In-person meetings are held in AP&M 7321 and Zoom meetings are held here.
If you would like to be included or removed from our email announcements, please email Brandon Seward.
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
Winter 2024 Speakers Past Speakers
- Jan. 11: Pieter Spaas (University of Copenhagen)
Location: AP&M 7321
Title: Local Hilbert-Schmidt stability
Abstract:
We will introduce a local notion of Hilbert-Schmidt stability (HS-stability), partially motivated by the recent introduction of local permutation stability by Bradford. We will discuss some basic properties, and then establish a local character criterion for local HS-stability of amenable groups, by analogy with the character criterion for HS-stability of Hadwin and Shulman. We will then discuss further examples of (flexible versions of) local HS-stability. Finally, we show that infinite sofic (resp. hyperlinear) property (T) groups are never locally permutation (resp. HS-) stable, answering a question by Lubotzky. This is based on joint work with Francesco Fournier-Facio and Maria Gerasimova.
- Jan. 18: Francis Wagner (Ohio State University)
Location: AP&M 7321
Title: Computational Models and Embeddings of Groups
Abstract:
In the late 1990s, Sapir, Birget, and Rips introduced a particular computational model, called an S-machine, which is closely related to (multi-tape, nondeterministic) Turing machines yet works with group words. Per the construction, there are finitely presented groups associated to each S-machine whose relational and geometric structures are tied to the computational structure of the machine. As such, this model has proved exceedingly useful in the construction of finitely presented groups with desirable properties, forming the foundation for the solution to several problems in group theory. We introduce a new graphical framework for this model, giving a condensed visual structure that also allows for improved estimates of the computational complexity of the machines. We then touch on various consequences involving group embeddings. This is joint work with Bogdan Chornomaz.
- Jan. 25: Jane Wang (University of Maine)
Location: Zoom
Title: The topology of the moduli space of dilation surfaces
Abstract:
Translation surfaces are geometric objects that can be defined as a collection of polygons with sides identified in parallel opposite pairs by translation. If we generalize slightly and allow for polygons with sides identified by both translation and dilation, we get a new family of objects called dilation surfaces. While translation surfaces are well-studied, much less is known about dynamics on dilation surfaces and their moduli spaces. In this talk, we will survey recent progress in understanding the topology of moduli spaces of dilation surfaces. We will do this by understanding the action of the mapping class group on the moduli space of dilation surfaces. This talk represents joint work with Paul Apisa and Matt Bainbridge.
video
- Feb. 1 (4:00 pm): Jinho Jeoung (Seoul National University)
Location: Zoom
Title: PGL_2(Q_p)-orbit closures on a p-adic homogenenous space of infinite volume
Abstract:
We proved closed/dense dichotomy of PGL_2(Q_p)-orbit closures in the renormalized frame bundle of a p-adic homogeneous space of infinite volume. Our result is a generalization of Ratner’s theorem and the result of McMullen, Mohammadi, and Oh in 2017 into non-Archimedean local fields.
Let K be an unramified quadratic extension of Q_p. Our homogeneous space is a quotient space of K by a certain class of Schottky subgroups. Using the main tools of McMullen, Mohammadi, and Oh, we introduced the necessary properties of Schottky subgroups and used the Bruhat-Tits tree PGL_2. In this talk, we introduce the highly-branched Schottky subgroups and steps for the proof of the main theorem.
This is a joint work with Seonhee Lim.
- Feb. 8: Darren Creutz (U.S. Naval Academy)
Location: AP&M 7321
Title: Word complexity cutoffs for mixing properties of subshifts
Abstract:
In the setting of zero-entropy transformations, the class of subshifts--closed shift-invariant subsets $X$ of $\mathcal{A}^{\mathbb{Z}}$ for a finite alphabet $\mathcal{A}$--possesses a quantitative measure of complexity: the number of distinct `words' of a given length $p(q) = |\{ w \in \mathcal{A}^{q} : \exists x \in X s.t. w is a substring of x \}|$.
I will discuss my work, some joint with R. Pavlov, pinning down the relationship between this quantitative notion of complexity with the qualitative dynamical complexity properties of probability-preserving systems known as strong and weak mixing.
Specifically, I will present results that strong mixing can occur with word complexity arbitrarily close to linear but cannot occur when $\liminf p(q)/q < \infty$ and that weak mixing can occur when $\limsup p(q)/q = 1.5$ but cannot occur when $\limsup p(q)/q < 1/5$.
The condition that $\limsup p(q)/q < 1.5$ is a (much) stronger version of zero entropy. A corollary of our work is that the celebrated Sarnak conjecture holds for all such systems.
- Feb. 15: Ben Hayes (University of Virginia)
Location: AP&M 7321
Title: Growth dichotomy for unimodular random rooted trees
Abstract:
We show that the growth of a unimodular random rooted tree (T,o) of degree bounded by d always exists, assuming its upper growth passes the critical threshold of the square root of d-1. This complements Timar's work who showed the possible nonexistence of growth below this threshold. The proof goes as follows. By Benjamini-Lyons-Schramm, we can realize (T,o) as the cluster of the root for some invariant percolation on the d-regular tree. Then we show that for such a percolation, the limiting exponent with which the lazy random walk returns to the cluster of its starting point always exists. We develop a new method to get this, that we call the 2-3-method, as the usual pointwise ergodic theorems do not seem to work here. We then define and prove the Cohen-Grigorchuk co-growth formula to the invariant percolation setting. This establishes and expresses the growth of the cluster from the limiting exponent, assuming we are above the critical threshold.
- Feb. 22: Bradley Zykoski (Northwestern University)
Location: Zoom
Title: Strongly Obtuse Rational Lattice Triangles
Abstract:
The dynamics of a billiard ball on a triangular table can be studied by considering geodesic trajectories on an associated singular flat metric structure called a translation surface when the angles of the triangle are commensurable with pi. In the case of the isosceles right triangle, this surface is a torus, whose geodesic trajectories in any direction are either all periodic or all uniquely ergodic. Triangles satisfying such a dichotomy are called lattice triangles, and our work contributes to the ongoing classification of such triangles. We make use of a number-theoretic criterion of Mirzakhani and Wright to classify such triangles with a large obtuse angle. This work is joint with Anne Larsen and Chaya Norton.
video
- Feb. 29: Carsten Petersen (Paderborn University)
Location: Zoom
Title: Quantum ergodicity on the Bruhat-Tits building for PGL(3) in the Benjamini-Schramm limit
Abstract:
Originally, quantum ergodicity concerned equidistribution
properties of Laplacian eigenfunctions with large eigenvalue on
manifolds for which the geodesic flow is ergodic. More recently, several
authors have investigated quantum ergodicity for sequences of spaces
which "converge" to their common universal cover and when one restricts
to eigenfunctions with eigenvalues in a fixed range. Previous authors
have considered this type of quantum ergodicity in the settings of
regular graphs, rank one symmetric spaces, and some higher rank
symmetric spaces. We prove analogous results in the case when the
underlying common universal cover is the Bruhat-Tits building associated
to PGL(3, F) where F is a non-archimedean local field. This may be seen
as both a higher rank analogue of the regular graphs setting as well as
a non-archimedean analogue of the symmetric space setting.
video
- March 7: Michael Zshornack (UC Santa Barbara)
Location: AP&M 7321
Title: Twist flows and the arithmetic of surface group representations
Abstract:
Margulis's work on lattices and a number of questions on the existence of surface subgroups motivate the need for understanding arithmetic properties of spaces of surface group representations. In recent work with Jacques Audibert, we outline one possible approach towards understanding such properties for the Hitchin component, a particularly nice space of representations. We utilize the underlying geometry of this space to reduce questions about its arithmetic to questions about the arithmetic of certain algebraic groups, which in turn, allows us to characterize the rational points on these components. In this talk, I'll give an overview of the geometric methods behind the proof of our result and indicate some natural questions about the nature of the resulting surface group actions that follow.
- March 14: David Gao (UCSD)
Location: AP&M 7321
Title: Sofic actions on sets and applications to generalized wreath products
Abstract:
Inspired by the work of Hayes and Sale showing wreath products of two sofic groups are sofic, we define a notion of soficity for actions of countable discrete groups on countable discrete sets. We shall prove that, if the action \alpha of G on X is sofic, G is sofic, and H is sofic, then the generalized wreath product H \wr_\alpha G is sofic. We shall demonstrate several examples of sofic actions, including actions of sofic groups with locally finite stabilizers, all actions of amenable groups, and all actions of LERF groups. This talk is based on joint work with Srivatsav Kunnawalkam Elayavalli and Gregory Patchell.
Spring 2024
- April 4: Sam Freedman (Brown University)
Location: Zoom
- April 11: Tariq Osman (Brandeis University)
Location: Zoom
- April 18:
- April 25:
- May 2:
- May 9:
- May 16: Qingyuan Chen (UCSD)
Location: AP&M 7321
- May 23: Joshua Bowman (Pepperdine University)
Location: AP&M 7321
- May 30: Carlos Ospina (University of Utah)
Location: AP&M 7321
- June 6: