Max Dehn Seminar
on Geometry, Topology, Dynamics, and Groups
Spring 2017 Wednesday 3:15  4:15, Room LCB 215
Date  Speaker  Title click for abstract (if available) 
August 31
RTG seminar 
Jenny Wilson
Stanford University 
Representation theory and higherorder stability in the
configuration spaces of a manifold
Let F_k(M) denote the ordered kpoint
configuration space of a connected open manifold M. Work of Church
and others shows that for a given manifold, as k increases, this
family of spaces exhibits a phenomenon called homological
"representation stability" with respect to the natural symmetric
group actions. In this talk I will explain what this means, and
describe a higherorder "secondary stability" phenomenon among the
unstable homology classes. The project is work in progress, joint
with Jeremy Miller.

September 7

Adam Kanigowski
Penn State 
Slow entropy for smooth flows on surfaces
Slow entropy is an useful invariant when
dealing with systems of
intermediate (polynomial) growth. The most classic examples are:
horocycle
flows, time changes of nilflows and mixing smooth flows on
surfaces (with
finitely many fixed points). In the talk we will focus mostly on
computing
slow entropy for the class of smooth flows on twotorus with one
fixed
point. As a consequence we get that such flows never are rank one
and that
the order of degeneracy of the fixed point is an invariant.
Moreover, we
establish variational principle for slow entropy in this class.

September 19 at 3
JWB 308 *Note unusual time and place* 
Sebastian
Hensel
University of Bonn 
Rigidity and Flexibility for the Handlebody Group
The handlebody group H_g is the subgroup of
the mapping class group Mod_g of a surface formed by all those
elements which extend to a given handlebody. In this talk we will
first show that finite index subgroups of this group are rigid:
any inclusion into Mod_g is conjugate to the standard inclusion.
We then discuss flexible behaviour: the existence of inclusion of
H_g into Mod_h whose image is not conjugate into any handlebody
subgroup of Mod_h.

October 12

No Seminar: Fall break  
October 19

Daniel Bernazzani
Rice University 
Centralizers in the Group of Interval Exchange Transformations
In this talk, I will explain why a typical
interval exchange transformation does not commute with any other
interval exchanges except for its powers.

November 2

Daniel Studenmund
University of Utah 
Semiduality from products of trees
A duality group has a pairing exhibiting
isomorphisms
between its homology and cohomology groups. Many naturally
occurring
groups fail to be duality groups, but are morally very close. In
this
talk we make this precise with the notion of a semiduality group
and
show that the lamplighter group is a semiduality group. We'll
finish
by stating a conjecture for semiduality of arithmetic groups over
function fields and a positive result for arithmetic groups acting
on
products of trees. This talk covers work joint with Kevin Wortman.

November 9

Julien Paupert
Arizona State University 
Rank 1 deformations of noncocompact hyperbolic lattices
Let X be a negatively curved symmetric space and Gamma a
noncocompact lattice in Isom(X). We show that small,
parabolicpreserving deformations of Gamma into a negatively
curved symmetric space containing X remain discrete and faithful.
(The cocompact case is due to Guichard.)
This applies in particular to a version of Johnson and Millson's
bending deformations, providing for all n infinitely many
noncocompact lattices in SO(n,1) which admit discrete and faithful
deformations into SU(n,1). We also produce deformations of the
figure8 knot group into SU(3,1), not of bending type, to which
the result applies. This is joint work with Sam Ballas and Pierre
Will.

November 23

No Seminar: Thanksgiving  
December 5 at 3 in LCB 219 *Note unusual time*

Aaron Brown
University of Chicago 
Zimmer’s conjecture for cocompact lattices
For n at least 3, consider a lattice in Sl(n,R). Zimmer’s
conjecture asserts that every action of the lattice on a manifold
of dimension at most n2 is finite. Recently, D. Fisher and S.
Hurtado, and I established Zimmer’s conjecture under the
additional assumption that the lattice is cocompact. I will give
some background and motivation for the conjecture. I will outline
our proof and explain a number of tools we use: strong Propterty
(T), cocycle superrigidity, Ratner’s measure classification
theorem, and smooth ergodic theory of Z^d actions.

December 7

Martin Deraux
Grenoble 
Nonarithmetic lattices
I will present joint work with Parker and
Paupert, that allowed us to
exhibit new commensurability classes of nonarithmetic lattices in
the
isometry group of the complex hyperbolic plane. If time permits, I
will also explain close ties between our work and the theory of
discrete reflection groups acting on other 2dimensional complex
space
forms.

January 18

Gordan Savin
University of Utah 
Affine buildings as sets of lattice functions
Let V be a finite dimensional vector space over a padic field.
The affine building of GL(V) can be constructed as the set of all lattice
functions on V. Let G be a Chevalley group attached to a simple, split,
Lie algebra over the padic field. I will explain how the affine building
of G can be constructed as the set of (some) lattice functions on the Lie
algebra.

January 25

Mladen Bestvina
University of Utah 
Boundary amenability for Out(F_n)
The motivation for the talk is the recent result, joint with Vincent
Guirardel and Camille Horbez, that Out(F_n) admits a topologically
amenable action on a Cantor set. This implies the Novikov conjecture
for Out(F_n) and its subgroups. Most of the talk will be an
introduction to boundary amenability and ways to prove it for simpler
groups.

February 1

Ioannis Konstantoulas
University of Utah 
Discrepancy of general symplectic lattices
The statistics of lattice points in Borel sets have been studied
extensively, both for single lattices like the integral points in
Euclidean space and on average over the space of lattices. The
magnitude of the error term in the approximation is related to
problems in spectral theory and number theory and good error terms
have been obtained for typical lattices using tools from
representation theory. However, averages over closed subspaces over
the space of all lattices are far less accessible and the only
discrepancy results so far have been associated to rank one subgroups
of GL(n). In this work, joint with J. Athreya, we provide power
savings bounds for the number of lattice points of a typical lattice
from the general symplectic ensemble in a nested family of Borel sets.
This is the first example of lattice point statistics for a higher
rank group other than the full GL(n) and SL(n).

February 15

Ken Bromberg
University of Utah 
Univalent maps and renormalized volume
We will discuss some classical results on univalent maps and their applications to the renormalized volume of hyperbolic 3manifolds. This is joint work with M. Bridgeman and J. Brock.

February 22

Derrick Wigglesworth
University of Utah 
Distortion and Abelian Subgroups
This talk will focus on abelian subgroups of the mapping class group and Out(F_n). After relating some structural results, we'll discuss how the intrinsic Euclidean geometry of abelian subgroups relates to the geometry of the ambient group.

March 1

Bruce Kleiner
New York University 
Ricci flow, singularities, stability, and the topology of 3manifolds
Ricci flow is a geometric PDE that has had a profound impact on 3dimensional topology. Like many geometric evolution equations, its solutions develop singularities, and their study has been crucial part of the story. Soon after introducing Ricci flow in 1982, Hamilton defined a notion of Ricci flow with surgery, a regularization scheme that allows one to avoid singularities. Building on many contributions of Hamilton, in 2003 Perelman used Ricci flow with surgery to prove Thurston’s Geometrization Conjecture, which includes the 3dimensional Poincare Conjecture as special case. At the same time, Perelman drew attention to the ad hoc character of Ricci flow with surgery, and conjectured the existence of "Ricci flow through singularities”, which would be a canonical evolution for any Riemannian 3manifold. Recently, Richard Bamler and I have proven Perelman’s conjecture, and used it to obtain new information about diffeomorphism groups of 3manifolds.

March 8

Matthew Stover
Temple University 
Proper actions on products of trees
Does a surface group act properly on a finite product of finitevalence trees? I don't know. I'll discuss two results motivated by this question, joint with David Fisher, Michael Larsen, and Ralf Spatzier. One hints toward a positive answer: faithful representations of surface groups into arithmetic groups in characteristic p. The other result is a structure theorem for proper actions of CAT(0) groups on products of trees that hints toward a negative answer; for example, these methods prove that rightangled Artin groups admitting such proper actions are products of free and free abelian groups.

March 29

Radhika Gupta
University of Utah 
Intersection form for relative currents and relative outer space
Using the intersection number between curves, the space of laminations acts as its own dual space. For free groups, the space of currents acts as dual to the closure of outer space via the intersection form defined by Kapovich and Lustig. This intersection form can be used to show that a fully irreducible outer automorphism acts loxodromically on the free factor complex. With the goal of understanding reducible outer automorphism, in this talk I will define relative currents, relative outer space, discuss the intersection form between them and mention an application to the relative free factor complex.

April 5

James Farre
University of Utah 
Unbounded geometry in bounded cohomology
We explore the bounded cohomology of closed surface groups whose actions on hyperbolic 3space may have unbounded geometry. The isometry types of marked hyperbolic 3manifolds are classified in terms of their end invariants. We discuss how the classification gives us a criterion for distinguishing bounded classes in degree 3 for surface groups and, more generally, finitely generated Kleinian groups without parabolics.

April 12

Bei Wang
University of Utah 
Convergence between Categorical Representations of Reeb Space and Mapper
The Reeb space, which generalizes the notion of a Reeb graph, is one
of the few tools in topological data analysis and visualization
suitable for the study of multivariate scientific datasets. First
introduced by Edelsbrunner et al., it compresses the components of the
level sets of a multivariate mapping and obtains a summary
representation of their relationships. A related construction called
mapper, and a special case of the mapper construction called the Joint
Contour Net have been shown to be effective in visual analytics.
Mapper and JCN are intuitively regarded as discrete approximations of
the Reeb space, however without formal proofs or approximation
guarantees. An open question has been proposed by Dey et al. as to
whether the mapper construction converges to the Reeb space in the
limit.
In this work, we are interested in developing the theoretical
understanding of the relationship between the Reeb space and its
discrete approximations to support its use in practical data analysis.
Using tools from category theory, we formally prove the convergence
between the Reeb space and mapper in terms of an interleaving distance
between their categorical representations. Given a sequence of refined
discretizations, we prove that these approximations converge to the
Reeb space in the interleaving distance; this also helps to quantify
the approximation quality of the discretization at a fixed resolution.
Joint work with Elizabeth Munch, University at Albany – SUNY Albany.

April 13, 2pm, JWB 333 Note unusual time and place 
Colin Adams
Williams College 
Multicrossing Number of Knots: Turning Knots into Flowers
Knots have traditionally been depicted using projections with crossings where two stands cross. But what if we allow three strands to cross at a crossing? Or four strands? Can we find projections of any knot with just one of these multicrossings? We will discuss these generalizations of traditional invariants to multicrossing numbers, ubercrossing numbers and petal numbers and their relation to hyperbolic invariants.

April 19

Leonard Carapezza
University of Utah 
Minimality detection between finite graphs
An infinite sequence in a compact metric space X is called
minimal if it satisfies a weak periodicity condition. A compact metric
space Y is called a minimality detector for X if every non minimal
sequence in X can be taken to a non minimal sequence in Y by a continuous
function. a priori whether Y is a minimality detector for X depends on if
we are considering halfinfinite or biinfinite sequences, but I show this
not to be the case. The question of when one finite graph is a minimality
detector for another finite graph turns out to be the only interesting
case, and some necessary conditions in this way will be presented.

October 16

Marco
Lopez
University of North Texas 
TBA
TBA

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You may also be interested in the RTG Seminar
Max Dehn Seminar is organized by Mladen Bestvina, Ken Bromberg, Jon Chaika,
Thibaut Dumont Ioannis Konstantoulas, Evelyn Lamb, Donald Robertson, Daniel Studenmund,
Domingo Toledo, and Kevin Wortman.
This web page is maintained by Jon Chaika.