by Thurston geodesics and
(ii) some descriptions of the envelope of geodesics
connecting pairs of points in Teichmuller space by Thurston
geodesics. The results hinge on a generalization of the
Jenkins-Serrin theory to minimal surfaces in singular
spaces.
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- Thursday December 7 at 3:30-4:30pm in Ry 178
- Jenny Wilson, Michigan
- The high-degree rational cohomology of the special linear group
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Abstract: Church--Farb--Putman conjectured that the
rational cohomology of SLn(Z) vanishes in a range
close to its rational cohomological dimension. In this talk
I will describe some current efforts to understand these
high-degree cohomology groups, and more generally the
rational cohomology of SLn(R) when R is a number
ring. Although the groups SLn(R) do not satisfy
Poincare duality, they do satisfy a twisted form of duality,
called (virtual) Bieri--Eckmann duality. Consequently, their
high-degree rational cohomology is governed by an
SLn(R)-representation called the Steinberg
module. The key to understanding these representations is
through studying the topology of certain associated
simplicial complexes. I will survey some results,
conjectures, and ongoing work on the Steinberg modules, and
the implications for the cohomology of the special linear
groups. This talk includes work joint with Bruck, Kupers,
Miller, Patzt, Sroka, and Yasaki.
Due to the high number of requests, we are no longer accepting speakers via self-invitations.
For questions, contact