If you are interested in our seminar you might also be interested in
the following seminars:
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July 22, 2021 16:00 PDT: Random walks on Gromov hyperbolic spaces and Teichmüller spaces
[abstract]
In this talk, I will discuss random walks
on Gromov hyperbolic spaces. Due to the hyperbolicity of the spaces,
random walks exhibit behaviors that differ from the classic (Euclidean)
ones. These behaviors include the escape to infinity, central limit
theorems when centered at the escape rate, and geodesic tracking. I will
explain how one can sharpen these behaviors based on the recent
observations by Gouëzel and Baik-Choi-Kim. If time allows, I will also
explain how one can implement this theory on (non-hyperbolic)
Teichmüller spaces.
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May 27, 2021: The extremal length systole of the Bolza surface
[abstract]
Extremal length is a conformal invariant
that plays an important role in Teichmüller theory. For each essential
closed curve on a Riemann surface, it furnishes a function on the
Teichmüller space. The extremal length systole of a Riemann surface is
defined as the infimum of extremal lengths of all essential closed
curves. Its hyperbolic analogue is the hyperbolic systole: the infimum
of hyperbolic lengths of all essential closed curves. While the latter
has been studied profusely, the extremal length systole remains widely
unexplored. For example, it is known that in genus 2, the hyperbolic
systole has a unique global maximum: the Bolza surface. In this talk we
introduce the extremal length systole and show that in genus two it
attains a strict local maximum at the Bolza surface, where it takes the
value square root of 2. This is joint work with Maxime Fortier Bourque
and Franco Vargas Pallete.
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May 20, 2021: The Manhattan curve and rough similarity rigidity
[abstract]
For every non-elementary hyperbolic
group, we consider the Manhattan curve, which was originally introduced
by M. Burger (1993), associated to any pair of (say) word metrics. It
is convex; we show that it is continuously differentiable and moreover
is a straight line if and only if the corresponding two metrics are
roughly similar, that is, they are within bounded distance after
multiplying by a positive constant.
I would like to explain how it is related to central limit theorem for
uniform counting measures on spheres, to ergodic theory of topological
flows built on general hyperbolic groups, and to multifractal structure
of Patterson-Sullivan measures.
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May 13, 2021: Towards optimal spectral gaps in large genus
[abstract]
I'll discuss recent joint work with
Alex Wright
(arXiv: 2103.07496)
showing that typical large genus hyperbolic surfaces have first
Laplacian eigenvalue at least 3/16−ϵ.
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May 6, 2021 10:00 PDT: Random hyperbolic surfaces via flat geometry
[abstract]
Mirzakhani gave an inductive procedure to
build random hyperbolic surfaces by gluing together smaller random
pieces along curves. She proved that as the length of the gluing curve
grows, these families equidistribute in the moduli space of hyperbolic
surfaces. In this talk, I’ll explain how the conjugacy (exposited in
James’s talk) between the earthquake and horocycle flows provides a
template for translating equidistribution results for flat surfaces
into equidistribution results for hyperbolic ones. Using this
correspondence, we address Mirzakhani’s twist torus conjecture and
exhibit new limiting distributions for hyperbolic surfaces built out of
symmetric pieces. This is joint work (in progress) with James Farre.
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April 29, 2021 10:00 PDT: Conjugating flows on the moduli of hyperbolic and flat surfaces
[abstract]
A measured geodesic lamination on a
hyperbolic surface encodes the horizontal trajectory structure of
certain quadratic differentials. Thurston’s earthquake flow along
such a lamination induces a dynamical system on the moduli space of
hyperbolic surfaces sharing many properties with the classical
Teichmüller horocycle flow. Mirzakhani gave a dynamical
correspondence between the earthquake and horocycle flows, defined
Lebesgue-almost everywhere. In this talk, we extend Mirzakhani’s
conjugacy and define an extension of the earthquake flow to an action
of the upper triangular group P in PSL(2,R) mapping certain flow lines
to Teichmüller geodesics. We classify the P-invariant ergodic
probability measures as those coming from affine invariant measures on
quadratic differentials and show that our map is a measurable
isomorphism between P actions with respect to these measures.
This is joint work with Aaron Calderon.
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March 11, 2021, 14:30 PDT: Ergodic theorems along trees
[abstract]
In the classical pointwise ergodic
theorem for a probability measure preserving (pmp) transformation $T$,
one takes averages of a given integrable function over the intervals
$\{x, T(x), T^2(x), \hdots, T^n(x)\}$ in the forward orbit of the
point $x$. In joint work with Jenna Zomback, we prove a “backward”
ergodic theorem for a countable-to-one pmp $T$, where the averages
are taken over subtrees of the graph of $T$ that are rooted at $x$
and lie behind $x$ (in the direction of $T^{-1}$). Surprisingly,
this theorem yields (forward) ergodic theorems for countable groups,
in particular, one for pmp actions of free groups of finite rank
where the averages are taken along subtrees of the standard Cayley
graph rooted at the identity. For free group actions, this
strengthens the best known result in this vein due to Bufetov (2000).
After reviewing the subject history and discussing the statements
of our theorems in the first half of the talk, we will highlight
some ingredients of proofs in the second half.
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January 21 (14:30), January 28 (14:00), February 4 (14:00), February 11 (14:30), February 18 (14:00), 2021
Note variable starting times
Five-lecture series: Large orbit closures of translation surfaces are strata or loci of double covers
[abstract]
Any translation surface can be presented
as a collection of polygons in the plane with sides identified. By
acting linearly on the polygons, we obtain an action of GL(2,R) on
moduli spaces of translation surfaces. Recent work of Eskin,
Mirzakhani, and Mohammadi showed that GL(2,R) orbit closures are
locally described by linear equations on the edges of the polygons.
However, which linear manifolds arise this way is mysterious.
In this lecture series, we will describe new joint work that shows that
when an orbit closure is sufficiently large it must be a whole moduli
space, called a stratum in this context, or a locus defined
by rotation by π symmetry.
We define "sufficiently large" in terms of rank, which is the most
important numerical invariant of an orbit closure, and is an integer
between 1 and the genus g. Our result applies when the rank is at least
1+g/2, and so handles roughly half of the possible values of rank.
The five lectures will introduce novel and broadly applicable
techniques, organized as follows:
- An introduction to orbit closures, their rank, their
boundary in the WYSIWYG partial compactification, and cylinder
deformations.
- Reconstructing orbit closures from their boundaries (this
talk will explicate a preprint of the same name).
- Recognizing loci of covers using cylinders (this talk will
follow a preprint titled “Generalizations of the
Eierlegende-Wollmilchsau”).
- An overview of the proof of the main theorem; marked points
(following the preprint “Marked Points on Translation
Surfaces”); and a dichotomy for cylinder degenerations.
- Completion of the proof of the main theorem.
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July 16, 2020: Stationary measure and orbit closure classification for random walks on surfaces
[abstract]
We study the problem of classifying
stationary measures and orbit closures for non-abelian action on
surfaces. Using a result of Brown and Rodriguez Hertz, we show that
under a certain average growth condition, the orbit closures are
either finite or dense. Moreover, every infinite orbit equidistributes
on the surface. This is analogous to the results of Benoist-Quint and
Eskin-Lindenstrauss in the homogeneous setting, and the result of
Eskin-Mirzakhani in the setting of moduli spaces of translation
surfaces.
We then consider the problem of verifying this growth condition in
concrete settings. In particular, we apply the theorem to two
settings, namely discrete perturbations of the standard map and the
\Out(F_2)-action on a certain character variety. We verify the growth
condition analytically in the former setting, and verify numerically
in the latter setting.
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July 2, 2020: Quantitative weak mixing for random substitution tilings
[abstract]
"Quantitative weak mixing" is the term
used to bound the dimensions of spectral measures of a
measure-preserving system. This type of study has gained popularity
over the last decade, led by a series of results of Bufetov and
Solomyak for a large class of flows which include general
one-dimensional tiling spaces as well as translation flows on flat
surfaces, as well as results on quantitative weak mixing by Forni.
In this talk I will present results which extend the results for
flows to higher rank parabolic actions, focusing on quantitative
results for a broad class of tilings in any dimension. The talk
won't assume familiarity with almost anything, so I will define all
objects in consideration.
(talk postponed from June 25)
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June 18, 2020: Counting social interactions for discrete subsets of the plane
[abstract]
Given a discrete subset V in the plane,
how many points would you expect there to be in a ball of radius 100?
What if the radius is 10,000? Due to the results of Fairchild and
forthcoming work with Burrin, when V arises as orbits of non-uniform
lattice subgroups of SL(2,R), we can understand asymptotic growth rate
with error terms of the number of points in V for a broad family of
sets. A crucial aspect of these arguments and similar arguments is
understanding how to count pairs of saddle connections with certain
properties determining the interactions between them, like having a
fixed determinant or having another point in V nearby. We will spend
the first 40 minutes discussing how these sets arise and counting
results arise from the study of concrete translation surfaces.
The following 40 minutes will be spent highlighting the proof strategy
used to obtain these results, and advertising the generality and
strength of this argument that arises from the computation of all
higher moments of the Siegel--Veech transform over quotients of
SL(2,R) by non-uniform lattices.
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June 11, 2020: There exists a weakly mixing billiard in a polygon
[abstract]
This main result of this talk is that
there exists a billiard flow in a polygon that is weakly mixing with
respect to Lebesgue measure on the unit tangent bundle to the
billiard. This strengthens Kerckhoff, Masur and Smillie's result that
there exists ergodic billiard flows in polygons. The existence of a
weakly mixing billiard follows, via a Baire category argument, from
showing that for any translation surface the product of the flows in
almost every pair of directions is ergodic with respect to Lebesgue
measure. This in turn is proven by showing that for every translation
surface the flows in almost every pair of directions do not share
non-trivial common eigenvalues. This talk will explain the problem,
related results, and approach. The talk will not assume familiarity
with translation surfaces.
This is joint work with Giovanni Forni.
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June 4, 2020, 13:00 PDT: Arithmetic and geometric properties of planar self-similar sets
[abstract]
Furstenberg's conjecture on the dimension
of the intersection of x2,x3-invariant Cantor sets can be restated as
a bound on the dimension of linear slices of the product of
x2,x3-Cantor sets, which is a self-affine set in the plane. I will
discuss some older and newer variants of this, where the self-affine
set is replaced by a self-similar set such as the Sierpinski triangle,
Sierpinski carpet or (support of) a complex Bernoulli convolution.
Among other things, I will show that the intersection of the
Sierpinski carpet with circles has small dimension, but on the other
hand the Sierpinski carpet can be covered very efficiently by linear
tubes (neighborhoods of lines). The latter fact is a recent result
joint with A. Pyörälä, V. Suomala and M. Wu.
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May 28, 2020: Almost-Prime Times in Horospherical Flows
[abstract]
There is a rich connection between
homogeneous dynamics and number theory. Often in such applications
it is desirable for dynamical results to be effective (i.e. the
rate of convergence for dynamical phenomena are known). In the first
part of this talk, I will provide the necessary background and
relevant history to state an effective equidistribution result for
horospherical flows on the space of unimodular lattices in R^n.
I will then describe an application to studying the distribution of
almost-prime times (integer times having fewer than a fixed number of
prime factors) in horospherical orbits and discuss connections of this
work to Sarnak’s Mobius disjointness conjecture. In the second part
of the talk I will describe some of the ingredients and key steps that
go into proving these results.
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May 21, 2020: A Bratteli-Vershik model for Z^2 actions, or how cohomology can help us make dynamical systems
[abstract]
The Bratteli-Vershik model is a method of
producing minimal actions of the integers on a Cantor set. It was
given by myself, Rich Herman and Chris Skau, building on seminal ideas
of Anatoly Vershik, over 30 years ago. Rather disappointingly and
surprisingly, there isn't a good version for Z^2 actions. I'll report
on a new outlook on the problem and recent progress with Thierry
Giordano (Ottawa) and Christian Skau (Trondheim). The new outlook
focuses on the model as an answer to the question: which cohomological
invariants can arise from such actions? I will not assume any
familiarity with either the original model or the cohomology. The
first half of the talk will be a gentle introduction to the Z-case
and the second half will deal with how to adapt the question to get
an answer for Z^2.
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May 14, 2020: Gaps of saddle connection directions for some branched covers of tori
[abstract]
Translation surfaces given by gluing
two identical tori along a slit have genus two and two cone-type
singularities of angle $4\pi$. There is a distinguished set of
trajectories called saddle connections that are the straight lines
trajectories between cone points. We can associate a
holonomy vector in the plane to each saddle
connection whose components are the horizontal and vertical
displacement of the saddle connection. How random is the planar set of
holonomy of saddle connections? We study this question by computing
the gap distribution for slopes of saddle
connections for these and other related classes of translation surfaces.
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May 7, 2020: Factors of Gibbs measures on subshifts
[abstract]
Classical results of Dobrushin and
Lanford-Ruelle establish, in rough terms, that for a local energy
function on a subshift without too much long-range order, the
translation-invariant Gibbs measures are precisely the equilibrium
measures. There are multiple definitions of a Gibbs measure in
the literature, which do not always coincide. We will discuss two
of these definitions, one introduced by Capocaccia and the other used
by Dobrushin-Lanford-Ruelle, and outline a proof (available at
arxiv.org/abs/2003.05532)
that they are equivalent.
We will also discuss forthcoming work, in which we show that
Gibbsianness is preserved by pushforward through a certain kind of
almost invertible factor map. As an application in one dimension,
we show that for a sufficiently regular potential, any equilibrium
measure on an irreducible sofic shift is Gibbs. As far as we know,
this is the first reasonably general result of the Lanford-Ruelle
type for a class of subshifts without the topological Markov property.
Joint work with Luísa Borsato, with extensive advice from Brian Marcus
and Tom Meyerovitch.
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April 30, 2020: Quantum Unique Ergodicity
[abstract]
In the first half I'll give a
colloquium-style introduction to the
equidistribution problem for Laplace eigenfucntions on Riemannian
manifolds, with emphasis on the locally symmetric spaces. I will
introduce positive results for exact eigenfunctions (with and
without reference to the number-theoretic symmetries of the manifold),
and negative results for approximate eigenfunctions. I will present
results (independenlty) joint with A. Venkatesh, N. Anantharaman,
and S. Eswarathasan. In the second half I'll answer questions and
provide details as requested by the audience.
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April 23, 2020: Effective equidistribution of horospherical flows in infinite volume
[abstract]
The horospherical flow on finite-volume hyperbolic
surfaces is well-understood. In particular, effective equidistribution
of non-closed horospherical orbits is known. New difficulties arise
when studying the infinite-volume setting. We will discuss the setting
in finite- and infinite-volume manifolds, and the measures that play
a crucial role in the latter. This is joint work with Jacqueline
Warren.
