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
pseudo-random 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 pseudo-random 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 quasi-isometries 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
non-autonomous dynamics.
In analogy to the set of
well-approximable 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 non-autonomous iterated function systems.
|
October 25
|
Sang-hyun
Kim
Seoul National University |
Free products in Diff(S1)
We prove that if G is a finitely
generated non-virtually-abelian
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
non-dense 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 higher-rank
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 low-dimensional 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 non-integer 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 3-manifolds, 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 3-sphere whose R-thick part has
volume (1 − r)vol(M). Purcell and I find knots in the 3-sphere 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
ergodic-theoretic 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 non-polynomial 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 11
|
Donald
Robertson
University of Utah |
The Erdos sumset conjecture
Erdos conjectured that every set of
natural numbers with positive density contains B+C for some infinite
sets B,C of natural numbers. In this talk I will describe joint work
with J Moreira and F. Richter on resolving this conjecture using ideas
from ergodic theory.
|
April 18
|
Kelly
Yancey
Institute for Defense Analyses |
Rigid Substitutions
Rigidity is an important property of a
large set of dynamical systems; in fact the property is generic. During
this talk I will discuss substitution systems that are rigid. I will
introduce a matrix and show how to determine rigidity based on the
spectrum of the matrix for constant length substitutions. I will also
link this with cutting and stacking transformations. This is joint work
with Jon Fickenscher.
|
April 25
|
Peter
Smillie
Harvard |
Entire spacelike surfaces of constant curvature in
Minkowski 3-space
We prove that every regular domain in
Minkowski 3-space which is not a wedge contains a unique entire
spacelike surface with constant intrinsic curvature equal to -1. This
completes the classification of such surfaces in terms of their domains
of dependence, for which partial results were obtained by Li,
Guan-Jian-Schoen, and Bonsante-Seppi. Using this result, we obtain an
analogous classification of entire spacelike surfaces with constant mean
curvature (CMC). We'll apply these ideas to the Minkowski problem of
prescribed curvature and to the construction CMC times in 2+1
relativity, and we'll see what we can say about the problem of deciding
when the induced hyperbolic metric on an entire surface is complete.
Everything is joint with Francesco Bonsante and Andrea Seppi.
|
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