Midwest Summer School in Geometry, Topology, and Dynamics

The RTG group in Geometry, Group Actions, and Dynamics at UW-Madison is organizing the Midwest Summer School in Geometry, Topology, and Dynamics on June 2-6, 2025 in Madison, WI.

The summer school will have 3 mini courses on Teichmuller dynamics , Discrete Subgroups of Lie Groups, Coarse geometry and Gromov hyperbolicity. In addition, we will have ten 35-minute talks by junior participants in related areas.

Link to the conference schedule. (All the talks are in B130 and breaks are in 901.)

Schedule of Lightning Talks

Titles and Abstracts for mini-courses

Speaker: Jon Chaika
Title: Translation surfaces: From rotations of the circle to the horocycle flow.
Videos: Lecture #1, Lecture #2, Lecture #3, Lecture #4
Abstract: This mini-course will be an idiosyncratic approach to translation surfaces. It has a two-fold objective.
a) To present a bit about the roots of the subject coming from rotations of the circle. These beautiful systems have many connections and have powerful tools to understand them. I believe that rotations of the circle and straight line flows on translation surfaces are valuable sources of examples for dynamicists and geometers.

b) After this the minicourse will turn to the space of translation surfaces and the SL(2,R) action on it, especially the horocycle flow.

Lecture 1: Rotations of the circle

This lecture will begin the mini-course. It will introduce some basics of dynamical systems before moving to one of the most elementary and I would say important dynamical systems: irrational rotations of the circle. The focus of the treatment will be the continued fraction expansion of the angle of the rotation and its connections to the dynamics of the rotation.

Lecture 2: Veech’s Example of a minimal and not uniquely ergodic interval exchange transformation.

We will present an example of Veech of a minimal and not uniquely ergodic interval exchange transformation arising as a Z/2Z skew product of a rotation where the skewing function is an interval. This construction is based on the continued fraction expansion discussed in the previous lecture. If time permits we will describe how this arises from a billiard in a rectangle with a barrier parallel to one of the sides.

Lecture 3: Translation surfaces

This talks will introduce translation surfaces which will hopefully have appeared at least briefly in the previous two talks. It will describe the straight line flow on translation surfaces, the space of translation surfaces and lastly the SL(2,R) action on the space of translation surfaces.

Lecture 4: The horocycle flow on strata of translations surfaces.

This talk will describe results about the horocycle flow on strata of translation surfaces:
1) There are points generic for ergodic measures which they are not in the support of.
2) A dense G_delta set of points is not generic for any ergodic measure

Time permitting we will describe a horocycle orbit closure of non-integer Hausdorff dimension. Open questions will also be presented.

Speaker: Matthew Stover
Title: Discrete subgroups of Lie groups
Videos: Lecture #1, Lecture #2, Lecture #3, Lecture #4
Abstract: This will be an introduction to geometric and dynamical methods for studying discrete subgroups of Lie groups, with an emphasis on “rigidity” and “flexibility” phenomena. Topics will include geometric structures, boundary maps, invariant measures, Weil/Mostow/Margulis rigidity, and deformations.

Speaker: Jing Tao
Title: Introduction to hyperbolic groups
Lecture notes: Hyperbolic groups
Abstract: This mini-course offers an introduction to Gromov-hyperbolic groups, emphasizing the geometric and algebraic structures that arise from negative curvature in a coarse setting. Starting from the slim triangle condition in metric spaces, we develop the notion of hyperbolicity and explore its consequences for finitely generated groups via their Cayley graphs. Topics include quasi-isometries, Morse lemma, the word problem and Dehn functions, finiteness properties, and the boundary at infinity. No prior knowledge is assumed beyond basic familiarity with metric spaces and group theory.

Titles and Abstracts for research talks

Speaker: Fernando Al Assal
Title: Asymptotically geodesic surfaces
Abstract: Let M be a hyperbolic 3-manifold. We say a sequence of distinct essential closed surfaces in M is asymptotically geodesic if their principal curvatures go uniformly to zero. When M is closed, these sequences exist abundantly by the Kahn-Markovic surface subgroup theorem, and we will discuss the fact that such surfaces are always asymptotically dense, even though they might not equidistribute. We will also talk about the fact that such sequences do not exist when M is geometrically finite of infinite volume. Finally, time permitting, we will discuss some partial answers to the question: does the existence of asymptotically geodesic surfaces in a negatively-curved 3-manifold imply the manifold is hyperbolic? This is joint work with Ben Lowe.

Speaker: Francisco Arana-Herrera
Title: Polygonal Billiards: Classic and Quantum
Abstract: We discuss two related problems on the dynamics of polygonal billiards. First, we discuss joint work with Chaika and Forni describing how chaos, i.e., sensitivity to initial conditions, arises in the setting of rational billiards. Second, we discuss ongoing joint work with Athreya and Forni studying quantizations of these systems. No previous knowledge on the subject will be assumed.

Speaker: Karen Butt
Title: Entropies of negatively curved surfaces
Abstract: I will discuss two notions of entropy for the geodesic flow: the topological entropy and the measure-theoretic entropy with respect to the Liouville measure. How these dynamical invariants relate to the underlying Riemannian metric has long been of interest. For negatively curved surfaces, Katok proved that equality of the Liouville and topological entropies characterizes metrics of constant negative curvature. In this setting, Manning asked how these two entropies vary along the normalized Ricci flow; more specifically, he proved the topological entropy is monotonic. The main result of this talk, joint with Erchenko, Humbert, and Mitsutani, is that the same is true for the Liouville entropy. In addition to geometric and dynamical methods, we use microlocal analysis to differentiate the Liouville entropy with respect to a conformal perturbation of the metric.

Speaker: Tam Cheetham-West
Title: Finite covers and boundary slopes of cusped hyperbolic 3-manifolds
Abstract: In joint work with Youheng Yao, we discuss how to relate the strongly detected boundary slopes of cusped hyperbolic 3-manifolds with the same finite quotients.

Speaker: Max Lahn
Title: Block Deformations of Anosov Representations
Abstract: Over the last two decades, the framework of Anosov representations has emerged as a generalization of the theory of convex cocompactness for Fuchsian and Kleinian groups to the setting of discrete subgroups of higher rank Lie groups. We present an in-depth look at a type of deformation of Anosov representations of a non-elementary word hyperbolic group Γ in a Lie group G called block deformations. These block deformations are in correspondence with the group cohomology H^1(Γ, D) for some finite-dimensional real vector space D related to the Langlands decomposition of a parabolic subgroup of G. We describe these block deformations, and show that within H^1(Γ, D), the loci of Anosov representations form bounded and convex open subsets which generalize metric open balls for the stable norm.

Speaker: Homin Lee
Title: Random dynamics on surfaces
Abstract: In this talk, we will discuss smooth random dynamical systems and group actions on surfaces. Random dynamical systems, especially understanding stationary measures, can play an important role to understand group actions. We will discuss absolute continuity of stationary measures, classification of orbit closure, and exact dimensionality of stationary measures. This talk will be mostly about the joint work with Aaron Brown, Davi Obata, and Yuping Ruan.

Speaker: Tina Torkaman
Title: Random Measured Laminations and Teichmüller Space
Abstract: In this talk, we introduce a canonical geodesic current KX​ for each X∈Tg​, representing a randomly chosen simple closed geodesic on X. We establish results analogous to Bonahon’s work on the Liouville measure. In particular, we show that the map X↦KX​ defines a proper embedding of Teichmüller space Tg​ into the space of geodesic currents. This embedding leads to a compactification of Tg​ that differs from Thurston’s compactification, which Bonahon’s results yield via the Liouville measure. We will discuss the construction of KX​ and its geometric properties. This is joint work with Curt McMullen.

Speaker: Brandis Whitfield
Title: Short curves of end-periodic mapping tori
Abstract: One creates a fibered 3-manifold by thickening a surface by the interval and gluing its ends by a surface homeomorphism. In the finite-type setting, much is known about how the topological data of the gluing homeomorphism determine geometric information about the hyperbolic 3-manifold. Currently, there is a lot of research activity surrounding end-periodic homeomorphisms of infinite-type surfaces. As an “infinite type” analogue to work of Minsky in the finite-type setting, we show that given a subsurface Y of S, the subsurface projections between the “positive” and “negative” Handel-Miller laminations provide bounds for the geodesic length of the boundary of Y as it resides in a hyperbolic end-periodic mapping torus.

Speaker: Yandi Wu
Title: Essential systoles of hyperbolic link complements
Abstract: The systole of a 3-manifold is the length of the shortest closed geodesic. Given a closed 3-manifold M and link L such that M\L is hyperbolic, the essential systole of M\L is the length of the shortest closed geodesic which is not nullhomotopic in M. In this talk, we will discuss and motivate the study of essential systoles of hyperbolic link complements, including their application towards answering a question of Freedman and Krushkal about the existence of “filling links” in closed 3-manifolds. This is joint work with Chris Leininger.

Speaker: Bradley Zykoski
Title: Contractibility of L^2 Isodelaunay Regions of Translation Surfaces
Abstract: Translation surfaces are parametrized by moduli spaces called strata, and these spaces are covered by the closures of finitely many L^2 isodelaunay regions: open subspaces that parametrize surfaces with a common L^2 Delaunay triangulation. These regions have been used e.g. by Masur-Smillie to analyze collapsing of flat structures and by Bowman to associate to each Teichmüller curve a tessellation of the hyperbolic plane. The topology of these regions, however, has remained largely mysterious; understanding their topology and the pattern in which they meet each other would provide a concrete manner by which to study the topology of strata. In this talk, we discuss a crucial step along these lines: demonstrating the contractibility of a large class of L^2 isodelaunay regions, namely those associated to square-tiled surfaces arranged “in the shape of a tree.” This work is joint with Sam Freedman.

This conference is supported by the RTG: Geometry, Group Actions, and Dynamics at Wisconsin.