Max Dehn Seminar
on Geometry, Topology, Dynamics, and Groups
Fall 2017 Wednesday 3:15  4:15, Room LCB 215
Date  Speaker  Title click for abstract (if available) 
August 30

Moon Duchin
Tufts University 
Discrete curvature, with applications
TBA

September 6

Matthew Smith
University of Utah 
Unique Ergodicity for Quadratic Differentials
A quadratic differential on a surface
determines a singular
foliation equipped with a natural transverse invariant measure. Whether
this is the only measure or not has important consequences for the
dynamics of the foliation. We will discuss a new sufficient condition
for
unique ergodicity, which is the case when there is exactly one measure.
For special classes of quadratic differentials, we obtain stronger
results. We will also discuss the limitations of the theorem and some
new
examples.

September 13

Alena
Erchenko
Pennsylvania State University 
Flexibility of some dynamical and geometrical data
We introduce the flexibility program
proposed by A. Katok and discuss first results. We show the
flexibility of the entropy with respect to the Liouville measure and
topological entropy for geodesic
flow on negatively curved surfaces with fixed genus and total area
(joint with A. Katok). Also, we
point out some restrictions which come from additionally fixing a
conformal class of metrics (joint
with T. Barthelm´e). If time permits, we describe a flexibility result
for Lyapunov exponents for
smooth expanding maps on a circle of fixed degree.

September 20

Florian
Richter
The Ohio State University 
The dichotomy between structure and randomness in
multiplicative number theory
We will begin the talk by discussing a
dichotomy theorem in multiplicative number theory which asserts that any
multiplicative function (that satisfies certain minor regularity
conditions) is either a (special kind of) almost periodic function or a
pseudorandom function. Then we will explore how this phenomenon extends
to other classical objects coming from multiplicative number theory. In
particular, we will study the combinatorial and dynamical properties of
level sets of multiplicative functions and I will present a structure
theorem which says that for any level set E of an arbitrary
multiplicative function there exists a highly structured superset R such
that E is a pseudorandom subset of R.

September 27

Chris Cashen
University of Vienna 
The contracting boundary of a group
We construct a bordification of a
proper geodesic metric space by
adding a ‘contracting boundary’ consisting of equivalence classes of
rays satisfying a contraction property enjoyed by rays in a hyperbolic
space. We think of these as the distinct ways of going to infinity
through hyperbolic directions. The topology we introduce on this
contracting boundary is invariant under quasiisometries and is
homeomorphic to the Gromov boundary when the space is hyperbolic. If
the space admits a geometric group action then our topology on the
boundary is metrizable.
This is joint work with John Mackay.

October 18

Marco
Lopez
University of North Texas 
Dimension of shrinking target sets arising from
nonautonomous dynamics.
In analogy to the set of
wellapproximable numbers in Diophantine approximation, a shrinking
target set is defined as the set of points in a metric space, X, whose
orbits under a dynamical system on X hit infinitely often a ball of
radius shrinking to zero. Using techniques from thermodynamic formalism
we establish a formula for the Hausdorff dimension of such sets in the
context of nonautonomous iterated function systems.

October 25

Sanghyun
Kim
Seoul National University 
Free products in Diff(S1)
We prove that if G is a finitely
generated nonvirtuallyabelian
group, then (G X Z) * Z does not embed into Diff^2(S^1). In
particular, the class of subgroups of Diff^r(S^1) is not closed under
taking free products for each r >= 2. We complete the classification
of RAAGs embeddable in Diff^r(S^1) for each integer r, answering a
question in a paper of M. Kapovich. (Joint work with Thomas Koberda)

November 15

Shariar
Mirzadeh
Brandeis University 
Dimension estimates for the set of points with
nondense orbit in homogeneous spaces.
In this talk we study the set of points in a homogeneous space whose
orbit escapes the complement of a fixed compact subset. We find an upper
bound for the Hausdorff dimension of this set. This extends the work of
Kadyrov, where he found an upper bound for the Hausdorff dimension of
the set of points whose orbit misses a fixed ball of sufficiently small
radius in a compact homogeneous space. We can also use our main result
to produce new applications to Diophantine approximation. This is joint
work with Dmitry Kleinbock.

November 29

Zhenqi Wang
Michigan State University 
Smooth Local rigidity of algebraic actions.
At first, we will introduce the
background of algebraic actions and
give some interesting examples, next,
we will review of various smooth rigidity results for higherrank
algebraic actions
and recent progress. Finally, we will talk about future directions.

January 8 at 4pm in LCB 219 *Note unusual time and place* 
Laura
Schaposnik
University of Illinois at Chicago 
On Cayley and Langlands type correspondences for Higgs
bundles
The Hitchin
fibration is a natural tool through which one can understand the moduli space of Higgs bundles and its
interesting subspaces (branes). After reviewing the type of
questions
and methods considered in the area, we shall dedicate this talk to
the
study of certain branes which lie completely inside the singular
fibres
of the Hitchin fibrations. Through Cayley and Langlands type
correspondences, we shall provide a geometric description of these
objects, and consider the implications of our methods in the context
of
representation theory, Langlands duality, and within a more generic
study of symmetries on moduli spaces.

January 10

Jennifer Wilson
Stanford University 
Stability in the homology of Torelli groups
The Torelli subgroups of mapping
class groups are fundamental objects in lowdimensional topology,
through some basic questions about their structure remain open. In this
talk I will describe these groups, and how to use tools from
representation theory to establish patterns their homology. This project
is joint with Jeremy Miller and Peter Patzt. These “representation
stability” results are an application of advances in a general algebraic
framework for studying sequences of group representations.

January 31

Barak
Weiss
Tel Aviv University 
New examples for the horocycle flow on the moduli space of
translation surfaces
The moduli space of translation
surfaces is a space on which SL(2,R) acts via its action on the complex
plane, viewed as the plane R^2. A folklore and imprecise conjecture is
that this action has many features in common with actions of Lie groups
generated by unipotent elements on homogeneous spaces, which were shown
to be rigid in fundamental work of Ratner. Work of Eskin, Mirzakhani and
Mohammadi, concerning action of the entire group SL(2,R), justified this
expectation. The restriction of the action to upper triangular unipotent
matrices is called the horocycle flow. For this flow a similar folklore
conjecture could be made. We provide new examples showing that the
horocycle flow exhibits features absent from analogous homogeneous
flows, e.g. orbits which are not generic for any measure, and orbit
closures with noninteger Hausdorff dimension. Joint work with Jon
Chaika and John Smillie.

February 2 at 3:05 in LCB 219 *Note unusual time and place* 
Sebastian
Hurtado
University of Chicago 
Burnside problem on diffeomorphism groups
Suppose G is a finitely generated
group such that every element has finite
order. Must G be a finite group?
This is known as the burnside problem, it was formulated around 1902 by
Burnside himself and it was central in the development of group theory
during the 20th century. The answer in general turned out to be
negative, G
might be infinite. Nonetheless, if one restricts G to be a linear group
(group of matrices), the answer is positive (Schur, 1911).
The problem remains open if we assume G is a group of homeomorphisms of
a
surface or a manifold in general. I will talk about the case where G is
a
group of diffeomorphisms of a surface.

February 21

Dmitry
Kleinbock
Brandeis University 
Hyperbolic dynamics and intrinsic Diophantine approximation
Dynamics on homogeneous spaces of Lie
groups has been a useful tool in solving many previously intractable
Diophantine problems. In this talk I will describe some existing
connections between homogeneous dynamics and Diophantine approximation,
and then show how a similar approach can help quantify the density of
rational points on quadric hypersurfaces (intrinsic approximation
problems). The case of spheres is reduced to dynamics on hyperbolic
manifolds. Joint with Lior Fishman, Keith Merrill and David Simmons.

March 14

Brian
Collier
University of Maryland 
Special deformations of Fuchsian representations into
SO(p,q)
The space of discrete and faithful
representations of a closed surface
group into Isom(H^2)=SO(1,2) defines a connected component of the
character variety.
While deformations of such representations into SO(1,3) define
interesting
hyperbolic 3manifolds, the space of such geometrically interesting
representations
is not closed in the SO(1,3) character variety. In fact, such
representations can be
continuously deformed to have compact Zariski closure. In this talk we
will define
analogous connected components of the SO(p,p+1) character variety and
consider their
deformations into the SO(p,p+2) character variety (and more generally
SO(p,q)).
Unlike the p=1 case, such representations cannot be deformed compact
representations. This leads to a dichotomy for the connected components
of the
SO(p,q) character variety which we will relate to recent notions of
positivity
introduced by Guichard and Wienhard.

March 7 Starting at 3pm

Autumn Kent
University of Wisconsin 
Spacious knots
Brock and Dunfield showed that there
are integral homology spheres whose thick parts are very thick and take
up most of the volume. Precisely, they show that, given R big and r
small, there is an integral homology 3sphere whose Rthick part has
volume (1 − r)vol(M). Purcell and I find knots in the 3sphere with this
property, answering a question of Brock and Dunfield.

March 28

Joel Moreira
Northwestern University 
Multiple recurrence along sparse sequences over thick sets
Khintchine’s recurrence theorem states
that the set of optimal return times in a measure preserving dynamical
system is syndetic, i.e., has bounded gaps. In 1977, as part of his
ergodictheoretic proof of Szemeredi’s theorem on arithmetic
progressions, Furstenberg established a partial extension of
Khintchine’s result by showing that the set of multiple return times is
also syndetic. Multiple recurrence has since been established along many
different types of sequences, including polynomial sequences and
sequences derived from functions in a Hardy field. However, they don’t
always lead to syndetic return time sets. In my talk I will describe
joint work with Vitaly Bergelson and Florian Richter where we establish
that for a general class of nonpolynomial sparse sequences, the set of
return times still possesses interesting combinatorial properties, and
in particular it satisfies a weak form of syndeticity and is thick, i.e.
contains arbitrarily long intervals of integers. Via Furstenberg’s
correspondence principle our work leads to novel variants of Szemeredi’s
theorem.

April 4

Roger
Baker
Bringham Young University 
Dependent variables that miss fixed targets.
Let n(1), n(2),... be a strictly
increasing sequence of integers. The dependent
variables in question are n(1)x, n(2) x,... where x is in the interval
(0,1]. As for
missed targets, an interval modulo one never entered by the fractional
parts of the
dependent variables is such a target.
Now things depend in a rather baffling fashion on the particular
sequence of
integers. The set of x for which a target is missed is certainly of
measure zero,
but does it have dimension 1, or some dimension between 0 and 1, or
indeed is it
finite and can we bound the number of x in this set? I will survey what
is known
about this and propose some conjectures.

April 18

Kelly
Yancey
Institute for Defense Analyses 
TBA
TBA

April 25

Peter
Smillie
Harvard 
TBA
TBA

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