Talks will be given on zoom, if you have the password you can join here. Please send an email to one of the organizers to get the the password.
If
you would like to give a talk, please send the title, abstract and
related papers (if available) of your proposed talk to one of the
organizers by email.
Organizers: Amir Mohammadi, Anthony Sanchez, Brandon Seward
Spring 2022 Speakers Past Speakers
- March 31: Andy Zucker (University of California San Diego)
Location: AP&M 6402 and on Zoom
Title: Minimal subdynamics and minimal flows without characteristic measures
Abstract:
Given a countable group G and a G-flow X, a probability measure on X is
called characteristic if it is Aut(X, G)-invariant. Frisch and Tamuz
asked about the existence, for any countable group G, of a minimal
G-flow without a characteristic measure. We construct for every
countable group G such a minimal flow. Along the way, we are motivated
to consider a family of questions we refer to as minimal subdynamics:
Given a countable group G and a collection F of infinite subgroups of
G, when is there a faithful G-flow for which the action restricted to
any member of F is minimal? Joint with Joshua Frisch and Brandon Seward.
video
- April 7: Frank Lin (Texas A&M University)
Title: Entropy for actions of free groups under bounded orbit equivalence
Abstract:
Joint work with Lewis Bowen. The f-invariant is a notion of
entropy for probability measure preserving (pmp) actions of free
groups. It is invariant under measure conjugacy and has some
similarities to Kolmogorov-Sinal entropy. Two pmp actions are
orbit equivalent if their orbits can be matched almost everywhere in a
measurable fashion. Although entropy in general is not invariant
under orbit equivalence, we show that the f-invariant is invariant
under the stronger notion of bounded orbit equivalence.
- April 14: Srivatsa Srinivas (University of California San Diego)
Location: AP&M 6402 and on Zoom
Title: An Escaping Lemma and its implications
Abstract:
Let $\mu$ be a measure on a finite group $G$. We define the spectral
gap of $\mu$ to be the operator norm of the map that sends $\phi \in
L^2(G)^{\circ}$ to $\mu * \phi$. We say that $\mu$ is symmetric if
$\mu(x) = \mu(x^{-1})$. Now fix $G = SL_2(\mathbb{Z}/n\mathbb{Z})
\times SL_2(\mathbb{Z}/n\mathbb{Z})$, with $n \in \mathbb{N}$ being
arbitrary. Suppose that $\mu$ is a measure on $G$ such that it's
pushforwards to the left and right component have spectral gaps lesser
than $\lambda_0 < 1$ and $\mu$ takes a minimum of $\alpha_0$ on it's
support. Further suppose that the support of $\mu$ generates $G$. Then
we show that there are constants $L, \beta > 0$ depending only on
$\lambda_0,\alpha_0$ such that $\mu^{(*)L\log |G|}(\Gamma) \leq
\frac{1}{|G|^{\beta}}$, where $\Gamma$ is the graph of any automorphism
of $SL_2(\mathbb{Z}/n\mathbb{Z})$. We will discuss this result and its
implications. This talk is based on joint work with Professor Alireza
Salehi-Golsefidy.
video
- April 21: Seonhee Lim (Seoul National University)
Location: AP&M 6402 and on Zoom
Title: Complex continued fractions and central limit theorem for rational trajectories
Abstract:
In this talk, we will first introduce the complex continued fraction
maps associated with some imaginary quadratic fields (d=1,2,3,7,11) and
their dynamical properties. Baladi-Vallee analyzed (real) Euclidean
algorithms and proved the central limit theorem for rational
trajectories and a wide class of cost functions measuring algorithmic
complexity. They used spectral properties of an appropriate bivariate
transfer operator and a generating function for certain Dirichlet
series whose coefficients are essentially the moment generating
function of the cost on the set of rationals. We extend the work of
Baladi-Vallee for complex continued fraction maps mentioned above.
(This is joint work with Dohyeong Kim and Jungwon Lee.)
- April 28: Osama Khalil (University of Utah)
Location: Zoom
Title: Mixing, Resonances, and Spectral Gaps on Geometrically Finite Manifolds
Abstract:
I will report on work in progress showing that the geodesic flow on any
geometrically finite, rank one, locally symmetric space is
exponentially mixing with respect to the Bowen-Margulis-Sullivan
measure of maximal entropy. The method is coding-free and is instead
based on a spectral study of transfer operators on suitably constructed
anisotropic Banach spaces, ala Gouezel-Liverani, to take advantage of
the smoothness of the flow. As a consequence, we obtain more precise
information on the size of the essential spectral gap as well as the
meromorphic continuation properties of Laplace transforms of
correlation functions.
- May 5: Matthew Welsh (University of Bristol)
Location: AP&M 6402 and on Zoom
Title: Bounds for theta sums in higher rank
Abstract:
In joint work with Jens Marklof, we prove new upper bounds for theta
sums -- finite exponential sums with a quadratic form in the
oscillatory phase -- in the case of smooth and box truncations. This
generalizes results of Fiedler, Jurkat and Körner (1977) and Fedotov
and Klopp (2012) for one-variable theta sums and, in the multi-variable
case, improves previous estimates obtained by Cosentino and Flaminio
(2015). Key inputs in our approach include the geometry of Sp(n, Z) \
Sp(n, R), the automorphic representation of theta functions and their
growth in the cusp, and the action of the diagonal subgroup of Sp(n, R).
video
- May 12: Yair Hartman (Ben-Gurion University)
Location: Zoom
Title: Tight inclusions
Abstract:
We discuss the notion of "tight inclusions" of dynamical systems which
is meant to capture a certain tension between topological and
measurable rigidity of boundary actions, and its relevance to
Zimmer-amenable actions. Joint work with Mehrdad Kalantar.
video
- May 19: Robin Tucker-Drob (University of Florida)
Location: AP&M 6402 and on Zoom
Title: Amenable subrelations of treed equivalence relations and the Paddle-ball lemma
Abstract:
We give a comprehensive structural analysis of amenable subrelations of
a treed quasi-measure preserving equivalence relation. The main
philosophy is to understand the behavior of the Radon-Nikodym cocycle
in terms of the geometry of the amenable subrelation within the tree.
This allows us to extend structural results that were previously only
known in the measure-preserving setting, e.g., we show that every
nowhere smooth amenable subrelation is contained in a unique maximal
amenable subrelation. The two main ingredients are an extension of
Carričre and Ghys's criterion for nonamenability, along with a new
Ping-Pong-style argument we call the "Paddle-ball lemma" that we use to
apply this criterion in our setting. This is joint work with Anush
Tserunyan.
video
- May 26: Dami Lee (University of Washington)
Location: AP&M 6402 and on Zoom
Title: Computation of the Kontsevich--Zorich cocycle over the Teichmüller flow
Abstract:
In this talk, we will discuss the dynamics on Teichmüller space and
moduli space of square-tiled surfaces. For square-tiled surfaces, one
can explicitly write down the SL(2,R)-orbit on the moduli space. To
study the dynamics of Teichmüller flow of the SL(2,R)-action, we study
its derivative, namely the Kontsevich--Zorich cocycle (KZ cocycle). In
this talk, we will define what a KZ cocycle is, and by following
explicit examples, we will show how one can compute the KZ monodromy.
This is part of an ongoing work with Anthony Sanchez.
- June 2: Israel Morales Jimenez (Universidad Nacional Autónoma de México)
Location: Zoom
Title: Big mapping class groups and their conjugacy classes
Abstract:
The mapping class group, Map(S), of a surface S, is the group of all
isotopy classes of homeomorphisms of S to itself. A mapping class group
is a topological group with the quotient topology inherited from the
quotient map of Homeo(S) with the compact-open topology. For surfaces
of finite type, Map(S) is countable and discrete. Surprisingly, the
topology of Map(S) is more interesting if S is an infinite-type
surface; it is uncountable, topologically perfect, totally
disconnected, and more importantly, has the structure of a Polish
group. In recent literature, this last class of groups is called “big
mapping class groups”. In this talk, I will give a brief introduction
to big mapping class groups and explain our results on the topological
structure of conjugacy classes. This was a joint work with Jesús
Hernández Hernández, Michael Hrušák, Manuel Sedano, and Ferrán Valdez.
video